François Béguin
email: beguin@math.univ-paris13.fr

Tom Dutilleul
email: tom.dutilleul@gmail.com.

Chaotic dynamics of spatially homogeneous spacetimes

In the 1970's, Belinskii, Khalatnikov and Lifshitz have proposed a conjectural description of the asymptotic geometry of cosmological models in the vicinity of their initial singularity. In particular, it is believed that the asymptotic geometry of generic spatially homogeneous spacetimes should display an oscillatory chaotic behaviour modeled on a discrete map's dynamics (the so-called Kasner map). We prove that this conjecture holds true, if not for generic spacetimes, at least for a positive Lebesgue measure set of spacetimes.

In the context of spatially homogeneous spacetimes, the Einstein field equations can be reduced to a system of differential equations on a finite dimensional phase space: the Wainwright-Hsu equations. The dynamics of these equations encodes the evolution of the geometry of spacelike slices in spatially homogeneous spacetimes. Our proof is based on the non-uniform hyperbolicity of the Wainwright-Hsu equations. Indeed, we consider the return map of the solutions of these equations on a transverse section and prove that it is a non-uniformly hyperbolic map with singularities. This allows us to construct some local stable manifolds à la Pesin for this map and to prove that the union of the orbits starting in these local stable manifolds cover a positive Lebesgue measure set in the phase space. The chaotic oscillatory behaviour of the corresponding spacetimes follows.

Introduction 1.The BKL conjecture for Bianchi spacetimes

Bianchi spacetimes

In classical General Relativity, spacetime is modeled as a smooth 4-dimensional Lorentz manifold (M, g) verifying the Einstein field equations

Ric g + Λ - 1 2 Scal g g = T (1.1)
where Ric g is the Ricci curvature tensor, Scal g is the scalar curvature, Λ is the cosmological constant and T is the stress-energy tensor, which encodes the presence of matter, radiation and non-gravitational force fields. Assuming that the gravitational force field only self-interacts and Λ is zero, (1.1) reduces to the vacuum Einstein field equations

Ric g = 0 (1.2)
Informally, a Bianchi spacetime (also called Bianchi cosmological model) is a spacetime which is spatially homogeneous. We will work with the following formal definition: a Bianchi spacetime is a Lorentzian manifold of the form (M, g) = (I × G, -ds 2 + h s ) where I is an interval of the real line, G is a simply-connected 3-dimensional real Lie group, s is a coordinate on I and h s is a left-invariant Riemannian metric on {s} × G ≃ G for every s ∈ I. If the Lie group G is unimodular 1 , then the Bianchi spacetime is said to be of class A, otherwise it is said to be of class B. We say that a Bianchi spacetime is maximal if it cannot be embedded isometrically as a strict submanifold of another Bianchi spacetime.

In this work, we will restrict our attention to maximal vacuum (with zero cosmological constant) class A Bianchi spacetimes 2 , that is, maximal class A Bianchi spacetimes solution to the vacuum Einstein field equations (1.2). It is well known (see e.g. [START_REF] Collins | Singularities in Bianchi Cosmologies[END_REF]) that, up to a change of time orientation, almost all maximal vacuum class A Bianchi spacetimes 3 admits an initial singularity 4 (often called Big-Bang). We are mostly interested in the description of the past-asymptotic geometry of maximal vacuum class A Bianchi spacetimes, i.e., in the description of their behaviour near their initial singularity.

BKL picture

In a series of papers, Belinskii, Khalatnikov and Lifshitz (see [START_REF] Belinskii | A General Solution of the Einstein Equations with a Time Singularity[END_REF] and [START_REF] Belinskii | Oscillatory Approach to a Singular Point in the Relativistic Cosmology[END_REF]) explained with heuristic arguments that general singularities should have the following properties:

1. As a first order approximation, the behaviour of the curvature of a spacetime near its initial singularity is dominated by the behaviour of its "spatially homogeneous part".

2. Solutions of the Einstein field equations with matter are well approximated, in the vicinity of their initial singularity, by solutions of the vacuum Einstein field equations. As the saying goes, near the initial singularity, "matter does not matter".

3. The geometry of the spatial hypersurfaces "oscillates" in a chaotic manner at the approach of the initial singularity.

What precedes is often referred to as the BKL picture or the BKL conjecture.

Wainwright-Hsu equations

The Einstein field equations are, in local coordinates, a system of non linear partial differential equations of order 2 about the coefficients of the lorentzian metric g. For a Bianchi spacetime, the metric g is fully characterized by a single variable function s ↦ h s valued in the finite dimensional space of 1 A Lie group is called unimodular if its left invariant Haar measure is also right invariant.

2 For some literature on class B Bianchi spacetimes, we refer to [START_REF] Hewitt | A Dynamical Systems Approach to Bianchi Cosmologies: Orthogonal Models of Class B[END_REF], [START_REF] Hewitt | Asymptotic Dynamics of the Exceptional Bianchi Cosmologies[END_REF] and [START_REF] Radermacher | Strong Cosmic Censorship in Orthogonal Bianchi Class B Perfect Fluids and Vacuum Models[END_REF]. 3 More precisely, all Bianchi spacetimes, except the simplest ones, namely Minkowski spacetimes and Taub-NUT spacetimes. 4 We say that a maximal vacuum class A Bianchi spacetime (M, g) = (I × G, -ds 2 + h s ) admits an initial singularity if I =]s -, s + [ with s -> -∞. If this is the case, the curvature blows up when the time tends to s -(see [START_REF] Ringström | Curvature Blow up in Bianchi VIII and IX Vacuum Spacetimes[END_REF]).

left-invariant Riemannian metrics on a Lie group. Therefore, in the specific context of Bianchi spacetimes, vacuum Einstein field equations should translate as a system of ordinary differential equations (abbreviated as ODEs) on a finite-dimensional phase space B. This allows one to study the vacuum Einstein field equations restricted to Bianchi spacetimes with classical dynamical systems methods. The first step to explicit the vacuum Einstein field equations is to choose a particular frame field or, equivalently, a coordinates system. One of the first successful attempts to do so has been made by Bogoyavlenski (see [START_REF] Igorevich | Methods in the Qualitative Theory of Dynamical Systems in Astrophysics and Gas Dynamics[END_REF]). Later on, Ellis and MacCallum (see [START_REF] Ellis | A Class of Homogeneous Cosmological Models[END_REF]) and then Wainwright and Hsu (see [START_REF] Wainwright | A Dynamical Systems Approach to Bianchi Cosmologies: Orthogonal Models of Class A[END_REF]) introduced useful coordinates using the so-called orthonormal frame method.

In this work, we will use a Hubble-renormalized system of variables (N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) closely related to the one used by Wainwright and Hsu. These variables are dimensionless, which means that they will not change if the spacetime metric is rescaled. Since these variables do not see the rescaling operation, one can hope that they remain bounded in the vicinity of the singularity. We also choose a dimensionless time variable t and an "anti-physical" time orientation 5 , which means that the initial singularities are located in t = +∞.

Before we give more details about these variables, let us recall that the 3-dimensional real Lie algebras have been classified by Luigi Bianchi in 1898. This is the reason why the Bianchi spacetimes are called that way and why it is now standard to classify them according to their "Bianchi type" (see table 1 and, e.g., [START_REF] Ellis | A Class of Homogeneous Cosmological Models[END_REF], [START_REF] Heinzle | Mixmaster: Fact and Belief[END_REF] and [START_REF] Milnor | Curvatures of Left Invariant Metrics on Lie Groups[END_REF]).

The numbers N 1 (t), N 2 (t), N 3 (t) describe the intrinsic curvature of the spacelike hypersurface {t}×G (that is, the curvature of the left-invariant riemannian metric h t ) and its Bianchi type. Actually, these three numbers are, up to a renormalization, the structure constants of the Lie algebra of G in a special basis (which depends on the metric h t ). The numbers Σ 1 (t), Σ 2 (t), Σ 3 (t) describe the extrinsic curvature of the spacelike hypersurface {t} × G. These numbers verify two constraint equations:

Σ 1 + Σ 2 + Σ 3 = 0 (1.3a)
(this relation comes from the fact that the numbers Σ 1 (t), Σ 2 (t), Σ 3 (t) are the diagonal coefficients of the trace-free part of the second fundamental form of the spacelike hypersurface {t} × G) and 6 -Σ

2 1 + Σ 2 2 + Σ 2 3 - 1 2 N 2 1 + N 2 2 + N 2 3 + (N 1 N 2 + N 2 N 3 + N 3 N 1 ) = 0 (1.3b)
(this relation comes from the Gauss formula, which connects the intrinsic and the extrinsic curvatures of a given hypersurface to the curvature of the ambiant space, and the fact that the scalar curvature of the spacetime (M, g) is null). The left-hand side of (1.3b) can be thought as the renormalized density parameter, which is null in the context of vacuum spacetimes. We will denote by B the phase space, defined as the set of points (N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) ∈ R 6 verifying (1.3a) and (1.3b). In particular, B is a non-singular and non-compact 4-dimensional quadric.

When the vacuum Einstein field equations are written in this system of variables, it gives rise to an autonomous system of six differential equations called the Wainwright-Hsu equations:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ N ′ 1 = -(q + 2Σ 1 )N 1 N ′ 2 = -(q + 2Σ 2 )N 2 N ′ 3 = -(q + 2Σ 3 )N 3 Σ ′ 1 = (2 -q)Σ 1 + S 1 Σ ′ 2 = (2 -q)Σ 2 + S 2 Σ ′ 3 = (2 -q)Σ 3 + S 3 (1.3c)
where

q def = 1 3 (Σ 2 1 + Σ 2 2 + Σ 2 3 ) and S i def = 1 3 (2N 2 i -N 2 j -N 2 k + 2N j N k -N i N j -N i N k ), {i, j, k} = {1, 2, 3}
The numbers S 1 (t), S 2 (t), S 3 (t) are, up to renormalization, the components of the traceless Ricci tensor of the metric h t and q is called the deceleration parameter.

The vector field associated to the ODE system (1.3c) is called the Wainwright-Hsu vector field and is denoted by X . The first thing to remark is the fact that the Wainwright-Hsu equations (1.3c) respect the constraint equations (1.3a) and (1.3b), i.e. the quadric B is invariant under the action of the flow of the Wainwright-Hsu vector field.

Up to some technical details 6 , there is a one-to-one correspondence between maximal solutions of the Wainwright-Hsu equations contained in the phase space B and maximal vacuum class A Bianchi spacetimes (see Chapter 22 of [START_REF] Ringström | The Cauchy Problem in General Relativity[END_REF] or Chapter 2 of [START_REF] Dutilleul | Chaotic Dynamics of Spatially Homogeneous Spacetimes[END_REF] for a complete proof of this correspondence).

Remark 1.1. Recall that with our choice of an anti-physical time orientation, describing the pastasymptotic states of a vacuum class A Bianchi spacetime amounts to describe the future-asymptotic states (that is, the ω-limit set 7 ) of the corresponding orbit of the Wainwright-Hsu vector field.

Stratification of the phase-space

The classification of 3-dimensional Lie algebras induces a stratification of the phase space B in six strata invariant under the flow of the Wainwright-Hsu vector field X . This invariant stratification is nothing more than the formalization of a simple fact: the signs of the variables N i define a stratification and, according to the Wainwright-Hsu equations (1.3c), the signs of the variables N i are invariant along the orbits of the Wainwright-Hsu vector field. The different strata each correspond to a certain Bianchi type and will be called Bianchi type I (resp. II, VI 0 , VII 0 , VIII and IX) stratum. The orbits of the Wainwright-Hsu vector field X contained in the Bianchi type I (resp. II, VI 0 ,. . . ) stratum will be called type I (resp. II, VI 0 ,. . . ) orbits. The Bianchi type I stratum is an Euclidean circle, called the Kasner circle, and is denoted by K . There are three particular ellipsoids intersecting along their common equator, which happens to be the Kasner circle K . The Bianchi type II stratum is the union of these three ellipsoids minus the Kasner circle. Each one of these ellipsoids (minus the Kasner circle) is contained in a subset of the form N i ≠ 0, N j = 0, N k = 0, where {i, j, k} = {1, 2, 3}. These two strata are respectively of codimension three and two in the phase space B. The Bianchi type VI 0 and VII 0 strata are both of codimension one while the Bianchi type VIII and IX strata are Zariski open subsets of B. We refer to section 2.2 for further details. Remark 1.2. One major advantage of the Wainwright-Hsu presentation is to allow to study all the vacuum class A Bianchi spacetimes with the same equations (1.3c) and in the same phase space B ⊂ R 6 . It means that we can "compare" two different vacuum class A Bianchi spacetimes (even if these spacetimes are of different Bianchi types) using the metric of our choice in R 6 and this approach has proved to be successful in the past (see e.g. [START_REF] Wainwright | A Dynamical Systems Approach to Bianchi Cosmologies: Orthogonal Models of Class A[END_REF], [START_REF] Ringström | The Bianchi IX Attractor[END_REF], [START_REF] Liebscher | Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles[END_REF], [START_REF] Béguin | Aperiodic Oscillatory Asymptotic Behavior for Some Bianchi Spacetimes[END_REF] and [START_REF] Brehm | Bianchi VIII and IX Vacuum Cosmologies: Almost Every Solution Forms Particle Horizons and Converges to the Mixmaster Attractor[END_REF]).

translation. Maximal solutions of the Wainwright-Hsu equations are considered up to permutation of the indices 1, 2, 3, simultaneous sign reversal of the N i 's and time translation. The Minkowski spacetime does not correspond to any solution of the Wainwright-Hsu equations. Each Bianchi spacetime of type IX (see paragraph 1.1.4) splits into two halves (the expanding part and the contracting part), each of which correspond to a solution of the Wainwright-Hsu equation.

7 Precisely, the ω-limit set of an orbit O(t) is defined as the set ω(O)

def = ∩ s≥0 {O(t) | t ≥ s}.
If O converges to a point x in the future, then ω(O) = {x} and we say that x is the ω-limit point of O.

Mixmaster attractor and past-asymptotic dynamics of Bianchi spacetimes

The union of the Kasner circle and the Bianchi type II stratum is called the Mixmaster attractor and is denoted by A . Geometrically, it is the union of three ellipsoids intersecting along their common equator. The Mixmaster attractor is invariant under the flow of the Wainwright-Hsu vector field. The importance of this particular subset is expressed by the following theorem (see [START_REF] Ringström | The Bianchi IX Attractor[END_REF] and [START_REF] Brehm | Bianchi VIII and IX Vacuum Cosmologies: Almost Every Solution Forms Particle Horizons and Converges to the Mixmaster Attractor[END_REF] for further details).

Theorem 1.3 [START_REF] Ringström | The Bianchi IX Attractor[END_REF][START_REF] Brehm | Bianchi VIII and IX Vacuum Cosmologies: Almost Every Solution Forms Particle Horizons and Converges to the Mixmaster Attractor[END_REF]. For Lebesgue almost every point q in the phase space B, the distance between the orbit of the Wainwright-Hsu vector field with initial condition q and the Mixmaster attractor A converges to 0 in the future. For such an orbit, it means that its ω-limit set is included in A .

In view of this theorem, one may ask the following (vague) question: does the future-asymptotic behavior of generic orbits of the Wainwright-Hsu vector field reflect the dynamics of the Wainwright-Hsu vector field restricted to the Mixmaster attractor? A precise version of the question will be stated in the next paragraph. It should be though as a reformulation of the point 3 of the BKL picture.

Restriction of the phase space

From now on, we will restrict ourselves to the part of the phase space characterized by

N 1 ≥ 0, N 2 ≥ 0, N 3 ≥ 0
In particular, we will only state results for orbits that are contained in this subpart of the phase space, denoted by B + . Remark that • B + is invariant under the flow of the Wainwright-Hsu vector field.

• Generic orbits of B + are type IX orbits.

This restriction will greatly simplify the presentation of the main result of this article. In particular it allows us to use simplified notations. We refer to the appendix B for a description of the results in the full phase space B.

Basic facts about the dynamics of Bianchi spacetimes

We now state some well-known facts about the dynamics of the Wainwright-Hsu vector field in low dimensional strata, in particular in the Mixmaster attractor. Any point of the Kasner circle K = B I is a critical point of the Wainwright-Hsu vector field X . This means that type I orbits are reduced to stationnary points and correspond to self-similar spacetimes (see [START_REF] Wainwright | A Dynamical Systems Approach to Bianchi Cosmologies: Orthogonal Models of Class A[END_REF] and [START_REF] Douglas | Self-Similar Spacetimes: Geometry and Dynamics[END_REF]). More precisely they correspond to Kasner spacetimes. There are three special points in the Kasner circle, called the Taub points and denoted by T 1 , T 2 , T 3 , which will play a crucial role in the understanding of the behaviour of the solutions of the Wainwright-Hsu equations. Any type II orbit is a heteroclinic connexion between two points of the Kasner circle. This means that any type II orbit converges in the future to a point q ∈ K and in the past to a point p ∈ K . We will say that such an orbit starts at p and arrives at q. See figure 1. Of course, one should recall that type II orbits never reach the Kasner circle, so it is an abuse of terminology. Type II orbits are explicitly known. In particular, for every point p of the Kasner circle that is not a Taub point, there is exactly one type II orbit starting at p in B + . We refer to section 2.4 for further details. The future-asymptotic behaviour of type VI 0 or VII 0 orbits is well-understood. Given such an orbit, its ω-limit set is either a single point of the Kasner circle or a flat point of type VII 0 , the latter being only possible if the orbit is constant. We refer to [START_REF] Rendall | Global Dynamics of the Mixmaster Model[END_REF] for further details.

Kasner map, heteroclinic chains and shadowing

The fundamental tool to describe the dynamics of the Wainwright-Hsu vector field restricted to the Mixmaster attractor is the Kasner map. It is a map from the Kasner circle K to itself defined in such a way that it encodes the dynamics of type II orbits. More precisely, it is defined as follows. Let p be a point of the Kasner circle that is not a Taub point. The type II orbit starting at p converges to another point of the Kasner circle, denoted by F (p). We will denote this type II orbit by O p→F (p) . If p is a Taub point, set F (p) ∶= p. This defines a continuous map F ∶ K → K called the Kasner map, whose dynamics is well understood:

• The Kasner map is topologically conjugated to θ ↦ -2θ on the cirle R/Z (see [START_REF] Béguin | Aperiodic Oscillatory Asymptotic Behavior for Some Bianchi Spacetimes[END_REF]). In particular, its dynamics is chaotic.

• There is an explicit "conjugation" between the Kasner map and an avatar of the Gauss transformation on the continued fractions (see section 1.2.1 below).

• The Kasner map admits a very simple geometric construction (see section 2.5).

We refer to sections 2.5 and 2.7 for further details on the Kasner map. One may rephrase the question asked in a preceding paragraph as follows: is the future-asymptotic dynamics of a generic type IX orbit in B + "driven" by the Kasner map? We now introduce two concepts to make this question rigorous: heteroclinic chains and shadowing.

Let p be a point of the Kasner circle (such that, for every k ≥ 0, F k (p) is not a Taub point). The heteroclinic chain starting at p is the concatenation of the unique type II orbit starting at p and arriving at F (p), then the unique type II orbit starting at F (p) and arriving at F Let t ↦ O(t) be a type IX orbit in B + converging to the Mixmaster attractor, p be a point of the Kasner circle (such that, for every k ≥ 0, F k (p) is not a Taub point) and H be the heteroclinic chain (1.4) starting at p. Definition 1.4 (Shadowing). We say that O shadows H (or H attracts O) if there exists a strictly increasing sequence

(t n ) n∈N ⊂ R + such that 1. d(O(t n ), F n (p)) -----→ n→+∞ 0.
2. The Hausdorff distance between the orbit interval {O(t) | t n < t < t n+1 } and the type II orbit O F n (p)→F n+1 (p) tends to 0 when n → +∞.

See figure 2 for a schematical representation of the shadowing.

Given a type IX orbit in B + , the concept of shadowing formalizes the idea that its future-asymptotic dynamics is "driven" by the Kasner map. We can now refine our preceding questions: given a point p of the Kasner circle (such that, for every k ≥ 0, F k (p) is not a Taub point) and the heteroclinic chain H starting at p, what is the geometrical structure of the union of all the type IX orbits in B + shadowing the heteroclinic chain H ? Are "typical" orbits driven by the Kasner map? More precisely, does the union of all the type IX orbits in B + shadowing some heteroclinic chain has full Lebesgue measure in the phase space B + ? If not, is it a set of positive measure?

• p O p→F (p) •F (p) O F (p)→F 2 (p) F 2 (p) • O F 2 (p)→F 3 (p) • O(t 0 ) • O(t 1 ) • O(t 2 )
Figure 2: Schematical representation of the first part of a heteroclinic chain shadowed by a type IX orbit in B + .

Possible formalization of the BKL conjecture for Bianchi spacetimes

Using the preceding definitions, we propose8 the following rewording of item 3 of the BKL picture (subsubsection 1.1.2):

1. Almost every heteroclinic chain is shadowed by some type IX orbits in B + .

2. The union of all the type IX orbits in B + shadowing some heteroclinic chain has full Lebesgue measure in the phase space B + .

Statement of the results

In this work, we intend to give a proof of item 1 and a partial proof of item 2 above. Our results can be stated in the following terms:

Main theorem 1.5. For Lebesgue almost every point p of the Kasner circle, if H denotes the heteroclinic chain starting at p, then the union of all the type IX orbits shadowing H contains a 3dimensional Lipschitz immersed submanifold. Moreover, the union of all the type IX orbits shadowing some heteroclinic chain has positive Lebesgue measure. More precisely, for all subset E of the Kasner circle with positive 1-dimensional Lebesgue measure, the union of all the type IX orbits shadowing some heteroclinic chain starting at a point of E has positive 4-dimensional Lebesgue measure.

Remark 1.6. Informally, this means that if one picks randomly a spatially homogeneous spacetime, then this spacetime has a chaotic oscillatory past-asymptotic behaviour with nonzero probability.

The first part of Theorem 1.5 is a refinement of the work done by Reiterer & Trubowitz in [START_REF] Reiterer | The BKL Conjectures for Spatially Homogeneous Spacetimes[END_REF]. To our knowledge, the second part of Theorem 1.5 is entirely new. It should be considered as the main result of this article. See subsection 1.2.4 below for more comments on previous results.

The purpose of the next two subsections is to explain what are the heteroclinic chains that we manage to shadow with a sufficiently big set of type IX orbits. Let us say that a point p belonging to the Kasner circle is shadowable if the union of all the type IX orbits shadowing the heteroclinic chain starting at p contains a 3-dimensional Lipschitz immersed submanifold. Let p be a point of the Kasner circle. Roughly speaking, our proof of Theorem 1.5 shows that if the orbit of p under the Kasner map "does not come too fast too close to the Taub points", then p is admissible for the shadowing. We are now going to introduce some tools to make this statement more precise.

Kasner parameter and Gauss transformation

The Kasner parameter ω ∶ K /S 3 → [1, +∞], where S 3 is the group of permutations of {1, 2, 3}, is a bijective parametrization of K /S 3 by [1, +∞] satisfying the relation ω(T i ) = +∞, for any Taub point T i . In this parametrization, the Kasner map F becomes an avatar of the Gauss transformation on the continued fractions. More precisely, let us define

f (ω) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ω -1 if ω ≥ 2 1 ω-1 if 1 < ω ≤ 2 +∞ if ω = 1 ou ω = +∞
The Kasner parameter is a C 0 -conjuguacy from (K /S 3 , F ) to ([1, +∞], f ). It means that, for any given point p of the Kasner circle, the dynamical behaviour of its orbit under the Kasner map F depends on the continued fraction expansion of its Kasner parameter

ω(p) = [k 0 ; k 1 , k 2 , k 3 , . . . ] = k 0 + 1 k 1 + 1 k 2 + 1 k 3 + . . .
We refer to section 2.7 for further details, see also [START_REF] Bini | The Lifshitz-Khalatnikov Kasner Index Parametrization and the Weyl Tensor[END_REF].

Rephrasing of the results

Let p be a point of the Kasner circle and ω(p) = [k 0 ; k 1 , k 2 , k 3 , . . . ] be its Kasner parameter. According to the preceding paragraph, p is "close" to a Taub point if and only if k 0 is "large". Adopting the view-point of the continued fractions, we can say that, roughly speaking, a point p is admissible for the shadowing if the partial quotients k i of the continued fraction expansion of its Kasner parameter ω(p) do not blow up "too fast". A precise meaning is given by the following definition. The next lemma shows that the moderate growth condition is not too restrictive. A proof can be found in Appendix A. Define

K (M G) = {p ∈ K | ω(p) verifies (MG)}
Lemma 1.8. The set K (M G) is a full Lebesgue measure subset of K .

We are now able to give a more precise statement of Theorem 1.5.

Main theorem 1.9. Let p be a point of the Kasner circle. If ω(p) verifies the moderate growth condition (MG), then the union of all the type IX orbits shadowing the heteroclinic chain starting at p contains a 3-dimensional ball D(p) Lipschitz embedded in the phase space B + . Moreover, for any E ⊂ K (M G) of positive 1-dimensional Lebesgue measure, the union of all the balls D(p) for p ∈ E has positive 4-dimensional Lebesgue measure.

We do not know whether the union of the type IX orbits intersecting a ball D(p) for some p has full Lebesgue measure in the phase space. Hence, the following question remains open: Question 1. Is the union of all type IX orbits shadowing a heteroclinic chain of type II orbits a full Lebesgue measure subset of the phase space B + ?

Examples of dynamical and geometrical consequences

In order to know the past-asymptotic behaviour of a maximal vacuum class A Bianchi spacetime, it is of prime interest to describe the ω-limit set of the corresponding orbit of the Wainwright-Hsu vector field. Knowing that for almost all point p of the Kasner circle (with respect to Lebesgue measure), the heteroclinic chain starting at p (seen as a subset of the phase space) is dense in the Mixmaster attractor A , one gets the following result as a direct consequence of Theorem 1.9.

Corollary 1.10. Let q be a point of the phase space B + . With positive probability on q, the ω-limit set of the orbit of the Wainwright-Hsu vector field with initial condition q is the entire Mixmaster attractor A .

Theorem 1.9 says in particular that, with positive probability, a maximal vacuum class A Bianchi spacetime will have an oscillatory past-asymptotic behaviour. However, oscillatory has multiple meanings and they are not all equivalent. Corollary 1.11 below shows that, when heading towards the the initial singularity, there is an infinite alternation between periods where the spacelike slices are curved in a single direction and periods where the spacelike slices are curved in two or three directions.

Corollary 1.11. Let q be a point of the phase space B

+ and (M, g) = ]s -, s + [ × G, -ds 2 + h s be a maximal vacuum class A Bianchi spacetime corresponding to the orbit of the Wainwright-Hsu vector field with initial condition q, with s -> -∞. 2. For every n ≥ 0,

|θ max (s 2n+1 )| |θ min (s 2n+1 )| ≤ 3

Comparison with previous results

It was already known that some heteroclinic chains attract a three-dimensional submanifold of type VIII or IX orbits. In [START_REF] Liebscher | Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles[END_REF], Liebscher & al proved such a result for a periodic heteroclinic chain. Their method extends, with some technical work, to any heteroclinic chain which do not come arbitrarily close to a Taub point. F. Béguin proved a similar result for aperiodic heteroclinic chains in [START_REF] Béguin | Aperiodic Oscillatory Asymptotic Behavior for Some Bianchi Spacetimes[END_REF].

One should note that in both these papers, the set of heteroclinic chains that are shown to attract some type VIII or IX orbits correspond to a null measure subset of the Kasner circle. In the preprint [START_REF] Reiterer | The BKL Conjectures for Spatially Homogeneous Spacetimes[END_REF], Reiterer & Trubowitz show that the set of points p for which the heteroclinic chains attract some type VIII or IX orbits is a Lebesgue full measure subset of the Kasner circle. However, their result, while showing that the union of all the type VIII or IX orbits shadowing a generic heteroclinic chain is in some sense "3-dimensional", does not describe its geometry as precisely as in [START_REF] Liebscher | Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles[END_REF] and [START_REF] Béguin | Aperiodic Oscillatory Asymptotic Behavior for Some Bianchi Spacetimes[END_REF]. This is mainly due to the degeneration of hyperbolicity as one approaches the Taub points.

The first part of Theorem 1.9 is essentially equivalent to the theorem proved by Reiterer & Trubowitz. There are three main differences between these two results:

• We do not work with the same equations. Indeed, while we use the orthonormal frame method, they use the orthogonal frame method. It means that their variables are the diagonal coefficients of the metric h t and the diagonal coefficients of the second fundamental form of the spacelike hypersurface {t} × G, while with the orthonormal frame method, as its name seems to indicate, the metric h t is normalized (its diagonal coefficients are equal to 1).

• We do not obtain the same subsets of the Kasner circle. Indeed, the result of Reiterer & Trubowitz applies to any point p of the Kasner circle such that the sequence (k i ) of the partial quotients of the continued fraction expansion of the Kasner parameter of p grows at most polynomially, that is, such that the sequence (k i ) satisfies the subpolynomial growth condition there exists P ∈ R[X] such that for all n ∈ N, k n ≤ P (n)

One can remark that between the two conditions (MG) and (sPG), neither is stronger than the other one.

• We obtain a slightly finer description of the geometry of the union of all the type IX orbits shadowing a generic heteroclinic chain. In our result, this set is proved to contain a Lipschitz manifold immersed in the phase space B + . In Reiterer & Trubowitz's work, it is not clear if the set they obtain is Lipschitz regular. Moreover, our posture is quite different from Reiterer-Trubowitz's. In [START_REF] Reiterer | The BKL Conjectures for Spatially Homogeneous Spacetimes[END_REF], Reiterer and Trubowitz provide a proof of their main theorem as concise as possible. On the contrary, our choice was to carry out a rather complete and systematic investigation of the properties of the Wainwright-Hsu vector field from the viewpoint of non-uniformly hyperbolic systems theory. Theorem 1.9 appears as a kind of corollary of this investigation.

The second assertion of Theorem 1.9 is new. Our proof of this assertion relies on a rather precise geometrical description of (a subset of) the union of the orbits shadowing heteroclinic chains : we prove that this union contains a continuous foliations by codimension one Lipschitz submanifolds, we study the regularity of the holonomy of this foliation. We believe that our viewpoint is naturally suited to such geometrical description. Yet, all the arguments we use could probably be translated in Reiterer-Trubowitz's language, and therefore the second assertion of Theorem 1.9 could also probably be obtained using Reiterer-Trubowitz's viewpoint.

As we already stated in Theorem 1.3, it was already known that the ω-limit set of almost all the orbits of the Wainwright-Hsu vector field is contained in the Mixmaster attractor A . Moreover, Ringström ([Rin01]) and Brehm ([Bre16]) proved that the ω-limit set of a generic orbit is not reduced to a Taub point. This implies that almost all the orbits of the Wainwright-Hsu vector field have an oscillatory future-asymptotic behaviour (a generic orbit has at least three different ω-limit points in the Kasner circle), but this result does not give precise information about the oscillatory behaviour. In particular, using only this result, we do not know if these generic orbits shadow some heteroclinic chains or not.

Hence, we still do not know if corollary 1.10 holds for generic orbits of the Wainwright-Hsu vector field. The question whether or not it is true is of particular importance, so let us state this open question here.

Question 2. Is it true that for a generic point q of the phase space B + (with respect to Lebesgue measure), the ω-limit set of the orbit of the Wainwright-Hsu vector field with initial condition q coincides with the Mixmaster attractor A ?

Some interesting dynamical features of the Wainwright-Hsu vector field

Even if one forgets its physical origin, the Wainwright-Hsu vector field appears to be very interesting from a purely dynamical systems viewpoint.

A catalogue of classical examples of dynamical systems.

First of all, it is quite amusing that the Wainwright-Hsu equations somehow gathers in a single vector field several of the most classical examples of chaotic dynamical systems that are presented in most introductory courses.

• The behaviour of the type II orbits is described by the so-called Kasner map, which is an avatar of the most basic example of chaotic map: the angle-doubling map of the circle. More precisely, being a (non-uniformly) expanding map of degree -2 of the circle, the Kasner map is topologically conjugate to the map θ ↦ -2θ on R/Z.

• As explained in the previous pages, the Kasner parameter conjugate the Kasner map (modulo a finite quotient) to an avatar of the famous Gauss map x ↦ 1 x -⌊ 1 x ⌋. As an immediate consequence, the behaviour of the orbit of a point p under the Kasner map depends on the continued fraction development of the Kasner parameter of p. Some properties of the Gauss map will indeed play a crucial role in the proof of our main theorems (see Appendix A).

• Recall that the classical Bowen's eye-attractor is obtained by considering a vector field in the plane with an attracting cycle made of two heteroclinic orbits connecting two hyperbolic saddletype singularities (see figure 3). This example is well-known because it has a very bad statistical behaviour: the Birkhoff sums along any orbit in the interior eye do not converge. The reason is that such an orbit will spend some time close to the left corner of the eye, then a much longer time close to the right corner of the eye, then a much much longer time close to the left corner of the eye, etc. This behaviour forces the Birkhoff sums to oscillate. Now consider a periodic chain of type II orbits in the Mixmaster attractor. It is nothing but a cycle of heteroclinic orbits connecting (partially) hyperbolic saddle-type singularities. It was proved by Georgi, Härterich, Liebscher and Webster that there is a three-dimensional set of type VIII or IX orbits that are attracted by this cycle (see [START_REF] Liebscher | Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles[END_REF]). The same arguments as for the classical Bowen's eye-attractor show that the Birkhoff sum along these orbits do not converge. Hence, every periodic chain of type II orbits can indeed be considered as a "generalized Bowen's eye-attractor". Therefore, the Mixmaster attractor somehow contains a "bunch of infinitely many interlaced (generalized) Bowen's eye-attractors".

• Yet another classical system "hidden" in the Wainwright-Hsu vector field ! In some variables that we will not use in this article, the flow of the restriction of the Wainwright-Hsu vector field to the Mixmaster attractor becomes a billiard in an ideal hyperbolic triangle, the so-called cosmological billiard (see e.g. [Dam] and [START_REF] Heinzle | Mixmaster: Fact and Belief[END_REF]).

Non-linearizable degenerate partially hyperbolic singularities. When one tries to analyze in detail the behaviour of the Wainwright-Hsu vector field, one realizes that this vector field presents some unusually complicated dynamical features. For example, the starting point of the proof of our main theorems is the analysis of the dynamics of the Wainwright-Hsu vector field X in the neighbourhood of a point p of the Kasner circle. Recall that every such point p is a singularity of X . The eigenvalues of DX (p) vary with p, and there often appears some resonance between them. As a consequence, there is a dense set of points p in the neighbourhood of which the Wainwright-Hsu vector field is not linearizable. As a further consequence, we are forced to study the local dynamics of X in the neighbourhood of such points p by very basic methods (which roughly consist in using repeatedly Grönwall's lemma to bound the effect of the nonlinear terms). Note that the situation we face (the local dynamics of a non-linear vector field in the neighbourhood of a partially hyperbolic singularity in dimension 4, with arbitrarily bad resonances, the vector field being C

∞ flat in the central direction) seems to be more degenerate than what has been studied by experts.

Remark 1.12. A result of F. Takens [START_REF] Takens | Partially Hyperbolic Fixed Points[END_REF] implies that the dynamics of the Wainwright-Hsu vector field can be linearized in the neighbourhood of a point p of the Kasner circle that is not pre-periodic for the Kasner map (these are exactly the points whose eigenvalues are non-resonant). But this result does not provide any lower bound on the size of the linearization neighbourhood, nor any upper bound on the derivatives of the linearizing coordinates. As a consequence, this result can only be used in order to build some local stable manifolds for chains of type II orbits that do not accumulate on a periodic orbit of the Kasner map (this has been done by F. Béguin in [START_REF] Béguin | Aperiodic Oscillatory Asymptotic Behavior for Some Bianchi Spacetimes[END_REF]). Such chains are very rare: their union has zero Lebesgue measure in the Mixmaster attractor.

A non-uniformly hyperbolic return map with poor regularity. The proof of our main theorems relies on the non-uniformly hyperbolic behaviour of the Wainwright-Hsu vector field. In practice, we will consider the second iterate of the Poincaré return map Φ of the orbits of the Wainwright-Hsu vector field on a transverse section S. We will prove some uniformly hyperbolic properties for this return map Φ: for every point p in the intersection of the section S with the Mixmaster attractor A , if the return map Φ is defined at p, then it contracts uniformly the direction transverse to A at p, and expands uniformly the direction tangent to A . We insist on the fact that the contraction and expansion constants are independent of the point p. Moreover, the contraction in the direction transverse to A happens to be super-linear. Nevertheless, the map Φ should be considered as a non-uniformly hyperbolic map. Indeed, the size of the neighbourhood of the point p on which one can prove some contraction/expansion properties is not bounded from below uniformly in p. This is due to:

• the presence of singularities: the return map Φ is not defined everywhere (roughly speaking, an orbit which falls on a Taub point never comes back in the section);

• the lack of regularity of the return map Φ: we are only able to prove that Φ is Lipschitz. Actually, Φ might be C 1 , but some evidence indicate that the derivative of Φ, if it happens to exist, cannot be α-Hölder for some uniform α > 0.

As a consequence of this non-uniformity:

• we will be able to prove the existence of local stable manifolds for almost every orbit of Φ, but not for all orbits,

• the size of these stable manifolds will depend on the orbit, and will not be uniformly bounded from below.

Although we will prove some non-uniformly hyperbolic properties for the return map Φ, the classical Pesin's theory of non-uniformly hyperbolic maps (see e.g. [START_REF] Barreira | Introduction to Smooth Ergodic Theory[END_REF]) does not apply to Φ. The theory of non-uniformly hyperbolic maps with singularities, as developed by A. Katok and J.-M. Strelcyn (see [START_REF] Katok | Invariant Manifolds, Entropy and Billiards. Smooth Maps with Singularities[END_REF] or [START_REF] Evgueni | Invariant Measures for Hyperbolic Maps with Singularities[END_REF]) does not apply directly either. The reason is once again the lack of regularity of Φ. Indeed, the above-mentioned theories concern maps whose derivatives may explode when one approaches some singular set, but which are quite regular (at least C

2 ) far from the singular set. This is not the case of Φ: as explained above, we are not able to prove that Φ is differentiable. The hardest task in the proof of our main theorems is to obtain some hyperbolicity estimates for Φ, with some explicit controls of the size of the neighbourhoods where these estimates hold. It will cover sections 3 to 8. Once we have these estimates, we will need to « redo » Katok-Strelcyn's work in our specific context, using some Lipschitz estimates instead of the classical bounds on the first and second derivatives. Appart from the low regularity of our map, there is another important difference between Katok-Strelcyn's setting and ours:

• roughly speaking, Katok-Strelcyn's hypotheses are chosen so that the size of the neighbourhoods on which one gets various types of estimates is always polynomial with respect to the distance to the singularity;

• in our situation, we will often be forced to consider neighbourhoods with exponentially small size . . .

• . . . but the extremely small size of the neighbourhood on which we can prove interesting estimates will be balanced by the super-linear contraction in the direction transverse to the Mixmaster attractor.

Remark 1.13. Note that one really needs to use some specific properties of the Poincaré map Φ to compensate its poor regularity. Indeed, C. Bonatti, S. Crovisier and K. Shinohara have proved that generic C 1 non-uniformly hyperbolic diffeomorphisms (such diffeomorphisms are not C 1+α for any α > 0) do not admit non-trivial local stable manifolds (see [START_REF] Bonatti | The C 1+α Hypothesis in Pesin Theory Revisited[END_REF]).

Some unusual arithmetic conditions. In a non-uniformly hyperbolic system with singularities, it is not possible to construct non-trivial Pesin stable manifolds at every point p. A necessary condition (among others) is that the orbit of p should wait a long time before coming very close to the singularities. For the Wainwright-Hsu vector field, this means that we have to focus on points p of the Kasner circle whose orbits under the Kasner map will wait a long time before coming very close to the Taub points. Since the Kasner parameter turns the Kasner map into an avatar of the Gauss map, this naturally translates as a condition on the continued fraction development of the Kasner parameter of the point p. In other words, we will only be able to deal with points p whose Kasner parameter satisfies a certain arithmetic condition. Arithmetic conditions appear in various areas of dynamical systems. They are usually of one of the following two types:

• Either one needs to consider real numbers that are badly approximated by rational numbers (socalled Diophantine numbers and their generalizations). This is typically the case when one wants to prove KAM-type results, solve cohomological equations, prove the convergence of a renormalization scheme, etc. The terms (k n ) n≥0 of the continued fraction development of such numbers grow slowly with respect to n.

• Or one needs to consider real numbers that are very well-approximated by rational numbers (socalled Liouville or super-Liouville numbers). This is typically the case when one wants to construct exotic examples of elliptic dynamical systems as limits of periodic systems (for example, by using the so-called Anosov-Katok method). The terms (k n ) n≥0 of the continued fraction development of such numbers grow very fast with respect to n.

The arithmetic condition (MG) we need to consider in our proof (which we call moderate growth condition) is of neither of the two above types. The integers (k n ) n≥0 that appear in a continued fraction development satisfying this condition might grow either slowly or very fast with respect to n.

What is important is that the size of k n should be balanced by the size of k 1 , . . . , k n-1 . This is due to the competition between two phenomena. Consider a chain of type II orbits starting at some point p of the Kasner circle, a type IX orbit whose initial condition is at distance ≪ 1 of p and denote by (k n ) n≥0 the terms of the continued fraction development of the Kasner parameter of p.

• On the one hand, the contraction rate of the flow in the direction transversal to the Mixmaster attractor between a small transverse section close to p and a small transverse section close to

F k 1 +⋅⋅⋅+k n-1 (p) depends on k 1 + ⋅ ⋅ ⋅ + k n-1 .
• On the other hand, the size of the neighbourhood of F k 1 +⋅⋅⋅+k n-1 (p) where we have a good control of the behaviour of the orbits depends of k n . So, very roughly speaking, the orbits starting at distance 1 of the Mixmaster attractor will hit the neighbourhood of F k 1 +⋅⋅⋅+k n-1 (p) where we can control their behaviour provided that k n is small compared to k 1 + ⋅ ⋅ ⋅ + k n-1 (of course, we are oversimplifying). This is the reason why the moderate growth condition (MG) comes into the game.

Remark 1.14. Proving that Lebesgue almost every real number satisfies the moderate growth condition (MG) (see Lemma 1.8) is not that easy. The argument that was suggested to us by S. Gouëzel uses some rather sophisticated properties of the Gauss map (namely, the existence of a spectral gap for the transfer operator associated with the Gauss map, acting on the space of L ∞ functions with bounded essential variation).

A complicated statistical behaviour. We have explained above that a periodic chain of type II orbits of the Wainwright-Hsu vector field can be thought as a generalized Bowen's eye-attractor. But the global statistical behaviour of the Wainwright-Hsu vector field is certainly much more complicated than those of a Bowen's eye-attractor.

Indeed, for a classical Bowen's eye-attractor, the set of all the limit points (in the space of probability measures) of the Birkhoff sums is rather small: it is exactly the affine segment whose ends are the Dirac masses supported by the two eye corners. Now consider a non-periodic chain of type II orbits in the Mixmaster attractor. Such a chain will almost surely be dense in the Mixmaster attractor, i.e. the corners of the chain will almost surely be dense in the Kasner circle. Moreover, Theorem 1.9 shows that such a chain will almost surely be shadowed by a three-dimensional set of type IX orbits of the Wainwright-Hsu vector field. We are not able to compute exactly the set of limit points of the Birkhoff sums along such orbits (in the general case). But some informal arguments show that this set should typically be infinite dimensional. In any case, it is clear that the behaviour of the Birkhoff sums along most orbits of the Wainwright-Hsu vector field must be very wild.

P. Berger has introduced a quantity which quantifies the statistical complexity of a dynamical system: the emergence of the system. Roughly speaking, it measures the growth rate, as goes to 0, of the number of probability measures that are necessary to -approximate the set of all limit points of the Birkhoff sums along almost all the orbits (see [START_REF] Berger | Emergence and Non-Typicality of the Finiteness of the Attractors in Many Topologies[END_REF] for a precise definition). It is known that there exists systems with arbitrarily large emergence (such systems are actually locally generic). But the constructions rely on Baire arguments, and do not yield explicit examples. We guess that the Wainwright-Hsu vector field might be an explicit example of a dynamical system with very large emergence. So we conclude this section by the following problem:

Question 3. Is it possible to compute the emergence of the Wainwright-Hsu vector field? Is it exponential?

A high emergence rate for the Wainwright-Hsu vector field would bring another evidence that explicit models of physical systems might display a very wild dynamical behaviour . . .

Heuristic arguments underlying the proof of the main theorem

The proof of Theorem 1.9 (as well as Reiterer-Trubowitz's proof in [START_REF] Reiterer | The BKL Conjectures for Spatially Homogeneous Spacetimes[END_REF]) is based on the following heuristic argumentation, which can be attributed to Belinskii, Khalatnikov and Lifshitz (except for the very last part concerning the moderate growth condition).

Consider a point p of the Kasner circle, so that p is not one of the Taub points. The point F (p) (the image of p under the Kasner map) is a partially hyperbolic singularity of the Wainwright-Hsu vector field X . More precisely, the linear part of X at F (p) has two negative eigenvalues -µ s 1 and -µ s 2 (with µ s 2 ≥ µ s 1 ), one zero eigenvalue (corresponding to the direction tangent to the Kasner circle), and one positive eigenvalue µ u . The eigendirections associated with the two stable eigenvalues, -µ s 1 and -µ s 2 , are tangent to the two type II orbits arriving at F (p) (hence, one of them, say the one associated with -µ s 1 , is tangent to the type II orbit O p→F (p) ). The eigendirection associated with the unstable eigenvalue µ u is tangent to the type II orbit O F (p)→F 2 (p) going from F (p) to F 2 (p). Consider a type IX orbit O traveling very close to the type II orbit O p→F (p) . After some time, it will enter a neighbourhood B 1 of F (p). Let d 1 be the distance between the orbits O and O p→F (p) when they enter in B 1 . The orbit O will continue to follow O p→F (p) until it comes very close to the point F (p) (going slower and slower since F (p) is a singularity). Then it will start to follow the unstable manifold of F (p), that is, to follow the type II orbit O F (p)→F 2 (p) . Now, suppose for a moment that one could neglect the non-linear part of X . Then we can compute explicitly the flow of X , and we see that the orbit O will exit B 1 roughly at distance d

µ s1 /µ u 1 from the orbit O F (p)→F 2 (p)
. The crucial point is that the stable eigenvalues of the point of the Kasner circle are "stronger" than the unstable one. In other words, µ s 1 /µ u is greater than 1 and therefore d Of course, this very rough heuristic argument dramatically oversimplifies the situation (otherwise the proof of Theorem 1.9 would not fill so many pages !). Yet it will serve us as a guideline, and our task will be to turn it into a rigorous proof.

µ s1 /µ u 1 is much smaller than d 1 . Now,
The main difficulties that we will face are the following. When we analyze the behaviour of the orbit O inside a neighbourhood of F (p), we need to take into account the effect of the non-linear part of X . These non-linear terms will in particular induce a drift in the central direction, i.e. in the direction of the Kasner circle. So the orbit O will deviate from the heteroclinic chain of type II orbits, and we shall need to control this deviation, and prove that it is somehow balanced by the very strong contraction due to the linear part of the vector field. We also have to take into account the fact that the stable and unstable eigenvalues -µ s 1 , -µ s 2 and µ u at the point F (p) critically depend on the position of this point on the Kasner circle: -µ s 1 and µ u tend to zero as the point F (p) approaches one of the Taub points. This means that, when F (p) is very close to a Taub point, the hyperbolicity of the linear part of X at F (p) is very weak, and therefore can only compensate the effect of the non-linear part in an extremely small neighbourhood B of F (p).

So there will be a competition. On the one hand, if the orbit O falls successively in the neighbourhooods B 1 , B 2 , . . . , B n of the points F 1 (p), F 2 (p), . . . , F n (p), then the distance between O and

• p O p→F (p) •F (p) O F (p)→F 2 (p) B 1 F 2 (p) • O F 2 (p)→F 3 (p) B 2 O Figure 4:
The orbit O successively enters the neighbourhoods B 1 , B 2 ,. . . Each time it passes inside one of these neighbourhoods, it gets much closer to the heteroclinic chain starting at p, due to the super-linear contraction.

the heteroclinic chain of type II orbits O p→F (p) , O F (p)→F 2 (p) , . . . will undergo a very strong contraction. Therefore the orbit O will have more chance to enter the neighbourhood B n+1 of the point F n+1 (p). On the other hand, if the point F n+1 (p) happens to be very close to one of the Taub points, then the neighbourhood B n+1 will be extremely small and it is quite likely that the orbit O will fail to enter this neighbourhood, in which case the future behaviour of O will get out of control. This is the reason why we will not always be able to prove the existence of type IX orbits shadowing the heteroclinic chain O p→F (p) , O F (p)→F 2 (p) , . . . . Roughly speaking, we will need this heteroclinic chain to "wait enough time before going close to the Taub points".

In order to be more quantitative, let us consider the continued fraction expansion [1; k 1 , k 2 , . . . ] of the Kasner parameter of the point p. On the one hand, if the orbit O falls in the neighbourhoods B 1 , B 2 , . . . , B n , then the contraction of the distance between O and the heteroclinic chain will roughly be controlled by k 5 1 + ⋅ ⋅ ⋅ + k 5 n . On the other hand, the size of the neighbourhood B n+1 will roughly be controlled by k 4 n+4 . So we will be able to keep some control on the behaviour of the orbit O if and only if the continued fraction expansion satisfies the moderate growth condition (MG). Once again, we are oversimplifying, but this is indeed the origin of the moderate growth condition.

Strategy of the proof of the main theorem and organization of the article

In the next few pages, we will describe the content of the different sections of this article. We hope that the strategy of the proof of our main theorems will arise from this description.

The Wainwright-Hsu vector field and the Mixmaster attractor. In Section 2, we describe the dynamics of the Wainwright-Hsu vector field X in restriction to the Mixmaster attractor (linear part of X at points of the Kasner circle, explicit expression of the type II orbits, Kasner map, Kasner parameter, etc.). This dynamics is well-known. The only original part of Section 2 is the description of a finite quotient of the classical phase space in which we shall work.

Local expression of the Wainwright-Hsu vector field in the neighbourhood of a point of the Kasner circle. As explained above in heuristic terms, the proof of Theorem 1.9 is based on the analysis of the local dynamics of the Wainwright-Hsu vector field X in the neighbourhood of a point p of the Kasner circle. To carry out this analysis, we use a quite standard strategy: we first construct some coordinates in which the vector field X has the simplest possible expression, and then, we use this expression to control the deviation of the true orbits of X from those of the linear part DX (p) of X . Hence, our first task is to find a "nice" local coordinate system in the neighbourhood of a point p of the Kasner circle (which is not one of the Taub points). Actually, the only property we need for this coordinates system is that it straightens the stable, central and unstable manifold of X at the point p. So the coordinates system will be provided by the stable manifold theorem. Yet we need a quite precise version of this result: in particular, we need some lower bounds on the size of the neighbourhoods on which the straightening coordinates are defined, and some upper bounds on the norm of the derivative of these coordinates, with some explicit dependence on a parameter. Once we have the suitable statement of the stable manifold theorem, we apply it three times (together with some other easy coordinate change) to get a local coordinate system straightening the strong stable, weak stable, central and strong unstable manifolds of p. Then we write the local expression of X in this "nice" local coordinate system, providing some upper bounds on the non-linear terms showing up in this expression. This is done in Section 3.

Local sections and transition maps.

In Section 4, we define some sections transverse to the Wainwright-Hsu vector field X . For every point p in the Kasner circle (which is not one of the Taub points), we define a local section S s p that will be intersected by the orbits of X when they arrive in a small neighbourhood of p. Similarly, we define a local section S u p that will be intersected by the orbits of X when they get out from a small neighbourhood of p. The size of these sections (in the different directions), as well as their distance to the point p, depend on several parameters. We also define a global section S which is intersected by all the type IX orbits that could possibly shadow some heteroclinic chain of type II orbits. Moreover, in order to understand the dynamics of the orbits traveling between two sections, we are led to define some transition maps. The transition map from a section S 1 to a section S 2 encodes, for an orbit O of X starting in S 1 , the first intersection point of O with the section S 2 .

Local dynamics in the neighbourhood of a point of the Kasner circle. In Section 5, we use the local expression of the Wainwright-Hsu vector field X in order to study the local dynamics of X in the neighbourhood of a point p of the Kasner circle. More precisely, we want to understand the transition map Υ p of the orbits of X from a local section S s p at the entrance of a neighbourhood of p to a local section S u p at the exit of the same neighbourhood. The task consists in controlling the effect of the non-linear terms in the local expression of X . The size of the neighbourhood of p, the size of the local sections S s p and S u p , and their distance to the point p, depend on the Kasner parameter of p. The outcome of the section is roughly the following: when the orbits of X cross a small neighbourhood of the point p, the distance from these orbits to the Mixmaster attractor undergoes a super-linear contraction, whereas the drift of the orbits in the direction tangent to the Mixmaster attractor is extremely small. In other words, the transition map Υ p is strongly contracting in the direction transverse to the Mixmaster attractor (the contraction is super-linear), and almost isometric in the direction tangent to the Mixmaster attractor. An important point is that the dependence of the contraction (resp. drift) rate with respect to the Kasner parameter of p is explicit. Note that to get this explicit dependence, we extend the methods employed in [START_REF] Liebscher | Ancient Dynamics in Bianchi Models: Approach to Periodic Cycles[END_REF].

Dynamics in the neighbourhood of a type II orbit. Consider again a point p on the Kasner circle. The purpose of Section 6 is to control the behaviour of a type IX orbit traveling very close to the type II orbit O p→F (p) . More precisely, we want to control the transition map Ψ p of the orbits of X from a local section S u p at the exit of a neighbourhood of p to a local section S s F (p) at the entrance of a neighbourhood of the point F (p). The estimates we obtain are very loose, since we are considering the long range behaviour of a non-linear vector field. The only thing we can do is to:

• find an upper bound of the travel time of the orbits between the sections S

(p). Observe that Φ p T 1 • T 2 • T 3 • ω = 1 • ω = 1 • ω = 1 • • p • • • • • F k 1 (p) • ω = 2 • ω = 2 •ω = 2
Figure 5: First era of the heteroclinic chain starting at p, represented in projection on the plane containing the Kasner circle.

is nothing else than the composition of the maps Υ p and Ψ p considered in Sections 5 and 6. So, we will only need to concatenate the estimates proven for the maps Υ p and Ψ p to obtain some estimates on Φ p . The only difficulty is to find some size of the sections S s p and S s F (p) so that the map Φ p is well-defined. This is done in Section 7. Once we know that Φ p is well-defined and is the composition of Υ p and Ψ p , we easily obtain some partial hyperbolicity properties for Φ p : it is super-contracting in the direction transverse to the Mixmaster attractor, and almost not contracting in the direction tangent to the Mixmaster attractor (this direction may be expanded, or very weakly contracted).

Dynamics along an era. Consider the region K ]1,2] of the Kasner circle where the Kasner parameter ranges between 1 and 2 (roughly speaking, this is the region of the Kasner circle which is far from the Taub points). Let p be a point in K ]1,2] , and denote by k 1 the first term in the continued fraction expansion of the Kasner parameter ω(p). The heteroclinic chain of type II orbits starting at p first goes close (roughly at distance 1 k 1 ) to one of the Taub points, say T 3 , then bounces k 1 -1 times from one side of T 3 to the other, slowly escaping from the vicinity of T 3 , until it comes back in K ]1,2] . An era is such a piece of heteroclinic chain, made of the concatenation of k 1 type II orbits, which starts and ends up in K ]1,2] . See figure 5. The purpose of Section 8 is to study the behaviour of the orbits of the Wainwright-Hsu vector field X along such an era. More precisely, we want to study the era transition map, i.e. the transition map Φp of the orbits of X from a local section S s p at the entrance of a neighbourhood of p to a local section S s F k1 (p) at the entrance of a neighbourhood of the point F k 1 (p). This map can be seen as the composition of the k 1 epoch transition maps Φ p , Φ F (p) , . . . , Φ F k1-1 (p) provided that we can find some size of sections so that this composition is well-defined. We indeed manage to set up an induction scheme, based on the estimates of Section 7, showing that the composition of the epoch transitions maps Φ p , Φ F (p) , . . . , Φ F k1-1 (p) is well-defined on a tiny local section close to p.

It is natural to expect some uniform hyperbolicity properties for Φp . Yet a minor (but quite annoying) technical difficulty shows up. One soon realizes that the map Φp cannot be uniformly expanding in the direction tangent to the Mixmaster attractor. This can be easily overcome by replacing Φp by the "double era transition map" Φp , which describes the behaviour of the orbits during two eras instead of a single one. We are indeed able to prove some hyperbolicity properties for this map: it contracts uniformly the direction transverse to the Mixmaster attractor and expands uniformly the direction tangent to this attractor.

Moreover, the Φp 's can be glued together, in order to get a global Lipschitz map Φ. This map is the second iterate of the Poincaré's return map of the orbits of the Wainwright-Hsu vector field X on a global section S. We call it the double era return map. The section S is intersected by all the orbits that could potentially shadow some heteroclinic chain. Yet, it is important to note that Φ is not well-defined on the whole section S. It is defined on a kind of hedge with variable height over the interval ]1, 2]: the height of the hedge over the point ω ∈ ]1, 2] depends on the four first terms of the continued fraction development of ω, and is equal to zero at certain points. The map Φ is uniformly hyperbolic on this hedge-shaped domain.

Construction of local stable manifolds for the double era return map. In Section 9, we use the hyperbolicity of the double era return map Φ, together with the usual graph transform mapping technique, in order to construct some local stable manifolds for Φ. The main difficulty is to find some domains where the graph transform mapping can be iterated (recall that the map Φ is not defined on the whole section S). This is where the moderate growth condition (MG) shows up. Roughly speaking, we can iterate the graph transform mapping over the orbit of a point p ∈ K ]1,2] if and only if the Kasner parameter of p satisfies the moderate growth condition. For such a point p, we obtain a non-trivial two-dimensional local stable manifold W s (p, Φ). The size of this local stable manifold depends on p. In particular, it depends on the time n 0 one has to wait in order to "see" the domination of k

4 n+4 by the sum ∑ n i=1 k 5 i for all n ≥ n 0 .
Shadowing of heteroclinic chains. Consider a point p of the Kasner circle having a non-trivial stable manifold W s (p, Φ) for the double era return map Φ, and a point q ∈ W s (p, Φ). In Section 10, we prove that the forward orbit of q (for the Wainwright-Hsu vector field) shadows (in the sense of definition 1.4) the heteroclinic chain O p→F (p) , O F (p)→F 2 (p) , . . . . This easily follows from what has been done earlier. Thanks to some estimates proven in Sections 5, 6 and 8, we know that, since q is close to p, the forward orbit of q will stay very close to the heteroclinic chain O p→F (p) , O F (p)→F 2 (p) , . . . during two complete eras. But since q is in the stable manifold W s (p, Φ), this orbit hits the section S very close to F (p) ∶= F k 1 +k 2 (p) (we call F the double era Kasner map). So, using again the estimates of Sections 5, 6 and 8, we obtain that the forward orbit of q stays very close to the heteroclinic chain during two more eras. Then it hits the section S even closer to F 2 (p) = F k 1 +k 2 +k 3 +k 4 (p). Iterating this argument, we obtain that the forward orbit of q shadows the entire heteroclinic chain O p→F (p) , O F (p)→F 2 (p) , . . . . At this point, we have proved the first part of Theorem 1.9, i.e. we have constructed a threedimensional set of type IX orbits that shadow the heteroclinic chain O p→F (p) , O F (p)→F 2 (p) , . . . for every point p whose Kasner parameter satisfies the moderate growth condition (MG).

Absolute continuity of the local stable manifolds foliation.

The second part of Theorem 1.9 is proven in Section 11. Namely, we consider a set E of positive one-dimensional Lebesgue measure in the Kasner circle, and we prove that the union of the type IX orbits shadowing a heteroclinic chain O p→F (p) , O F (p)→F 2 (p) , . . . with p ∈ E has positive four-dimensional Lebesgue measure in the phase space. Without loss of generality, one can assume that E ⊂ K ]1,2] . According to what has been explained above, it is enough to prove that the union of the local stable manifolds W s (p, Φ) when p ranges over E has positive three-dimensional Lebesgue measure in the transverse section S.

Remark 1.15. Readers that are not familiar with non-uniformly hyperbolic dynamics might think that this is a straightforward consequence of the W s (p, Φ)'s being two-dimensional submanifolds which depend continuously on p. However, a continuous dependence is not sufficient to apply a Fubini type argument (recall that a homeomorphism might send positive Lebesgue measure sets to zero Lebesgue measure sets). Examples of non-uniformly hyperbolic dynamical systems with pathological local stable manifold foliations do exist.

We use a well-known strategy due to Y. Pesin. Roughly speaking, given two (one-dimensional) local transversal T, T ′ to the "foliation" by local stable manifold, one considers the holonomy map which sends a point x ∈ T to the unique point of intersection of the local stable manifold passing through x with the transversal T ′ . If one can prove that this holonomy map is absolutely continuous (i.e. if it maps zero 1-dimensional Lebesgue measure sets in T to zero 1-dimensional Lebesgue measure sets in T ′ ) for any choice of T and T ′ , then it is easy to set up a Fubini type argument and prove that the union of the local stable manifolds has positive 4-dimensional Lebesgue measure. The absolute continuity of the holonomy map follows from estimates on the action of these maps on the volume of discs transverse to the local stable manifolds. Such estimates are trivial for "big" discs. The trick is to turn small discs into big ones using the map Φ. Indeed the discs are transversal to the stable manifold, hence essentially behave as unstable discs, and therefore, their images under Φn become larger and larger as n goes to infinity.

Our setting is easier than the usual general setting because the "foliation" by stable manifolds is transversally one-dimensional, and therefore the discs transverse to the foliation are just arcs, whose volume can be computed easily (in particular, it roughly coincide with the diameter of these arcs). On the other hand, our setting is also more tricky because we have to work with a map Φ which is not defined everywhere, so we have to be very careful when we consider large iterates of Φ to make the discs grow.

Remark 1.16. One could be worried since it is well-known that Pesin's absolute continuity techniques only work for C 1+α maps, and since we have explained previously that our map Φ is only Lipschitz. Actually, the C 1+α -regularity is used for two purposes in Pesin's proof. First, to find some lower bounds for the size of the neighbourhoods of the points of the attractor where certain hyperbolicity estimates hold. We already have computed such sizes in the previous sections. Second, to get some Hölder regularity on the unstable direction (tangent space to the attractor). In our case, this regularity is for free, since we know explicitly the attractor, and since the intersection of this attractor with the section S is extremely simple and regular: this is an affine interval in our local coordinates (which are at least C 4 ). Hence the low regularity of the map Φ is not a true problem for this precise proof.

Continued fractions. Some classical material about continued fractions and the Gauss map is gathered in Appendix A. This is also the place where we prove that the moderate growth condition is generic in the measure-theoretical sense.

Statement of the main theorem in the full phase space. For sake of simplicity, we have stated Theorem 1.9 in the restricted phase space B + (in particular, we have restricted ourselves to heteroclinic chains that can be shadowed by type IX orbits). Nevertheless, our proof also works in the full phase space, provided that we introduce a natural notion of coherent heteroclinic chain. The generalization of Theorem 1.9 to the full phase space B is stated in Appendix B.

Correspondence with Reiterer-Trubowitz's paper [RT10] .

Although based on a slightly different viewpoint and formulated in a somewhat different language, the strategy of our proof is quite parallel to those of Reiterer-Trubowitz's in [START_REF] Reiterer | The BKL Conjectures for Spatially Homogeneous Spacetimes[END_REF]:

• The "nice coordinates" that we introduce in Section 3 are reminiscent of the variables introduced in Definition 3.3 of [START_REF] Reiterer | The BKL Conjectures for Spatially Homogeneous Spacetimes[END_REF]. The Taylor expansion of the Wainwrigh-Hsu vector field in our "nice coordinate system" can be compared to the system of differential equations provided in Lemma 3.1 of [START_REF] Reiterer | The BKL Conjectures for Spatially Homogeneous Spacetimes[END_REF]. Some differences can nevertheless be noticed. Reiterer-Trubowitz's variables are specifically designed to study the evolution equations of the vacuum spatially homogeneous spacetimes.

A nice feature of these variables is that they are completely explicit, as well as their evolution equations. On the contrary, the "nice coordinates" that we consider are not explicit. Their existence follows from general results on partially hyperbolic system. The main advantage of such coordinates is that they diagonalize the linear part of the evolution equations, and show very clearly which non-linear terms should be controlled.

• In Section 4, we introduce several local sections of the Wainwright-Hsu flow, as well as the Poincaré return maps on these sections. One of the goals of Sections 5, 6 is to prove that these Poincaré return maps are well-defined on some explicit subsets of the local sections. This should be compared to the series of definitions and lemmas in [START_REF] Reiterer | The BKL Conjectures for Spatially Homogeneous Spacetimes[END_REF] which culminates into Proposition 3.3. Indeed the purpose of this part of Reiterer-Trubowitz's paper is to introduce a 3-dimensional discrete dynamical system which can be thought as a "section" of the flow of their 4-dimensional system of differential equations, and to find an explicit set on which this 3-dimensional discrete dynamical system is well-defined.

• Proposition 5.1 of [START_REF] Reiterer | The BKL Conjectures for Spatially Homogeneous Spacetimes[END_REF] is a general fixed point result which should be compared to the graph transform techniques that we use in Section 9 to prove the existence of stable manifolds. As always, our viewpoint is more geometric since we consider the action of our dynamical system on sequences of sub-manifolds rather than sequences of points. Yet, this is merely a matter of language, since the sequences of points considered by Reiterer and Trubowitz depend on real parameters which should be thought as the coordinates parametrizing our sub-manifolds. With some extra technical work, it should be possible to find some hypotheses which ensure that Reiterer-Trubowitz's fixed points depend nicely enough on the parameters to form some invariant sub-manifolds.

• The goal of Section 8 is to prove that the estimates on the epoch maps can be combined to show that the (double) era return map satisfies some hyperbolicity properties. Then, we use these properties in Section 9 to prove the existence of local stable manifolds for the (double) era return map. This should be compared to Theorems 6.1., 6.2 and 6.3. in [START_REF] Reiterer | The BKL Conjectures for Spatially Homogeneous Spacetimes[END_REF].

The Wainwright-Hsu vector field and the Mixmaster attractor

In this section, we will recall a number of well known facts about the Wainwright-Hsu vector field X , its dynamics in restriction to the Mixmaster attractor, the Kasner map and the Kasner parameter. A good reference for these facts is [START_REF] Heinzle | Mixmaster: Fact and Belief[END_REF]. However, there is something "new" in addition to what is presented in [START_REF] Heinzle | Mixmaster: Fact and Belief[END_REF]: we will define the quotient phase space and the induced Wainwright-Hsu vector field X on that space (see section 2.6), as they will be more convenient to work with in what follows.

The Wainwright-Hsu vector field X

Recall that we will be interested in describing the behaviour of solutions of the system of equations (1.3).

Phase space. Consider the phase space

B = (N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) ∈ R 6 satisfying (1.3a) and (1.3b)
Observe that B is a non-singular and non-compact 4-dimensional quadric in R 6 .

Wainwright-Hsu vector field The Wainwright-Hsu vector field, denoted by X , is defined as the vector field on B associated with the Wainwright-Hsu equations (1.3), that is,

X (N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ -(q + 2Σ 1 )N 1 -(q + 2Σ 2 )N 2 -(q + 2Σ 3 )N 3 (2 -q)Σ 1 + S 1 (2 -q)Σ 2 + S 2 (2 -q)Σ 3 + S 3 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (2.1)
The solutions of the system of equation (1.3) can be seen as the orbits of the flow of the Wainwright-Hsu vector field X on the phase space B.

Symmetries. The Wainwright-Hsu vector field is equivariant for the action of the permutations group S 3 :

σ.(N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) = (N σ(1) , N σ(2) , N σ(3) , Σ σ(1) , Σ σ(2) , Σ σ(3) ) (2.2)
and for the action of the group Z/2Z = {Id, } given by:

.(N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) = (-N 1 , -N 2 , -N 3 , Σ 1 , Σ 2 , Σ 3 )
One should remark that this implies that S 3 is acting on the space of the orbits of the Wainwright-Hsu vector field. Later on, to simplify the presentation, we will work in the quotient phase space B/S 3 (see section 2.6).

Stratification of the phase space B

Stratification of the phase space. According to (1.3c), the signs (positive, negative or null) of the variables N i are invariant along the orbits of the Wainwright-Hsu vector field. This fact leads to a stratification of the phase space B in six subsets which are invariant under the flow of the Wainwright-Hsu vector field X (see table 1). Recall that the variables N i are closely related to the structure constants of the Bianchi spacetime represented by the orbit. So this stratification is no more than a reinterpretation of the classification of the 3-dimensional unimodular Lie algebras. This stratification plays an important role in the study of the dynamics of the Wainwright-Hsu vector field X . This is thanks to the following facts: the dynamics on the low dimensional strata (1 and 2) can be described entirely explicitly and the reunion of these low dimensional strata forms an attractor on which almost every orbit of the Wainwright-Hsu vector field accumulate.

Restriction to the "positive" part of the phase space. As stated in the introduction, to avoid clutter with notations and to simplify the presentation, we will restrict our attention to the dynamics of the Wainwright-Hsu vector field in

B + def = {(N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) ∈ B | N 1 ≥ 0, N 2 ≥ 0, N 3 ≥ 0} Recall that • B
+ is invariant under the flow of the Wainwright-Hsu vector field.

• Generic orbits of B + are type IX orbits.

We will denote B + II ∶= B II ∩ B + and analogously for other stratas. Also, we will implicitly restrict the Wainwright-Hsu vector field to B

+ from now on.

The Kasner circle K . The stratum K = B I corresponding to Abelian Lie algebra is onedimensional. It is a Euclidean circle denoted by K and called the Kasner circle (because its points correspond to Kasner spacetimes, see [START_REF] Wainwright | A Dynamical Systems Approach to Bianchi Cosmologies: Orthogonal Models of Class A[END_REF]):

K = {(N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) ∈ B | N 1 = N 2 = N 3 = 0} = (0, 0, 0, Σ 1 , Σ 2 , Σ 3 ) ∈ R 6 | Σ 1 + Σ 2 + Σ 3 = 0, Σ 2 1 + Σ 2 2 + Σ 2 3 = 6 (2.
3)

The stratum B + II . The stratum B + II corresponding to Heisenberg Lie algebras is two-dimensional. It is the reunion of three open hemiellipsoids (see later figure 7), each having the Kasner circle as boundary:

B + II = B 1 II ⊔ B 2 II ⊔ B 3 II
where

B 1 II = {(N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) ∈ B | N 1 > 0, N 2 = N 3 = 0} = (N 1 , 0, 0, Σ 1 , Σ 2 , Σ 3 ) ∈ R 6 | N 1 > 0, Σ 1 + Σ 2 + Σ 3 = 0, Σ 2 1 + Σ 2 2 + Σ 2 3 + 1 2 N 2 1 = 6 (2.4)
The hemiellipsoids B

T 1 • T 2 • T 3 • Q 2 • Q 3 • Q 1 • µ 3 < µ 1 < 0 < µ 2 µ 1 < µ 3 < 0 < µ 2 µ 1 < µ 2 < 0 < µ 3 µ 2 < µ 1 < 0 < µ 3 µ 2 < µ 3 < 0 < µ 1 µ 3 < µ 2 < 0 < µ 1 K (213) K (231) K (321) K (312) K (132) K (123)
Figure 6: Order of the eigenvalues.

Linearization of the Wainwright-Hsu vector field along the Kasner circle

Critical points. The critical points of the Wainwright-Hsu vector field correspond to self-similarly expanding spacetimes (see [START_REF] Wainwright | A Dynamical Systems Approach to Bianchi Cosmologies: Orthogonal Models of Class A[END_REF]). Using (1.3c) and (2.3), one can see that any point of the Kasner circle K is a critical point. The goal of this section is to describe the eigenvalues of DX (p) for any point p of the Kasner circle. 6). There are three particular points in the Kasner circle called the Taub points:

Notations (see figure

T 1 = (0, 0, 0, 2, -1, -1)

T 2 = (0, 0, 0, -1, 2, -1)

T 3 = (0, 0, 0, -1, -1, 2)
These points split the Kasner circle in three open arcs K 1 , K 2 , K 3 defined as following: K i is the connected component of K \ {T 1 , T 2 , T 3 } admitting T j and T k as end points, where {i, j, k} = {1, 2, 3}. Alternatively, one can define K i as the subset of K where

Σ i < -1
We denote by Q 1 , Q 2 , Q 3 the diametrically opposite points of the Taub points in the Kasner circle, that is,

Q 1 = (0, 0, 0, -2, 1, 1) Q 2 = (0, 0, 0, 1, -2, 1) Q 3 = (0, 0, 0, 1, 1, -2) One can remark that Q i is the middle of the arc K i . As such, Q i divides K i in two open arcs K (ijk)
and K (ikj) with respective end points Q i , T k and Q i , T j . Alternatively, one can define K (ijk) as the subset of K where Σ i < Σ j < Σ k Eigenvalues of the linearized vector field at points of the Kasner circle. In Section 5, we will study the behaviour of the orbits passing close to a point p of the Kasner circle. The first step is to linearize the Wainwright-Hsu vector field at the points of the Kasner circle. Indeed, the local behaviour of the orbits is determined, at the first order, by the linear part of the vector field.

Let p = (0, 0, 0, Σ 1 , Σ 2 , Σ 3 ) be a point of the Kasner circle K . One can remark that

(Σ 3 -Σ 2 )∂ Σ 1 + (Σ 1 -Σ 3 )∂ Σ 2 + (Σ 2 -Σ 1 )∂ Σ 3
is tangent to K at p and that

(∂ N 1 , ∂ N 2 , ∂ N 3 , (Σ 3 -Σ 2 )∂ Σ 1 + (Σ 1 -Σ 3 )∂ Σ 2 + (Σ 2 -Σ 1 )∂ Σ 3 )
is a basis of T p B. In this basis, the matrix of DX (p) is

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ µ 1 0 0 0 0 µ 2 0 0 0 0 µ 3 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (2.5)
where

µ i = -(2 + 2Σ i ) (2.6)
We summarize the main properties of the eigenvalues µ i in Proposition 2.1 and figure 6.

Proposition 2.1. If {i, j, k} = {1, 2, 3}, then 1. On K (ijk) , we have µ k < µ j < 0 < µ i . Moreover, the unstable eigenvalue µ i is "weaker" than the stable eigenvalues µ j and µ k :

µ i < |µ j | and µ i < |µ k |.
2. At the point T i , we have µ i < 0 and µ j = µ k = 0.

At the point

Q i , we have µ i > 0 and µ j = µ k < 0.
Proof. This is a straightforward consequence of (2.6).

Remark 2.2. According to Proposition 2.1, the unstable direction and the weak stable direction swap at the Taub points while the weak stable direction and the strong stable direction swap at the points Q i , i = 1, 2, 3. Moreover, each Taub point has a one dimensional stable manifold and a three dimensional central manifold. Every other point of the Kasner circle has a two dimensional stable manifold, a one dimensional unstable manifold and a one dimensional central manifold.

Type II orbits

The orbits contained in the stratum B + II are called type II orbits. These orbits can be explicitly described in an easy manner. Let M 1 = (0, 0, 0, -4, 2, 2) 7). As a consequence, any type II orbit is a heteroclinic connexion9 between two points of the Kasner circle. One easy way to see this is to remark that, for a type II orbit contained in B i II , the Wainwright-Hsu equations (1.3c) lead to the conservation of the quantity Σ j -2 Σ k -2 along the orbit (see [START_REF] Ringström | The Bianchi IX Attractor[END_REF] for more details).

M 2 = (0, 0, 0, 2, -4, 2) M 3 = (0, 0, 0, 2, 2, -4) For i ∈ {1, 2,
B i II K • • • M i ∂ N i • • • • Figure 7: Type II orbits contained in B i II .
Local view-point. Let p ∈ K (ijk) . There are exactly three type II orbits which establish a heteroclinic connexion between p and another point of the Kasner circle. We are now going to determine the "time direction" of these orbits, that is, to determine whether they admit p as an ω-limit point or an α-limit point. Recall that the Wainwright-Hsu vector field admits three non trivial eigenvalues µ k < µ j < 0 < µ i at the point p. It follows that:

• The type II orbit contained in B i II , denoted by O u p , admits the point p as its α-limit point. We will say that this orbit starts at p. p ), admits the point p as its ω-limit point. We will say that these orbits arrive at p.

Global view-point. B

i II is foliated by type II orbits in a very specific way. Any type II orbit contained in B i II starts in K i and arrives in K j ∪ {T i } ∪ K k . More precisely, those starting in K (ijk) arrive in K j and the one starting at Q i arrives at T i . There is no type II orbit starting from a Taub point.

Projection view-point. Another way to describe the type II orbits is to give their projection on the (Σ 1 , Σ 2 , Σ 3 )-plane, that is, the two-dimensional plane containing the Kasner circle. Let P 1 i be the sheaf of one-dimensional affine lines passing through M i and contained in the (Σ 1 , Σ 2 , Σ 3 )-plane.

According to what precedes, the projections of the type II orbits contained in B i II are exactly the intersections of the open disc delimited by the Kasner circle and the lines of the sheaf P 1 i (see figure 8).

The Kasner map F

Geometrical construction of F (p).

Let p ∈ K i . The line (M i p) intersects the Kasner circle at two points. The closest to M i is p while the farthest is F (p) (see figure 9). One can remark that

F (Q i ) = T i for i ∈ {1, 2, 3}. T 1 • T 2 • T 3 • Q 2 • Q 3 • Q 1 • M 1 • M 2 • M 3 • T 1 • T 2 • T 3 • Q 2 • Q 3 • Q 1 • M 1 • M 2 • M 3 • T 1 • T 2 • T 3 • Q 2 • Q 3 • Q 1 • M 1 • M 2 • M 3 •
T 1 • T 2 • T 3 • M 1 • M 2 • M 3 • • F (p) • p • q • F (q)
Figure 9: The Kasner map.

T 1 • T 2 • T 3 • • p • • • • • • • • • • • •
Figure 10: The Kasner map is chaotic.

Dynamics of the Kasner map. The dynamics of the Kasner map on the circle is chaotic (see figure 10). One can verify that the Kasner map is C ∞ and of degree -2, is non uniformly expanding (the derivative is, in absolute value, strictly superior to 1 except at the Taub points where its absolute value is equal to 1). By a classical argument (see e.g. [KH97, Theorem 2.4.6], the Kasner map is topologically conjugate to θ ↦ -2θ (on the circle R/Z) which has a well understood dynamics 10 . In particular, F has the following properties:

• Periodic points of F are dense in K .

• There exists points in K whose forward orbit under F are dense in K . The set of all such points is a G δ dense.

• For every point p of the Kasner circle, the complete backward orbit of p under F is dense in K .

• The topological entropy of the Kasner map is positive (it is equal to log(2)).

• F possesses an invariant measure (of infinite mass) absolutely continuous with respect to Lebesgue measure.

We will come back to the dynamics of the Kasner map in section 2.7, after introducing the Kasner parameter and reducing the dynamics modulo symmetries.

Generalized heteroclinic chains.

Let p be a point of the Kasner circle. One can consider the orbit of p under the Kasner map p, F (p), F 2 (p), F 3 (p), . . . and a chain of heteroclinic connexions between the consecutive iterates of this sequence. This forms a continuous curve in the phase space B. At every step, the heteroclinic connexion is the type II orbit O F n (p)→F n+1 (p) .

The following notion of heteroclinic chain is standard.

Definition 2.4 (Heteroclinic chain). Let p be a point of the Kasner circle which is not one of the Taub points. The heteroclinic chain starting at p is the sequence If there exists n ∈ N * such that F n (p) is a Taub point, then the heteroclinic chain starting at p ends at that point.

H (p) def = O p→F (p) , O F (p)→F 2 (p) , O F 2 (p)
To simplify the definition of some transition maps that we will use later on, we extend the above definition.

Definition 2.5 (Generalized heteroclinic chain). Let q ∈ B + \ K . Denote by O(q) the forward X -orbit of q. The heteroclinic chain H (q) starting at q is defined as follows:

• If O(q) converges to a point p of the Kasner circle which is not a Taub point, then H (q) is the concatenation of O(q) with H (p):

H (q) def = O(q), O p→F (p) , O F (p)→F 2 (p) , . . . (2.8) • Otherwise, H (q) is simply the orbit O(q).
It is well known that type IX orbits cannot converge to a Taub point. Hence, if q ∈ B + IX , the heteroclinic chain starting at q is nothing but the forward X -orbit of q. Recall that we want to describe the heteroclinic chains starting at points of the Kasner circle which are shadowed by some type IX orbits (see definition 1.4).

Quotient phase space B

Recall that S 3 acts on B by permutation of the indices 1, 2, 3 (see (2.2)). From now on, we will make a systematic use of these symmetries. Let us define the quotient phase space and its positive part

B def = B/S 3 , B + def = B + /S 3
as well as the natural projection map π ∶ B → B (2.9)

Many results have a natural presentation in the quotient phase space B. The only case where it is better to work in the phase space B is when one needs to use precisely the Wainwright-Hsu equations. This will not happen often in our work. We will mainly use the properties described in sections 2.3 and 2.4: the behaviour of type II orbits and the eigenvalues of DX (p) for p ∈ K . In order to have a better understanding of the quotient, one needs to describe the orbits under the action of S 3 . Before going into the details in low dimensional strata, one can notice that S 3 acts freely and properly on B \ S where S is the singular set defined by

S def = (N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) ∈ B | ∃i ≠ j, N i = N j and Σ i = Σ j
Hence, B is a 4-dimensional orbifold with singular locus π (S ). The fact that B is singular is not a huge issue. Indeed, we will be interested in heteroclinic chains which are disjoint from the singular locus S . Since S is closed and invariant under the flow of the Wainwright-Hsu vector field, the orbits shadowing such heteroclinic chains will also be disjoint from S .

Let us define the regular part of the quotient phase space by

B reg def = B \ π (S )
It will be convenient to work on a smaller part of the quotient phase space, so we define

B 0 def = (N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) ∈ B | ∀i ≠ j, Σ i ≠ Σ j (2.10a) B + 0 def = (N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) ∈ B + | ∀i ≠ j, Σ i ≠ Σ j (2.10b) B 0 def = π (B 0 ) (2.10c) B + 0 def = π B + 0 (2.10d) Observe that B 0 is an open subset of B reg . Let B (123) def = B ∩ {Σ 1 < Σ 2 < Σ 3 } Proposition 2.6. The projection map π restricted to B (123) is a C ∞ -diffeomorphism from B (123) to B 0 . In particular, B (123) is a fundamental domain of π ∶ B 0 → B 0
Proof. π restricted to B (123) is injective and π is a local C ∞ -diffeomorphism everywhere on B (123) by definition of the quotient manifold structure so the result follows immediately.

Orbits under the action of S 3 on the Kasner circle. Let p ∈ K . If p is one of the three exceptional points T 1 , T 2 , T 3 (resp. Q 1 , Q 2 , Q 3 ), then the orbit of p under the action of S 3 is {T 1 , T 2 , T 3 } (resp. {Q 1 , Q 2 , Q 3 }). On the other hand, if p is not one of the above points, then the orbit of p under the action of S 3 contains six points, one in each sixth of the Kasner circle K (ijk) .

General orbits under the action of S 3 . Let p ∈ B. Similarly to the previous case, if p ∈ S , then its orbit under the action of S 3 contains three points. On the other hand, if p ∉ S , then its orbit under the action of S 3 contains six points.

Stratification of the quotient phase space. The stratification of the phase space B induces a stratification of the quotient phase space B (see table 2).

Bianchi type

Name of the stratum Dimension of the stratum

I K 1 II B II 2 VI 0 B VI 0 3 VII 0 B VII 0 3 VIII B VIII 4 IX B IX 4
Table 2: Stratification of the quotient phase space.

Induced Kasner segment

According to what precedes, the projection in B of the Kasner circle K

K def = K /S 3
is in fact a topological segment (hence we will speak of the Kasner segment K). The end points of this segment, denoted by T and Q, are respectively the projection of the Taub points and the projection of the points Q i in the quotient phase space B. Let ). They are defined as following: let x ∈ B 0 and choose a point y = (N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) ∈ B 0 in the fiber over x. Let {i, j, k} = {1, 2, 3} such that Σ i < Σ j < Σ k , then we define

K 0 def = K \ {T,
N u def = N i , N s 1 def = N j , N s 2 def = N k Σ u def = Σ i , Σ s 1 def = Σ j , Σ s 2 def = Σ k
This definition does not depend on the choice of y in the fiber of x, hence N u , N s 1 , N s 2 , Σ u , Σ s 1 , Σ s 2 are well defined on B 0 . One cannot extend them by continuity on B. In particular, beware of the fact that induced type II orbits in B are not contained in B 0 . This implies that N u , N s 1 , N s 2 , Σ u , Σ s 1 , Σ s 2 are not continuous functions along type II orbits in the quotient phase space (one cannot extend them by continuity when the orbit crosses B \ B 0 ).

Note that the map

x ↦ N u (x), N s 1 (x), N s 2 (x), Σ u (x), Σ s 1 (x), Σ s 2 (x)
is a diffeomorphism from B 0 to B 0 where

B 0 def = N u , N s 1 , N s 2 , Σ u , Σ s 1 , Σ s 2 ∈ R 6 | Σ u + Σ s 1 + Σ s 2 = 0, 6 -Σ 2 u + Σ 2 s 1 + Σ 2 s 2 - 1 2 N 2 u + N 2 s 1 + N 2 s 2 + N u N s 1 + N s 1 N s 2 + N s 2 N u = 0, Σ u < Σ s 1 < Σ s 2 Induced Wainwright-
F(T ) = F(Q) = T .

Quotient Mixmaster attractor. Let us denote by

A def = A /S 3 , A + def = A + /S 3
the quotient Mixmaster attractor and its "positive" part.

Induced type II orbits. One can remark that type II orbits in A + which do not arrive at some Taub point do not cross the singular set S . Hence they induce orbits of X in A + . We will use the following notations, where p ∈ K 0 and q is a lift of p:

O p→F (p) def = π O q→F (q) O * p def = π O * q
for * ∈ {u, s 1 , s 2 }. In the positive part of the quotient Mixmaster attractor, type II orbits look like a "loop" (see figure 11).

• T • Q K Figure 11:
Half of the quotient Mixmaster attractor and some type II orbits.

Induced heteroclinic chains.

Definition 2.7 (Induced heteroclinic chain). Let p ∈ B + \ {T } and q ∈ B + be a lift of p. The heteroclinic chain H (p) starting at p is the projection of H (q) by π.

For example, if p is a point of the Kasner interval K 0 such that, for every n ∈ N, F n (p) is not a Taub point, then

H (p) def = O p→F (p) , O F (p)→F 2 (p) , O F 2 (p)→F 3 (p) , . . .
(2.12)

Kasner parameter and Kasner map

The Kasner parameter. The main tool to study the dynamics of the Kasner map is the Kasner parameter. The Kasner parameter is a bijection K → [1, +∞] which conjugates the (induced) Kasner map F with the Gauss transformation on continued fractions (defined precisely in the next paragraph).

Definition 2.8 (Kasner parameter). For every p ∈ K, the Kasner parameter associated with p is defined by

ω(p) = µ s 2 (p) µ s 1 (p) ∈ ]1, +∞[ if p ≠ T , p ≠ Q (2.13) ω(Q) = 1 ω(T ) = +∞ This formula defines a bijection ω ∶ K → [1, +∞].
Let us also denote by µ * ( * ∈ {u, s 1 , s 2 }) the eigenvalue µ * as a function of the Kasner parameter so that, for every p ∈ K, µ * (ω(p)) ∶= µ * (p). Formally this is an abuse of notations, but it will not give rise to confusion. A simple computation shows that, for every ω ∈ [1, +∞],

µ u (ω) = 6ω 1 + ω + ω 2
(2.14a)

µ s 1 (ω) = 6(1 + ω) 1 + ω + ω 2 (2.14b) µ s 2 (ω) = 6ω(1 + ω) 1 + ω + ω 2 (2.14c)
We refer to [START_REF] Heinzle | Mixmaster: Fact and Belief[END_REF] for more details.

Conjugacy between the Kasner map and the Gauss transformation. As stated earlier, the Kasner parameter

ω ∶ K → [1, +∞] conjugates the Kasner map F ∶ K → K to the Gauss map f ∶ [1, +∞] → [1, +∞] defined by f (ω) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ω -1 if ω ≥ 2 1 ω-1 if 1 < ω ≤ 2 +∞ if ω = 1 or ω = +∞ (2.15)
We refer to [START_REF] Heinzle | Mixmaster: Fact and Belief[END_REF] for more details. This conjugacy can be represented by the commutative diagram

K K [1, +∞] [1, +∞] ω F ω f
We will also call f the Kasner map.

Remark 2.9. Every point q ∈ K has two pre-images under the Kasner map f . These two pre-images are the starting points of the two type II orbits O s 1 q and O s 2 q arriving at q. This allows to distinguish the two pre-images: let us denote them by p 1 and p 2 where O

s 1 q = O p 1 →q = O u p 1 and O s 2 q = O p 2 →q = O u p 2 .
Using the geometric description of the Kasner map F , one can check that ω(p 1 ) ≥ 2 and ω(p 2 ) ≤ 2. Reversing the viewpoint, one gets that :

for every point p ∈ f, if ω(p) > 2 then O u p = O s 1 F (p) , if ω(p) < 2 then O u p = O s 2 F (p) .

The era Kasner map. Let us define the era Kasner map

f ∶ ]1, 2[ → [1, 2[ by the formula f (ω) = f r(ω) (ω) (2.16)
where r(ω) = ⌊ 1 ω-1 ⌋ (here, ⌊.⌋ is the floor function).

Interpretation of the dynamics of the Kasner map in terms of continued fractions. Let

ω 0 ∈ ]1, +∞[. Let [k 0 ; k 1 , k 2 , k 3 , .
. . ] be the continued fraction expansion associated with ω 0 , that is, the only (finite or infinite) sequence of integers such that

ω 0 = k 0 + 1 k 1 + 1 k 2 +⋯ def = [k 0 ; k 1 , k 2 , k 3 , . . . ]
In terms of continued fractions, we have

f ([k 0 ; k 1 , k 2 , k 3 , . . . ]) = [k 0 -1; k 1 , k 2 , k 3 , . . . ] if k 0 ≥ 2 [k 1 ; k 2 , k 3 , . . . ] if k 0 = 1 and if 1 < ω 0 < 2 (i.e. k 0 = 1), we have f ([1; k 1 , k 2 , k 3 , . . . ]) = [1; k 2 , k 3 , . . . ]
In other words, the era Kasner map f is a left-shift on the continued fractions.

In what follows, we assume that ω 0 ∉ Q, so the continued fraction expansion associated with ω 0 is infinite.

Let (ω n ) n≥0 ∈ [1, +∞[
N be the sequence generated by the Kasner map f from ω 0 , i.e. ω n+1 = f (ω n ) for every n ≥ 0. Every term of this sequence is called an epoch. It is quite natural to consider the subsequence (ω n ) n≥1 defined by

ωn def = ω k 0 +k 1 +⋅⋅⋅+k n-1 -1 = [1; k n , k n+1 , . . . ]
This subsequence divides the sequence (ω n ) n≥0 in eras of the form:

f (ω n ) =[k n ; k n+1 , k n+2 , . . . ], [k n -1; k n+1 , k n+2 , . . . ],
. . . ,

[1; k n+1 , k n+2 , . . . ] = ωn+1
On each era, (ω n ) n≥0 is decreasing. Moreover, one can remark that f (ω n ) = ωn+1 for any n ≥ 1.

Local expression of the Wainwright-Hsu vector field near the Kasner circle

In Section 5, we will study the dynamics of the Wainwright-Hsu vector field X in a neighbourhood of a point of the Kasner interval K 0 . The aim of the present section is to describe a "nice" system of local coordinates ξ in a neighbourhood of K 0 and to write a workable local form of X in these "nice" coordinates. The key property of these coordinates is the fact that they straighten the stable and the unstable manifolds of the points belonging to K 0 for X . We now proceed to define those stable and unstable manifolds.

For any ω ∈ ]1, +∞[, let us denote by P ω the unique point of K 0 whose Kasner parameter is ω. Recall that for every ω ∈ ]1, +∞[, there exist:

• one type II orbit, denoted by O u ω , which converges to the point P ω as time goes to -∞. This orbit is asymptotically tangent to the direction ∂ N u ;

• two type II orbits, denoted by O Let us denote by

W u (P ω , X ) def = {P ω } ∪ O u ω
the union of the point P ω with the type II orbit which converges to the point P ω as time goes to -∞. This is a 1-dimensional smooth11 embedded submanifold of B + tangent to the direction ∂ N u at the point P ω . The notation W u (P ω , X ) comes from the fact that it is the unstable manifold of the point P ω for the vector field X . Indeed, it follows from the stable manifold theorem that the unstable manifold of the point P ω for the vector field X is 1-dimensional. Moreover, from what precedes, we get that W u (P ω , X ) is included in the unstable manifold of the point P ω for the vector field X . By dimension, this inclusion must be an equality. In other words,

W u (P ω , X ) = x ∈ B + | X t (x) ----→ t→-∞ P ω
Analogously, let

W s 1 (P ω , X ) def = {P ω } ∪ O s 1 ω W s 2 (P ω , X ) def = {P ω } ∪ O s 2 ω W s 1 (P ω , X ) (resp. W s 2 (P ω , X )) is a 1-dimensional smooth embedded submanifold of B + tangent to the direction ∂ N s1 (resp. ∂ N s2
) at the point P ω , called the "weak stable manifold" (resp. the "strong stable manifold"). Note that W s 1 (P ω , X ) cannot be characterized as a stable manifold. Indeed, W s 1 (P ω , X ) and W s 2 (P ω , X ) are both included in the stable manifold of the point P ω for the vector field X :

W s (P ω , X ) def = x ∈ B + | X t (x) ----→ t→+∞ P ω
It follows from the stable manifold theorem that the stable manifold of the point P ω for the vector field X is a 2-dimensional smooth embedded submanifold of B + . The submanifolds W u (P ω , X ) ω∈]1,+∞[ foliate the 2-dimensional submanifold

W u (K 0 , X ) def = ⨆ ω∈]1,+∞[ W u (P ω , X ) = B + 0 ∩ N s 1 = N s 2 = 0, N u ≥ 0 = x ∈ B + 0 | ∃p ∈ K 0 , X t (x) ----→ t→-∞ p
Analogously, the submanifolds W s (P ω , X ) ω∈]1,+∞[ foliate the 3-dimensional submanifold

W s (K 0 , X ) def = ⨆ ω∈]1,+∞[ W s (P ω , X ) = B + 0 ∩ N u = 0, N s 1 ≥ 0, N s 2 ≥ 0 = x ∈ B + 0 | ∃p ∈ K 0 , X t (x) ----→ t→+∞ p
In order to prove the last equality above, one just needs to notice that:

• B + 0 ∩ N u = 0, N s 1 ≥ 0, N s 2 ≥ 0 is clearly a X -invariant 3-dimensional submanifold of B + ; • W s (K 0 , X ) is also a X -invariant 3-dimensional submanifold of B + (this is consequence of K 0 being a 1-dimensional submanifold which is normally hyperbolic for X , see e.g. [HPS77]); • W s (K 0 , X ) contains B + 0 ∩ N u = 0, N s 1 ≥ 0, N s 2 ≥ 0 (see e.g. [Rin01, Proposition 10.2] ; also notice that the fact that W s (K 0 , X ) contains the intersection B + 0 ∩ N u = 0, N s 1 ≥ 0, N s 2 ≥ 0 with a neighbourhood of K 0 is a direct consequence of
the expression of X and the sign of the eigenvalues).

This three facts immediately imply the equality

B + 0 ∩ N u = 0, N s 1 ≥ 0, N s 2 ≥ 0 = W s (K 0 , X ).
B ω,C,n def = (x u , x s 1 , x s 2 , x c ) ∈ R + 3 × ]1, +∞[ | max(x u , x s 1 , x s 2 , |x c -ω|) ≤ 1 Cω n
the ball of center (0, 0, 0, ω) and radius 1

Cω n in R

+ 3 × ]1, +∞[ (for the sup-norm).
We now proceed to give formal statements of the main results of this section. We delay the proofs until the following sections.

Proposition 3.2 (System of local coordinates). There exist two constants

C > 0 and n ∈ N, an open neighbourhood U ξ of K 0 in B + 0 (see (2.10c)), an open neighbourhood U ξ of {0 R 3 } × ]1, +∞[ in R + 3 × ]1, +∞[ and a smooth system of local coordinates ξ = (x u , x s 1 , x s 2 , x c ) ∶ U ξ → U ξ
with the following properties:

1. We have (x u , x s 1 , x s 2 ) = (N u , N s 1 , N s 2 ) (3.1a) and x c = ω in restriction to K 0 (3.1b)
In particular, ξ maps the Kasner interval

K 0 to {x u = x s 1 = x s 2 = 0} ∩ U ξ : ξ (K 0 ) = {0 R 3 } × ]1, +∞[ = {x u = x s 1 = x s 2 = 0} ∩ U ξ (3.2)
2. The chart ξ straightens the stable and unstable manifold foliations along the Kasner interval K 0 . More precisely, for any ω ∈ ]1, +∞[, we have

ξ W u loc (P ω , X ) ∩ U ξ = {x s 1 = x s 2 = 0, x c = ω} ∩ U ξ (3.3a) ξ W s loc (P ω , X ) ∩ U ξ = {x u = 0, x c = ω} ∩ U ξ (3.3b) ξ W s 1 loc (P ω , X ) ∩ U ξ = {x u = x s 2 = 0, x c = ω} ∩ U ξ (3.3c) ξ W s 2 loc (P ω , X ) ∩ U ξ = {x u = x s 1 = 0, x c = ω} ∩ U ξ (3.3d)
3. The open sets U ξ and U ξ are "big enough": for any

ω ∈ ]1, +∞[, B ω,C,n ⊂ U ξ and B ω,C,n ⊂ U ξ (3.4)
4. The C 6 -norm of ξ restricted to a neighbourhood of P ω admits an upper bound which is polynomial in ω, and similarly for ξ -1 . More precisely, for any ω ∈ ]1, +∞[,

∥ξ∥ C 6 ≤ Cω n in restriction to B ω,C,n (3.5a) ξ -1 C 6 ≤ Cω n in restriction to B ω,C,n (3.5b)
From now on the system of local coordinates ξ given by Proposition 3.2 is fixed. We will use roman letters for objects viewed in the system of local coordinates ξ. For example, we will denote by X the vector field ξ * X . The Wainwright-Hsu vector field X has a "nice" expression in the local coordinates ξ:

• The fact that ξ straightens the stable and the unstable manifolds of X implies that a lot of non linear terms vanish in the development of X.

• The estimates on the C 6 norm of ξ and ξ -1 allow one to get analogous estimates on the C 3 norm of the non linear terms appearing in the development of X. These estimates will eventually lead to a C

1 control of the non linear terms appearing in the development of the renormalized vector field X ω (see Proposition 3.8).

Proposition 3.3 (Local expression of X).

There exist two constants C > 0 and n ∈ N such that the vector field X admits the following expression on the open set

⋃ ω∈]1,+∞[ B ω,C,n ⊂ U ξ : X(x) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ µ u (x c ) 0 0 0 0 -µ s 1 (x c ) 0 0 0 0 -µ s 2 (x c ) 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ x u x s 1 x s 2 x c ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ Xu,u u (x)x 2 u + Xu,s 1 u (x)x u x s 1 + Xu,s 2 u (x)x u x s 2 Xu,s 1 s 1 (x)x u x s 1 + Xs 1 ,s 1 s 1 (x)x 2 s 1 + Xs 1 ,s 2 s 1 (x)x s 1 x s 2 Xu,s 2 s 2 (x)x u x s 2 + Xs 1 ,s 2 s 2 (x)x s 1 x s 2 + Xs 2 ,s 2 s 2 (x)x 2 s 2 Xu,s 1 c (x)x u x s 1 + Xu,s 2 c (x)x u x s 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (3.6)
where, for every ω ∈ ]1, +∞[, the functions X * , * * (where * ∈ {u, s 1 , s 2 , c} and different occurrences of * are independent) appearing in the non linear part of (3.6) satisfy

∥ X * , * * ∥ C 3 ≤ Cω n on B ω,C,n (3.7) Remark 3.4. µ u (x c ), -µ s 1 (x c
) and -µ s 2 (x c ) defined in (2.14) are the nonzero eigenvalues of the derivative DX(0, 0, 0, x c ).

To further simplify the computations, we will renormalize the local vector field X (by multiplying it by a positive function γ ω ) in order to linearize the dynamics in the unstable direction. This trick will allow us to compute explicit travel time between two local sections.

Let ω ∈ ]1, +∞[. We define the renormalization function γ ω in the neighbourhood of (0, 0, 0, ω) by the formula

γ ω (x) = µ u (ω) µ u (x c ) + Xu,u u (x)x u + Xu,s 1 u (x)x s 1 + Xu,s 2 u (x)x s 2 (3.8)
The renormalization function γ ω is chosen so that, according to (3.6), the coordinate of (γ ω .X) in the direction ∂ x u is µ u (ω)x u . In other words, (γ ω .X) is "linear" in the direction ∂ x u .

Lemma 3.5 (Domain of γ ω ).

There exist two constants C > 0 and n ∈ N such that for every ω ∈ ]1, +∞[, for every x ∈ B ω,C,n , we have

µ u (x c ) + Xu,u u (x)x u + Xu,s 1 u (x)x s 1 + Xu,s 2 u (x)x s 2 > 0 (3.9)
In particular, γ ω is well defined and positive on B ω,C,n .

Definition 3.6. We define the local vector field

X ω def = γ ω .X (3.10)
on B ω,C,n for C, n large enough so that, for every ω ∈ ]1, +∞[, the conclusion of Lemma 3.5 is satisfied.

Remark 3.7. In restriction to B ω,C,n , the orbits of X ω are the same than the one of X, up to a time reparametrization.

Proposition 3.8 (Local expression of X ω

). There exist two constants C > 0 and n ∈ N such that for every ω ∈ ]1, +∞[, the local vector field X ω admits the following expression on the open ball B ω,C,n :

X ω (x) = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ µ u (ω) 0 0 0 0 -μ ω,s 1 (x c ) 0 0 0 0 -μ ω,s 2 (x c ) 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ x u x s 1 x s 2 x c ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 X u,s 1 ω,s 1 (x)x u x s 1 + X s 1 ,s 1 ω,s 1 (x)x 2 s 1 + X s 2 ,s 1 ω,s 1 (x)x s 2 x s 1 X u,s 2 ω,s 2 (x)x u x s 2 + X s 1 ,s 2 ω,s 2 (x)x s 1 x s 2 + X s 2 ,s 2 ω,s 2 (x)x 2 s 2 X u,s 1 ω,c (x)x u x s 1 + X u,s 2 ω,c (x)x u x s 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (3.11) where μω,s i (x c ) def = µ u (ω) µ u (x c ) µ s i (x c ) (3.12)
Moreover, the functions X * , * ω, * (where * ∈ {u, s 1 , s 2 , c} and different occurrences of * are independent) appearing in the non linear part of (3.11) satisfy

∥X * , * ω, * ∥ C 1 ≤ Cω n on B ω,C,n (3.13) Remark 3.9. µ u (ω), -μ ω,s 1 (x c ), -μ ω,s 2 (x c
) are the nonzero eigenvalues of DX ω (0, 0, 0, x c ).

A straightening theorem for a stable manifold foliation

In this subsection, we present a general result on vector fields, Theorem 3.10 and its addendum, that will be used to construct the system of local coordinates ξ given by Proposition 3.2. This result is a reformulation and a simplification of Theorem 1.4 and its addendum from the article [START_REF] Dutilleul | Some Estimates for the Stable Manifold Theorem[END_REF] in our current context. We refer to this article for an independent and complete proof of these general theorems.

The context is as follows. Let Ω be an open set of R n , G a linear subspace of R n and Y ∶ Ω → R n be a smooth vector field vanishing on Ω 0 ∶= Ω ∩ G ≠ ∅. Assume that there exists a complement F of G such that for every ω ∈ Ω 0 , F is stabilized and contracted by DY (ω) (Ω 0 is said to be "normally contracted"). Recall that we denote by W s (ω, Y ) the stable set of ω for Y , that is, the union of all the orbits of Y which converge to the point ω as time goes to +∞. According to the standard stable manifold theorem (see e.g.

[KH97], [Irw01], [Rue89], [Rob99], [BS02] and [HPS77]), W s (ω, Y
) is a smooth embedded submanifold passing through ω and the family of stable manifolds (W s (ω, Y )) ω∈Ω 0 is a smooth foliation of a small neighbourhood Ω of Ω 0 . Moreover, the stable foliation (W s (ω, Y )) ω∈Ω 0 can be straightened using smooth local charts.

The standard result explained above can be stated as follows:

Theorem 3.10 (Straightening of a stable foliation). Let Ω be an open set of R n , G be a linear subspace of R n and Y ∶ Ω → R n be a smooth vector field such that 1. Y vanishes on Ω 0 def = Ω ∩ G ≠ ∅;
2. There exists a complement F of G such that for every ω ∈ Ω 0 , the derivative

DY (ω) ∶ T ω R n ≃ R n → T ω R n ≃ R n stabilizes F and µ max (DY (ω)) |F < 0
where µ max (DY (ω)) |F denotes the maximum of the real parts of the eigenvalues of (DY (ω)) |F .

Let ω 0 ∈ Ω 0 . Then there exists a smooth local coordinate system ξ defined on a ball B ∶= B R n (ω 0 , R) such that the family of stable manifolds (W s (ω, Y )) ω∈Ω 0 ∩B foliates B and is straightened by ξ: for

every ω ∈ Ω 0 ∩ B, ξ W s (ω, Y ) ∩ B = (ω + F ) ∩ ξ (B)
We emphasize the fact that Theorem 3.10 is a straightforward consequence of the stable manifold theorem. The point of the article [START_REF] Dutilleul | Some Estimates for the Stable Manifold Theorem[END_REF] is to prove the following addendum, which provides some explicit estimates on the radius R and on the derivatives of all orders of ξ and ξ -1 : Addendum 3.11. For every r > 0 such that B R n (ω 0 , r) ⊂ Ω, one can find a radius R and a local coordinate system ξ on B R n (ω 0 , R) as above satisfying the following properties:

1. The radius R admits a lower bound which is

• linear in r,
• polynomial in the spectral gap µ max (DY (ω 0 )) |F ,

• inversely linear in the norm of the second derivative of Y on the closed ball B R n (ω 0 , r),

• inversely polynomial in the norm of DY (ω 0 ), -the angle between the generalized eigenspaces of DY (ω 0 ).

This lower bound depends only on the parameters cited above.

For every

> 0, ξ restricted to B R n (ω 0 , R) is -close to the identity in C 1 -norm.
3. The norms of the k-th derivatives of ξ and ξ -1 admit an upper bound which is

• polynomial in -the norm of DY (ω 0 ),
the angle between the generalized eigenspaces of DY (ω 0 ),

-the norms of the (k + 1) first derivatives of Y on the closed ball B R n (ω 0 , r) • inversely polynomial in -the spectral gap µ max (DY (ω 0 )) |F -r
This upper bound depends only on the parameters cited previously.

Moreover, identifying R n = F ⊕ G with F × G the local coordinate system ξ has the following form:

ξ(x, y) = (x, y + ξ(x, y))
where ξ(0, y) ≡ 0.

Finally, the different charts are compatible in the following sense: for any two charts ξ and ξ

′ defined respectively on B and B ′ , we have ξ = ξ ′ in restriction to B ∩ B ′ .
Remark 3.12. In order to get such estimates on R and ξ, one must choose a compact ball B(ω 0 , r) ⊂ Ω on which one controls the derivatives of all orders of Y .

System of local coordinates ξ

The existence of the system of local coordinates ξ which straightens the stable and the unstable foliations of X (see Proposition 3.2) is a consequence of Theorem 3.10 and Addendum 3.11. The proof will be divided into several steps. We first construct a chart which straightens K 0 , W s (K 0 , X ) and W u (K 0 , X ). This is done by using the Kasner parameter and the radial projection on the Kasner circle. We then apply Theorem 3.10 and Addendum 3.11 twice: in W s (K 0 , X ) to straighten the foliation

W s (P ω , X ) ω∈]1,+∞[ and in W u (K 0 , X ) to straighten the foliation W u (P ω , X ) ω∈]1,+∞[ . Finally, we
merge the two families of charts (of lower dimension since we restricted ourselves to submanifolds) obtained above into a unique chart straightening both the stable foliation and the unstable foliation.

Proof of Proposition 3.2. It will be convenient for this proof to work in B 0 instead of B + 0 . The reason is technical: the Kasner circle is in the boundary of B + 0 and Theorem 3.10 and Addendum 3.11 apply to a vector field defined on an open set of a vector space. Taking a chart of B + 0 in the neighbourhood of the Kasner circle, we cannot apply Theorem 3.10 and Addendum 3.11 to the push forward of X by this chart.

Step 1: Straightening of K 0 , W s (K 0 , X ) and W u (K 0 , X ). Recall from section 2.6 that the map

ξ 0 ∶ y ↦ N u (y), N s 1 (y), N s 2 (y), Σ u (y), Σ s 1 (y), Σ s 2 (y) is a diffeomorphism from B 0 (see (2.10c)) to B 0 where B 0 def = (N u , N s 1 , N s 2 , Σ u , Σ s 1 , Σ s 2 ) ∈ R 6 | Σ u + Σ s 1 + Σ s 2 = 0, 6 -Σ 2 u + Σ 2 s 1 + Σ 2 s 2 - 1 2 N 2 u + N 2 s 1 + N 2 s 2 + N u N s 1 + N s 1 N s 2 + N s 2 N u = 0, Σ u < Σ s 1 < Σ s 2
Let us identify the Kasner interval K 0 with the set

(Σ u , Σ s 1 , Σ s 2 ) ∈ R 3 | Σ u + Σ s 1 + Σ s 2 = 0, Σ 2 u + Σ 2 s 1 + Σ 2 s 2 = 6, Σ u < Σ s 1 < Σ s 2
The idea is to "straighten" B 0 into a subset of the product R 3 × K 0 . To do this, we use the radial projection from the sixth of the

(Σ u , Σ s 1 , Σ s 2 )-plane (Σ u , Σ s 1 , Σ s 2 ) ∈ R 3 | Σ u + Σ s 1 + Σ s 2 = 0, Σ u < Σ s 1 < Σ s 2
on the Kasner interval K 0 . In other words, we consider the chart

ξ 1 ∶ B 0 → ξ 1 (B 0 ) ⊂ R 3 × K 0 (N u , N s 1 , N s 2 , Σ u , Σ s 1 , Σ s 2 ) ↦ N u , N s 1 , N s 2 , Σ u q 2 , Σ s1 q 2 , Σ s2 q 2
where q is the deceleration parameter (see (1.1.3)) and more explicitly

q 2 = 1 6 Σ 2 u + Σ 2 s 1 + Σ 2 s 2
Note that ξ 1 is well defined because q ≠ 0 on B 0 . Moreover, the equality

q 2 = 1 - 1 12 N 2 u + N 2 s 1 + N 2 s 2 + 1 6 N u N s 1 + N s 1 N s 2 + N s 2 N u
which holds true on B 0 shows that q 2 is entirely determined by N u , N s 1 and N s 2 . It follows that ξ 1 is invertible and its inverse is

ξ -1 1 ∶ ξ 1 (B 0 ) → B 0 (N u , N s 1 , N s 2 , Σ u , Σ s 1 , Σ s 2 ) ↦ N u , N s 1 , N s 2 , q 2 Σ u , q 2 Σ s 1 , q 2 Σ s 2
Now recall that the Kasner parameter (defined in (2.13)) is a diffeomorphism from K 0 to ]1, +∞[. The composition of the Kasner parameter with the charts ξ 1 and ξ 0 leads to a smooth chart

ξ 2 = (N u , N s 1 , N s 2 , ω)∶ B 0 → ξ 2 (B 0 ) ⊂ R 3 × ]1, +∞[ y ↦ N u (y), N s 1 (y), N s 2 (y), ω(π 2 • ξ 1 • ξ 0 (y)) (3.14)
where π 2 is the projection

R 3 × K 0 → K 0 . The chart ξ 2 straightens K 0 , W s (K 0 , X ) and W u (K 0 , X ). Remark that ξ 2 (B 0 ) contains the open set B R 3 (0, 1/2) × ]1, +∞[. Moreover, by a straightforward computation, there exist C 2 > 0 and n 2 ∈ N such that for every ω ∈ ]1, +∞[, we have ∥ξ 2 ∥ C 6 ≤ C 2 ω n 2 in restriction to B ω,C 2 ,n 2 (3.15a) ξ -1 2 C 6 ≤ C 2 ω n 2 in restriction to B ω,C 2 ,n 2 (3.15b)
Step 2: Straightening of the stable foliation of (

ξ 2 ) * X . Let Y s be the restriction of (ξ 2 ) * X to Ω s ∶= ξ 2 (B 0 ) ∩ {N u = 0}, that is, the stable manifold of {0 R 3 } × ]1, +∞[ for (ξ 2 ) * X . Identifying R 4 ∩ {N u = 0} with R 3 endowed with the coordinates N s 1 , N s 2 and ω, Ω s is an open set of R 3 . Since Ω s ∩ {N s 1 = N s 2 = 0} is the image of K 0 by ξ 2 , Y s vanishes on Ω s ∩{N s 1 = N s 2 = 0}. Let F ∶= R∂ N s1 ⊕R∂ N s2 and G ∶= R∂ ω .
According to (2.5) and (3.14), for every ω ∈ ]1, +∞[, the decomposition F ⊕ G = R 3 is stabilized by DY s (0, 0, ω) and the eigenvalues -µ s 1 and -µ s 2 of DY s (0, 0, ω) |F are both (strictly) negative.

For

ω ∈ ]1, +∞[, let r(ω) def = min 1 4 , ω -1 2 Observe that B R 3 ((0, 0, ω), r(ω)) ⊂ Ω s .
According to Theorem 3.10 and Addendum 3.11, there exist two constants C s > 0 and n s ∈ N such that, for any ω

0 ∈ ]1, +∞[, there exist an open set V s ω 0 ⊂ Ω s and a smooth chart ξ s 3,ω 0 ∶ V s ω 0 → ξ s 3,ω 0 V s ω 0 ⊂ R 3 (N s 1 , N s 2 , ω) ↦ (N s 1 , N s 2 , ω + ξs 2,ω 0 (N s 1 , N s 2 , ω))
where ξs 3,ω 0 (0, 0, ω) ≡ 0, such that ξ

s 3,ω 0 straightens the stable foliation of Y s in V s ω 0 : ξ s 3,ω 0 W s ((0, 0, ω), Y s ) ∩ V s ω 0 = (N s 1 , N s 2 , ω) | ω = ω ∩ ξ s 3,ω 0 V s ω 0 (3.16) Moreover, V s ω 0 and ξ s 3,ω 0 V s ω 0 both contain the open set. B s ω 0 def = B R 2 0, R s ω 0 × ω 0 -min R s ω 0 , ω 0 -1 2 , ω 0 + min R s ω 0 , ω 0 -1 2 where R s ω 0 ∶= 1 C s ω n s 0 , and ∥ξ s 3,ω 0 ∥ C 6 , ξ s 3,ω 0 -1 C 6 ≤ C s ω n s 0 (3.17)
Remark that the particular form of ξ s 3,ω 0 assures that the invariant manifolds W s 1 (P ω , X ) and W s 2 (P ω , X ) are both straightened automatically by the "composition" of ξ s 3,ω 0 with ξ 2 :

ξ s 3,ω 0 ξ 2 W s 1 (P ω , X ) ∩ V s ω 0 = (N s 1 , N s 2 , ω) | N s 2 = 0, ω = ω ∩ ξ s 3,ω 0 V s ω 0 (3.18a) ξ s 3,ω 0 ξ 2 W s 2 (P ω , X ) ∩ V s ω 0 = (N s 1 , N s 2 , ω) | N s 1 = 0, ω = ω ∩ ξ s 3,ω 0 V s ω 0 (3.18b)
Step 3: Straightening of the unstable foliation of (ξ 2 ) * X . This step will be treated analogously to step 2. Let Y u be the restriction of (ξ

2 ) * X to Ω u ∶= ξ 2 (B 0 ) ∩ {N s 1 = N s 2 = 0}, that is, the unstable manifold of {0 R 3 } × ]1, +∞[ for (ξ 2 ) * X . Remark that the unstable foliation of Y u is exactly the stable foliation of -Y u . Identifying R 4 ∩ {N s 1 = N s 2 = 0} with R 2 endowed with the coordinates N u and ω, Ω u is an open set of R 2 . Since Ω u ∩ {N u = 0} is the image of K 0 by ξ 2 , Y u vanishes on Ω u ∩ {N u = 0}. Let F ∶= R∂ N u and G ∶= R∂ ω .
According to (2.5) and (3.14), for every ω ∈ ]1, +∞[, the decomposition

F ⊕ G = R 2 is stabilized by D -Y u (0, ω) and the eigenvalue µ u of D -Y u (0, ω) |F is (strictly) negative.
According to Theorem 3.10 and Addendum 3.11, there exist two constants C u > 0 and n u ∈ N such that, for any

ω 0 ∈ ]1, +∞[, there exist an open set V u ω 0 ⊂ Ω u and a smooth chart ξ u 3,ω 0 ∶ V u ω 0 → ξ u 3,ω 0 V u ω 0 ⊂ R 2 (N u , ω) ↦ (N u , ω + ξu 3,ω 0 (N u , ω)) where ξu 3,ω 0 (0, ω) ≡ 0, such that ξ u 3,ω 0 straightens the stable foliation of -Y u in V u ω 0 : ξ u 3,ω 0 W s ((0, ω), -Y u ) ∩ V u ω 0 = {(N u , ω) | ω = ω} ∩ ξ u 3,ω 0 V u ω 0 (3.19) Moreover, V u ω 0 and ξ u 3,ω 0 V u ω 0 both contain the open set B u ω 0 def = -R u ω 0 , R u ω 0 × ω 0 -min R u ω 0 , ω 0 -1 2 , ω 0 + min R u ω 0 , ω 0 -1 2 where R u ω 0 ∶= 1 C u ω n u 0 , and ∥ξ u 3,ω 0 ∥ C 6 , ξ u 3,ω 0 -1 C 6 ≤ C u ω n u 0 (3.20)
Since a stable manifold of -Y u is an unstable manifold of (ξ 2 ) * X , it follows that ξ u 3,ω 0 straightens the unstable foliation of (ξ 2 ) * X restricted to Ω u .

Step 4: Straightening of both the stable and the unstable foliation of (ξ 2 ) * X .

Let ω 0 ∈ ]1, +∞[. Let V ω 0 = (N u , N s 1 , N s 2 , ω) | (N s 1 , N s 2 , ω) ∈ V s ω 0 , (N u , ω) ∈ V u ω 0 and let ξ 3,ω 0 ∶ V ω 0 → R 4 (N u , N s 1 , N s 2 , ω) ↦ (N u , N s 1 , N s 2 , ω + ξs 3,ω 0 (N s 1 , N s 2 , ω) + ξu 3,ω 0 (N u , ω)) (3.21)
According to Addendum 3.11, the map ξs 3,ω 0 (resp. ξu

3,ω 0 ) restricted to B s ω 0 (resp. B u ω 0 ) where R s ω 0 (resp. R u ω 0 ) is replaced by R s ω 0 (resp. R u ω 0 ) is -close to 0 with respect to the C 1 -norm. It follows that there exist two constants C 3 ≥ max(C s , C u ) and n 3 ≥ max(n s , n u ) such that for every ω 0 ∈ ]1, +∞[, ξ 3,ω 0 is invertible on B ω 0 ∶= B R 3 0, R ω 0 × ω 0 -min R ω 0 , ω 0 -1 2 , ω 0 + min R ω 0 , ω 0 -1 2 where R ω 0 ∶= 1 C 3 ω n 3 0
From now on, we make the abuse of notation to consider that ξ 3,ω 0 is restricted to B ω 0 . Using (3.17) and (3.20), we get

∥ξ 3,ω 0 ∥ C 6 , ξ 3,ω 0 -1 C 6 ≤ C 3 ω n 3 0 (3.22)
By local uniqueness (see Addendum 3.11) of the charts ξ s 3,ω 0 and ξ u 3,ω 0 , the charts {ξ 3,ω 0 } ω 0 ∈]1,+∞[ glue together and induce a global chart ξ 3 on the neighbourhood

V ∶= ⋃ ω 0 ∈]1,+∞[ B ω 0 ⊂ R 3 × ]1, +∞[ (3.23) One can remark that V contains the open set B R 3 0, 1 C 3 2 n3 × ]1
, 2[: the size of V does not shrink when ω → 1, it only shrinks when ω → +∞.

According to (3.16) and (3.19), ξ 3 straightens the stable and the unstable foliations of (ξ 2 ) * X :

ξ 3 W s ((0, 0, 0, ω), (ξ 2 ) * X ) ∩ V = (N u , N s 1 , N s 2 , ω) | N u = 0, ω = ω ∩ ξ 3 (V ) (3.24a) ξ 3 W u ((0, 0, 0, ω), (ξ 2 ) * X ) ∩ V = (N u , N s 1 , N s 2 , ω) | N s 1 = N s 2 = 0, ω = ω ∩ ξ 3 (V ) (3.24b)
Step 5:Straightening of both the stable and the unstable foliation of X . Let us define

ξ = (x u , x s 1 , x s 2 , x c ) def = ξ 3 • ξ 2 The chart ξ is well defined on the open set U ξ ∶= ξ -1 2 (V ) ⊂ B 0 . Let U ξ ∶= ξ U ξ .
We now proceed to check all the properties of ξ announced in Proposition 3. Remark 3.13. In step 3, we used the same argument as in step 2. Nevertheless, one does not need Theorem 3.10 and Addendum 3.11 to straighten the unstable foliation of (ξ 2 ) * X . Indeed, the leaves of this foliation are all type II orbits explicitly known: all the computations could be done explicitly without the help of a general result.

Proofs of the main results on the local expression of the Wainwright-Hsu vector field

Fix C 0 > 0 and n 0 ∈ N such that Proposition 3.2 holds true with these constants. We begin this subsection with a proof of Proposition 3.3.

Proof of Proposition 3.3. Denote by Xu , Xs 1 , Xs 2 , Xc the coordinates of X and let

U X def = ⋃ ω∈]1,+∞[ B ω,C 0 ,n 0
According to Proposition 3.2, ξ is smooth. Since X is also smooth, it follows that X is smooth. Using the invariance of the set {N u = 0} by the flow of X and (3.1a), we get that the set {x u = 0} is invariant by the flow of X. Using the standard Hadamard's lemma in differential calculus, we get the existence of some smooth functions Xu,u u , Xu,s 1 u and Xu,s 2 u defined on the open set U X such that

Xu (x) = µ u (x c )x u + Xu,u u (x)x 2 u + Xu,s 1 u (x)x u x s 1 + Xu,s 2 u (x)x u x s 2
Analogously, the sets {x s 1 = 0} and {x s 2 = 0} are invariant so (3.6) holds true for the first three coordinates. For any x c ∈ ]1, +∞[, the stable manifold of (0, 0, 0, x c ) for X is invariant by the flow of X. Using (3.3b), it follows that Xc (0, x s 1 , x s 2 , x c ) ≡ 0 and we get the existence of a smooth function Xu c defined on

U X such that Xc (x) = x u Xu c (x)
The unstable manifold of (0, 0, 0, x c ) for X being also invariant, it follows by (3.3a) that there exist two smooth functions Xu,s

1 c and Xu,s 2 c defined on U X such that Xu c (x) = Xu,s 1 c (x)x s 1 + Xu,s 2 c (x)x s 2
We can conclude that (3.6) holds true on U X . The functions X * , * * depend on the second derivatives of X. A C 3 control of these functions involves a C 5 control of X and a C 6 control of ξ. A C 6 control of ξ is given by (3.5) while a C 5 control of X is trivial: there exists a constant C 1 > 0 such that ∥X ∥ C 5 ≤ C 1 on U X . The conjunction of these two controls implies that (3.7) holds true for some C > 0, n ∈ N large enough.

We now give a proof of Lemma 3.5.

Proof of Lemma 3.5. For every C > 0 and n ∈ N, let us denote by E C,n the set of all (ω, x)

such that ω ∈ ]1, +∞[ and x ∈ B ω,C,n ⊂ U ξ . Recall that B ω,C,n = (x u , x s 1 , x s 2 , x c ) ∈ R + 3 × ]1, +∞[ | max(x u , x s 1 , x s 2 , |x c -ω|) ≤ 1 Cω n
According to 2.14a, for every ω ∈ ]1, +∞[ and for every x ∈ U ξ such that x c ≤ 2ω, we have

µ u (x c ) ≥ 1 ω (3.25)
According to (3.7), there exist C 1 ≥ C 0 and n 1 ≥ n 0 such that for every

(ω, x) ∈ E C 1 ,n 1 , we have | Xu,u u (x)x u + Xu,s 1 u (x)x s 1 + Xu,s 2 u (x)x s 2 | ≤ C 1 ω n 1 max(x u , x s 1 , x s 2 ) (3.26)
Inequalities (3.25) and (3.26) imply that for every

(ω, x) ∈ E 2C 1 ,n 1 +1 , we have | Xu,u u (x)x u + Xu,s 1 u (x)x s 1 + Xu,s 2 u (x)x s 2 | < µ u (x c )
which concludes the proof.

Next lemma gives estimates on γ ω and its derivatives, which will be useful to obtain estimates on X ω later on. Lemma 3.14 (Control of γ ω and its derivatives). There exist two constants C > 0 and n ∈ N such that for every ω ∈ ]1, +∞[, for every x ∈ B ω,C,n , we have

1 2 ≤ γ ω (x) ≤ 3 2 (3.27)
and, for every

1 ≤ k ≤ 3, D k γ ω (x) ≤ Cω N (3.28)
Proof of Lemma 3.14. For every C > 0 and n ∈ N, let us denote by E C,n the set of all (ω, x)

such that ω ∈ ]1, +∞[ and x ∈ B ω,C,n ⊂ U ξ . Fix 3/4 < α < 1. Proof of (3.27). Let Xu (x) ∶= Xu,u u (x)x u + Xu,s 1 u (x)x s 1 + Xu,s 2 u (x)x s 2 .
According to Lemma 3.5, there exist C 0 > 0 and n 0 ∈ N such that for every C ≥ C 0 , every n ≥ n 0 and every (ω, x) ∈ E C,n , (3.9) holds true and it follows that

1 2 ≤ γ ω (x) ≤ 3 2 ⟺ µ u (x c ) + Xu (x) ≤ 2µ u (ω) ≤ 3(µ u (x c ) + Xu (x)) ⟺ Xu (x) ≤ 2µ u (ω) -µ u (x c ) 2µ u (ω) -3µ u (x c ) ≤ 3 Xu (x)
According to 2.14a and the mean value theorem, for every ω ∈ ]1, +∞[ and for every

x ∈ U ξ such that |x c -ω| ≤ 1 6ω , we have |µ u (x c ) -µ u (ω)| ≤ 1 ω and it follows that 2µ u (ω) -µ u (x c ) ≥ 1 ω (3.29)
Analogously, for every ω ∈ ]1, +∞[ and for every x ∈ U ξ such that |x c -ω| ≤ 1 24ω , we have

|µ u (x c ) -µ u (ω)| ≤ 1 4ω and it follows that 2µ u (ω) -3µ u (x c ) ≤ - 1 2ω (3.30)
According to (3.7), (3.29) and (3.30), there exist

C 1 ≥ C 0 and n 1 ≥ n 0 such that for every C ≥ C 1 , every n ≥ n 1 and every (ω, x) ∈ E C,n , we have Xu (x) ≤ 2µ u (ω) -µ u (x c ) and 2µ u (ω) -3µ u (x c ) ≤ 3 Xu (x) so (3.27) holds true.
Proof of (3.28). Recall that

γ ω (x) = µ u (ω) µ u (x c ) + Xu (x)
and

Dγ ω (x) = -γ ω (x) 2 µ u (ω) (Dµ u (x c ) + D Xu (x)) (3.31)
According to (2.14a) and (3.7), there exist C 2 ≥ C 1 and n 2 ≥ n 1 such that for every C ≥ C 2 , every n ≥ n 2 , every (ω, x) ∈ E C,n and every 1 ≤ k ≤ 3, we have

D k µ u (x c ) ≤ C 2 ω n 2 (3.32) and D k Xu (x) ≤ C 2 ω n 2 (3.33)
Using (3.27), (3.31), (3.32), (3.33) and the inequality µ u (ω) ≥ 2 ω , we get that there exist

C 3 ≥ C 2 and n 3 ≥ n 2 such that for every C ≥ C 3 , every n ≥ n 3 , every (ω, x) ∈ E C,n and every 1 ≤ k ≤ 3, we have D k γ ω (x) ≤ C 3 ω n 3
so (3.28) holds true.

We now have everything to prove the main result on X ω .

Proof of Proposition 3.8. The expression (3.11) follows from (3.8) and computations analogous to the ones presented in the proof of Proposition 3.3. The estimate (3.13) follows from (3.5), Lemma 3.14, a C 5 control of X on an arbitrary compact neighbourhood of K and analogous computations to the ones detailed for Proposition 3.3.

Local sections and transition maps

The purpose of this section is to define some local Poincaré sections for the Wainwright-Hsu vector field, together with some transitions maps describing how the orbits of the flow travel from one section to another. All these sections will be located in the vicinity of the Kasner circle and will be defined in the local coordinate system ξ constructed in the previous section. The transition maps between the sections will play a central role in our investigation of the dynamics of the Wainwright-Hsu vector field.

We will first recall some properties of the local coordinate system ξ (section 4.1). Then we will define a "global section" S h (section 4.2). The dynamics of the Wainwright-Hsu vector field is almost completely captured by the return map Φ of the orbits of this vector field on the global section S h . Therefore, understanding the dynamical properties of Φ will be our long-term goal. But, since this goal cannot be achieved directly, it is necessary to decompose Φ as a product of a large number of "local transitions maps". This will lead us to introduce some local sections (section 4.3), and some transitions maps describing how the orbits move from one local section to another (section 4.4). In order to define the global and local sections, we first need to recall a few facts conncerning the "nice" local coordinate system ξ = (x u , x s 1 , x s 2 , x c ) constructed in the previous section, and to introduce some pseudo-norms and projections related to this coordinate system. Recall that the local coordinate system ξ

Reminder conncerning the local

= (x u , x s 1 , x s 2 , x c ) is defined on a neighbourhood U ξ of the Kasner interval K 0 in the quotient phase space B + . The range of ξ, denoted by U ξ , is a neighbourhood of {0 R 3 } × ]1, +∞[ in R + 3 × ]1, +∞[. The local coordinate system ξ maps the Kasner interval K 0 to the interval {0 R 3 } × ]1, +∞[. More- over,
in restriction to K 0 , the coordinate x c is nothing but the Kasner parameter. In other words, the coordinates of the point ω , arriving at P ω . These orbits are mapped by ξ to the straight lines The Mixmaster attractor A is mapped by ξ to the set

P ω ∈ K 0 are (x u , x s 1 , x s 2 , x c ) = (0, 0, 0, ω). For ω ∈]1, +∞[,
O u ω def = {x u > 0, x s 1 = x s 2 = 0, x c = ω} (4.1a) O s 1 ω def = {x s 1 > 0, x u = x s 2 = 0, x c = ω} (4.1b) O s 2 ω def = {x s 2 > 0, x u = x s 1 = 0, x c = ω} (4.1c) More precisely, the connected component of O u ω ∩ U ξ starting 12 at P ω is mapped to O u ω ∩ U ξ ,
A def = x u = x s 1 = 0 ∪ x u = x s 2 = 0 ∪ x s 1 = x s 2 = 0 (4.2) that is, ξ(A ∩ U ξ ) = A ∩ U ξ .
Recall that our goal is to compare the behaviour of the type IX orbits winding around the Mixmaster attractor with the dynamics on the Mixmaster attractor itself. In view of that goal, it will be convenient to project the type IX orbits (or at least some of their points) on the Mixmaster attractor. Definition 4.1 (Projection on the Mixmaster attractor). Let us denote by ∆ the set of all x = (x u , x s 1 , x s 2 , x c ) such that two of the three coordinates x u , x s 1 , and x s 2 are equal and larger than the third one, that is, the set Remark 4.2. According to the equation of the Mixmaster attractor in local coordinates (4.2), one can see that Proj A (x) is the closest point to x (both for the Euclidean standard norm and the sup-norm) belonging to the Mixmaster attractor A. This is why we say that Proj A (x) is the projection of x on the Mixmaster attractor.

(x u , x s 1 , x s 2 , x c ) | x u = x s 1 ≥ x s 2 or x u = x s 2 ≥ x s 1 or x s 1 = x s 2 ≥ x u

We define a projection Proj

A ∶ R + 3 × ]1, +∞[ \ ∆ → A by the formula Proj A (x u , x s 1 , x s 2 , x c ) def = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ (x u , 0, 0, x c ) if x u > max(x s 1 , x s 2 ) (0, x s 1 , 0, x c ) if x s 1 > max(x u , x s 2 ) (0, 0, x s 2 , x c ) if x s 2 > max(x u , x s 1 ) (4.3) 12 Any connected component of O u ω ∩ U ξ is
For x = (x u , x s 1 , x s 2 , x c ) ∈ R 4 , we denote ∥x∥ ∞ = max |x u | , |x s 1 | , |x s 2 | , |x c |
It will be convenient to discriminate the direction ∂ x c from the other directions, for dynamical reasons. This leads us to introduce two pseudo-norms.

Definition 4.3 (Pseudo-norms). For any x = (x u , x s 1 , x s 2 , x c ) ∈ R 4 , we define ∥x∥ ⊥ def = max |x u | , |x s 1 | , |x s 2 | ∥x∥ / / def = |x c | Remark 4.4. For any x ∈ R 4 , ∥x∥ ∞ = max(∥x∥ ⊥ , ∥x∥ / / ).
Remark 4.5. If the projection Proj A (x) of x on the Mixmaster attractor is well defined (see definition 4.1), then ∥x -Proj A (x)∥ ⊥ = ∥x -Proj A (x)∥ ∞ is the distance between x and the Mixmaster attractor A.

The global section S h , the era return map Φh and the double era return map Φh

Definition 4.6 (Global section). For h > 0, we define the global section S h ∶= S

s 1 h ∪ S s 2 h where S s 1 h def = x = (x u , x s 1 , x s 2 , x c ) | x s 1 = h, 0 ≤ x u ≤ h, 0 ≤ x s 2 ≤ h, 1 < x c < 2 (4.4)
and analogously for S s 2

h . If h is small enough, the global section is included in the range U ξ of the local coordinate system ξ. In this case, we consider the geometric global section

S h def = ξ -1 (S h )
Suppose that h is small enough, so that the geometric global section S h is well-defined. On the one hand, for every ω ∈]1, 2[, the two type II orbits O

s 1 ω , O s 2
ω intersect the global section S h . On the other hand, formula (2.15) shows that, for every ω ∈]1, +∞[\Q, the forward orbit of ω under the Kasner map f passes infinitely many times in the interval ]1, 2[. It follows that every heteroclinic chain of type II orbits either converges to a Taub point, or crosses infinitely many times the global section S h . Hence all type IX orbits that possibly shadow a heteroclinic chain of type II orbits must cross infinitely many times the global section S h . This is the reason why we say that S h and S h are global sections.

The above discussion shows that our main Theorem 1.9 can be proved by investigating the dynamical properties of the return map of the orbits on the global section S h , called the era return map. We will now proceed to the formal definition of this map. This definition is not completely straightforward for two reasons:

• the global section S h was defined in the local coordinate system ξ but the orbit segments (or heteroclinic chains) travelling from S h to S h do not remain inside the open set where this local coordinate system is defined,

• we want to consider not only the returns of orbits, but also the return of heteroclinic chains.

Recall that, for every point q ∈ B + which is not a Taub point, we have defined a generalized heteroclinic chain H (q) starting at q (see definitions 2.5 and 2.7). In particular, H (q) is nothing but the forward orbit of q when q ∈ B + IX . Definition 4.7. For x = (x u , x s 1 , x s 2 , x c ) ∈ U ξ , we will denote by H (x) the heteroclinic chain starting at the point in B + of coordinates x.

Definition 4.8 (Era return map). Let h > 0 be small enough so that the global section S h is included in the range U ξ of the local coordinate system ξ. We define the era return map

Φh ∶ Domain( Φh ) ⊂ S h → S h
as follows. Let x ∈ S h . If the heteroclinic chain H (x) intersects the section S h , then Φh (x) is the 4-tuple of coordinates of the first intersection point of H (x) with S h . Otherwise, Φh is not defined at the point x.

If x u , x s 1 , x s 2 > 0, then the heteroclinic chain H (x) is nothing but the forward X -orbit of the point ξ -1 (x). As a consequence, in restriction to {x u , x s 1 , x s 2 > 0}, the era return map Φh is nothing but the first return map of the orbits of the Wainwright-Hsu vector field on the global section S h , expressed in local coordinates.

For technical reasons (namely, because we will discover that Φh fails to be uniformly expanding in the direction parallel to the Kasner interval), we will be led to replace Φh by its square. This motivates the following definition. Definition 4.9 (Double era transition map). Let h > 0 be small enough so that the global section S h is included in the range U ξ of the local coordinate system ξ. We define the double era return map Φh ∶ S h → S h by the formula Φh

def = Φh • Φh (4.5)
The goal of the remainder of the papeer is to find a subset of S h where the double era return map Φh is well-defined, to prove that Φh has nice hyperbolicity properties on this subset, to construct some local stable manifolds for the map Φh and finally to prove that the union of these local stable manifolds cover a subset of positive Lebesgue measure in S h . Our main Theorem 1.9 will follow easily. ω that are at distance h from the Kasner interval, that is, ω . These local sections will be crossed by the orbits traveling close to the heteroclinic chain passing through the point P ω . They will serve as gates controlling the entrance in (resp. the exit from) a neighbourhood of the point P ω . Definition 4.11 (Local sections). Let ω ∈ ]1, +∞[ and h = h, h ⊥ , h / / where h, h ⊥ and h / / are some positive numbers. We consider the local sections

The local sections S

P u ω,h def = (h, 0, 0, ω) , P s 1 ω,h def = (0, h, 0, ω) , P s 2 ω,h def = (0, 0, h, ω) If h is
S u ω,h def = x = (x u , x s 1 , x s 2 , x c ) | x u = h, ∥x -P u ω,h ∥ ⊥ ≤ h ⊥ , ∥x -P u ω,h ∥ / / ≤ h / / = x = (x u , x s 1 , x s 2 , x c ) | x u = h, 0 ≤ x s 1 ≤ h ⊥ , 0 ≤ x s 2 ≤ h ⊥ , ω -h / / ≤ x c ≤ ω + h / / and S s 1 ω,h def = x = (x u , x s 1 , x s 2 , x c ) | x s 1 = h, x -P s 1 ω,h ⊥ ≤ h ⊥ , x -P s 1 ω,h / / ≤ h / / = x = (x u , x s 1 , x s 2 , x c ) | x s 1 = h, 0 ≤ x u ≤ h ⊥ , 0 ≤ x s 2 ≤ h ⊥ , ω -h / / ≤ x c ≤ ω + h / /
The local section S s 2 ω,h is defined analogously, permuting the roles of s 1 and s 2 . See figure 12

S u ω,h def = ξ -1 S u ω,h ⊂ B + , S s i ω,h def = ξ -1 S s i ω,h ⊂ B + A A ξ(K 0 ) x s 1 x u x c h S s 1 ω,h s S u ω,h u h / / h ⊥ • x x -P s 1 ω,h / / x -P s 1 ω,h ⊥ • P s 1 ω,h • P ω • P u ω,h O s 1 ω O u ω
Figure

ω,h . • Assume that 0 < h ⊥ < min(h, h / / ). The (in)equalities max(x u , x s 2 ) = x -P s 1 ω,h ⊥ ≤ h ⊥ < h = x s 1
show that the projection Proj A (x) of x on the Mixmaster attractor is well defined, and

x -P s 1 ω,h ⊥ = ∥x -Proj A (x)∥ ⊥ . In other words, x -P s 1
ω,h ⊥ is the distance from x to the Mixmaster attractor. Hence, h ⊥ can be seen as the size of the section in the direction transversal to the Mixmaster attractor A.

• Assume again that 0 < h ⊥ < min(h, h / / ). The section S s 1 ω,h is a 3-dimensional "rectangle" in R + 3
× ]1, +∞[. Using the fact that the Kasner interval corresponds to x u = x s 1 = x s 2 = 0 and the preceding item, we see that h is the distance from the section S

Transition maps

We will now construct some transition maps between the local sections that were defined in section 4.3. These maps describe the behaviour of the orbits of the Wainwright-Hsu vector field in some specific regions (in the neighbourhood of a point of the Kasner circle, in the neighbourhood of a type II orbit, etc). Our goal is to decompose the era return map Φh as a product of elementary transition maps, that are easier to understand than Φh itself.

The era transition map Φω,h and the double era transition map Φω,h

The orbits of the Wainwright-Hsu vector field can follow very different routes between two intersections with the global section S h . For example, some orbits come back rather quickly to the global section S h , whereas some others will spend a very long time oscillating in the vicinity of the Taub point T before coming back to S h . For that reason, we cannot study the era return map Φh globally: we need to define some localized version of Φh (and Φh ). Definition 4.14 (Era transition map Φω,h and double era transition map Φω,h ). Let ω ∈ ]1, 2[ and h = (h, h ⊥ , h / / ) so that the sections S s ω,h and S h are included in the range U ξ of the local coordinate system ξ. We define the era transition map

Φω,h ∶ S s ω,h ∩ S h → S h
as the restriction of Φh to the section S s ω,h . See figure 13. Analogously, we define the double era transition map Φω,h ∶ S s ω,h ∩ S h → S h as the restriction of Φh to the section S s ω,h .

• Q • T K ]1,2[ S h O • P ω S s ω,h 0 • P f (ω) • P f 2 (ω) • P f 3 (ω) • P f k1-1 (ω) • P f (ω)
= [1; k 1 , k 2 , . . .]. The local sections S s ω,h 0 , S s f (ω),h 1 , . . . , S s f (ω),h k1 are represented in green, while the local sections S u ω,h ′ 0 , S u f (ω),h ′ 1 , . . . , S u f (ω),h ′ k1 are represented in blue.
The era transition map Φω,h 0 encodes the travel of the orbit O between the local section S s ω,h 0 and the global section S h (in yellow). One can decompose the era transition map into several epoch transition maps, encoding the travel of the orbit O between two consecutive green sections.

= Φω,h • Proj A Consider a point x ∈ S s ω,h ∩ S h ∩ B + IX .
On the one hand, Φh (x) is the first intersection point of the forward orbit of the point x with the section S h . On the other hand, ΦA ω,h (x) is the first intersection point of the heteroclinic chain H (Proj A (x)) with the section S h . Since the point Proj A (x) belongs to the Mixmaster attractor, H (Proj A (x)) is a heteroclinic chain of type II orbits. As a consequence, the comparison between the maps Φω,h (x) and ΦA ω,h (x) will allow us to understand whether the type IX orbits follow (or deviate from) the heteroclinic chains of type II orbits.

Remark 4.16. The return map ΦA ω,h admits an explicit expression. First recall that the era Kasner map f admits the folllowing explicit expression:

f ([1; k 1 , k 2 , . . . ]) = [1; k 2 , k 3 , . . . ]
(see (2.16) for more details). Now, since both f and ΦA ω,h encode the behaviour of heteroclinic chains of type II orbits, these two maps are naturally related. To be more precise, recall that the two type II orbits O

s 1 f (x c ) and O s 2 f (x c ) arriving at the point P f (x c ) cross the section S s ω,h respectively at (0, h, 0, f (x c ))
and (0, 0, h, f (x c )) (this is a direct consequence of the explicit expression of the type orbits II orbits in local corrdinates, see (4.1b) and (4.1c)). Now, recall that

f (x c ) = f k 1 (x c ) (x c ).
Hence, the k 1 (x c ) th type II orbit in the heteroclinic chain starting at

P x c is either O s 1 f (x c ) or O s 2 f (x c )
. More precisely, according to Remark 2.9, this k 1 (x c ) th orbit is

O s 1 f (x c ) if f k 1 (x c )-1 (x c ) > 2, O s 2 f (x c ) if f k 1 (x c )-1 (x c ) < 2.
Lastly, observe that

f k 1 (x c )-1 (x c ) > 2 if k 1 (x c ) ≥ 2, f k 1 (x c )-1 (x c ) < 2 if k 1 (x c ) = 1.
Putting everything together, we get the following explicit expression for the return map ΦA ω,h :

ΦA ω,h (x) = (0, h, 0, f (x c )) if k 1 (x c ) ≥ 2 (0, 0, h, f (x c )) if k 1 (x c ) = 1 where x c = [1; k 1 (x c ), k 2 (x c ), . . . ] (4.6) Analogously, ΦA ω,h (x) = (0, h, 0, f (x c )) if k 2 (x c ) ≥ 2 (0, 0, h, f (x c )) if k 2 (x c ) = 1 where x c = [1; k 1 (x c ), k 2 (x c ), . . . ] (4.7)
where f is the double era Kasner map defined by

f (ω) def = f • f (ω) S s ω,h • P ω S u ω,h ′′ O O P ω →P f (ω) S s f (ω),h ′ Υ ω,h,h ′′ Ψ ω,h ′′ ,h ′ Φ ω,h,h ′ • P f (ω)

If the parameters h, h

′ and h ′′ are well-chosen (roughly speaking, h ′ must be small enough, h ′′ much smaller than h ′ and h much smaller than h ′ ; explicit bounds will be given in the next sections), the orbit O will first cross the section S u ω,h ′′ and then cross the section S s f (ω),h ′ .

The epoch transition map Φ ω,h,h ′

Consider a type IX orbit orbit traveling between S s ω,h ∩ S h and S h . Typically, this orbit stays close to the piece of heteroclinic chain connecting successively the points P ω , P f (ω) , . . . , P f (ω) . Since the global behaviour of this orbit is rather complex, it is a good idea to focus on a smaller part of this travel, namely a transition between a neighbourhood of a point P f j (ω) and a neighbourhood of the point P f j+1 (ω) . This leads to the definition of the epoch transition map.

Definition 4.17 (Epoch transition map Φ

ω,h,h ′ ). Let ω ∈ ]1, +∞[ \ {2}, h = (h, h ⊥ , h / / ) and h ′ = (h ′ , h ′ ⊥ , h ′ / / ) so that the sections S s ω,h and S s f (ω),h ′ are included in U ξ . We define the epoch transition map Φ ω,h,h ′ ∶ S s ω,h → S s f (ω),h ′ as usual: if the heteroclinic chain H (x) intersects the section S s f (ω),h ′ , then Φ ω,h,h ′ (x)
is the 4-tuple of coordinates of the first intersection point of H (x) with S s f (ω),h ′ , otherwise Φ ω,h,h ′ is not defined at the point x. See figures 13 and 14.

Remark 4.18. Consider a point x in the section S u ω,h . For some time, the orbit of x remain close to the type II orbit O P ω →P f (ω) , and therefore will pass close to the point P f (ω) . Nevertheless, it might happen that the orbit of x misses the section S s f (ω),h ′ (this typically happens if h ′ is very small). Then the orbit of x will diverge away from the type II orbit O P ω →P f (ω) , and "fly around" the Mixmaster attractor. It is perfectly possible that, after a long and complicated flight around the Mixmaster, the orbit of x happens to hit the section S s f (ω),h ′ . In this case, the transition map Φ ω,h,h ′ is defined at x.

Nevertheless, it will be impossible to prove any estimate concerning Φ ω,h,h ′ (x) in such a situation. To get estimates, one needs some precise information about the orbit segment going from x to Φ ω,h,h ′ (x). In section 5, 6 and 7, we will prove that, for appropriate choices of h and h ′ , the orbit starting at some point x ∈ S u ω,h will remain close to the type II orbit O P ω →P f (ω) until it intersects the section S s f (ω),h ′ . For such choices of h and h ′ , we will be able to get some estimates on the transition map Φ ω,h,h ′ .

When restricteed to the set {x u > 0}, the epoch transition map Φ ω,h,h ′ is simply the transition map of the orbits of the Wainwright-Hsu vector field X between the section S s ω,h and the section S s f (ω),h ′ , expressed in local coordinates.

If x u = 0, then x is contained in the stable manifold of the point P x c = (0, 0, 0, x c ). So the heteroclinic chain H (x) is the concatenation of the orbit of x, the type II orbit

O P xc →P f (xc ) , the type II orbit O P f (xc ) →P f 2 (xc ) ,
Φ ω,h,h ′ (0, x s 1 , x s 2 , x c ) = (0, h ′ , 0, f (x c )) if ω > 2 (0, 0, h ′ , f (x c )) if 1 < ω < 2 (4.8) Indeed, if ω > 2, then O u x c = O s 1 f (x c
) and in that case, the first intersection point of O u x c with the section

S s f (ω),h ′ is in S s 1 f (ω),h ′ , otherwise it is in S s 2 f (ω),h ′ . Definition 4.19 (Map Φ A ω,h,h ′ ). Let ω ∈ ]1, +∞[ \ {2}, h = (h, h ⊥ , h / / ) and h ′ = (h ′ , h ′ ⊥ , h ′ / /
) so that the sections S s ω,h and S s f (ω),h ′ are included in U ξ and so that Proj A is well defined on the section S s ω,h . We define the map Φ

A ω,h,h ′ ∶ S s ω,h → S s f (ω),h ′ by the formula Φ A ω,h,h ′ (x) def = Φ ω,h,h ′ • Proj A (x)
Remark 4.20. According to (4.8), the map Φ A ω,h,h ′ admits an explicit expression. For any reasonable choice of the parameters h and h ′ ,

Φ A ω,h,h ′ (x) = (0, h ′ , 0, f (x c )) if ω > 2 (0, 0, h ′ , f (x c )) if 1 < ω < 2 (4.9)
Recall that S s ω,h is the "entry gate" (for the orbits of the Wainwright-Hsu vector field) of a neighbourhood of the point P ω while S s f (ω),h ′ is the "entry gate" of a neighbourhood of the point P f (ω) . As a consequence, when an orbit travels between S s ω,h and S s f (ω),h ′ , there is a first phase where it is close to the point P ω (and, a fortiori, it is close to the Kasner interval) and a second phase where it is far away from the Kasner interval but close to the type II orbit O P ω →P f (ω) . This leads us to introduce two more transition maps, Υ ω,h,h ′′ and Ψ ω,h ′′ ,h ′ , such that Φ ω,h,h ′ = Ψ ω,h ′′ ,h ′ • Υ ω,h,h ′′ . Each one of these maps captures the behaviour of the orbits during one of the two phases described above.

Note that until the end of this section, we will assume that all the local sections considered are included in U ξ . We will also implicitly assume that Proj A is well defined on these local sections. This is to avoid a lot of repetition in the following definitions, as they are all modeled on definitions 4.17 and 4.19.

The transition map Υ ω,h,h ′

We start with the transition map Υ ω,h,h ′ capturing the behaviour of the orbits in the neighbourhood of the point P ω .

Definition 4.21 (Transition map Υ

ω,h,h ′ ). Let ω ∈ ]1, +∞[, h = (h, h ⊥ , h / / ) and h ′ = (h ′ , h ′ ⊥ , h ′ / / ). We define the transition map Υ ω,h,h ′ ∶ S s ω,h → S u ω,h ′ as usual: if the heteroclinic chain H (x) intersects the section S u ω,h ′ , then Υ ω,h,h ′ (x) is the 4-tuple of coordinates of the first intersection point of H (x) with S u ω,h ′ , otherwise Υ ω,h,h ′ is not defined at the point x. See figure 14. Definition 4.22 (Map Υ A ω,h,h ′ ). Let ω ∈ ]1, +∞[, h = (h, h ⊥ , h / / ) and h ′ = (h ′ , h ′ ⊥ , h ′ / / ). We define the map Υ A ω,h,h ′ ∶ S s ω,h → S u ω,h ′ by the formula Υ A ω,h,h ′ (x) def = Υ ω,h,h ′ • Proj A (x) Remark 4.23. If h ′ / / ≥ h / / , then for every x ∈ S s ω,h such that x u = 0, we have Υ ω,h,h ′ (x) = (h ′ , 0, 0, x c ) (4.10)
In other words, Υ ω,h,h ′ (x) is the unique intersection point of the type II orbit

O u x c with S u ω,h ′ . As a consequence, the map Υ A ω,h,h ′ admits an explicit expression: if h ′ / / ≥ h / / , then Υ A ω,h,h ′ (x) = (h ′ , 0, 0, x c ) (4.11)

The transition map Ψ ω,h,h ′

We conclude with the transition map Ψ ω,h,h ′ capturing the behaviour of the orbits in the neighbourhood of the type II orbit

O P ω →P f (ω) . Definition 4.24 (Transition map Ψ ω,h,h ′ ). Let ω ∈ ]1, +∞[\{2}, h = (h, h ⊥ , h / / ) and h ′ = (h ′ , h ′ ⊥ , h ′ / / ). We define the transition map Ψ ω,h,h ′ ∶ S u ω,h → S s f (ω),h ′ as usual: if the heteroclinic chain H (x) intersects the section S s f (ω),h ′ , then Ψ ω,h,h ′ (x) is the 4-tuple of coordinates of the first intersection point of H (x) with S s f (ω),h ′ , otherwise Ψ ω,h,h ′ is not defined at the point x. See figure 14. Definition 4.25 (Map Ψ A ω,h,h ′ ). Let ω ∈ ]1, +∞[, h = (h, h ⊥ , h / / ) and h ′ = (h ′ , h ′ ⊥ , h ′ / / ). We define the map Ψ A ω,h,h ′ ∶ S u ω,h → S s f (ω),h ′ by the formula Ψ A ω,h,h ′ (x) def = Ψ ω,h,h ′ • Proj A (x)
Remark 4.26. The map Ψ A ω,h,h ′ admits an explicit expression. More precisely, if h

′ / / is larger than h / / , then for every x ∈ S u ω,h , Ψ A ω,h,h ′ (x) = (0, h ′ , 0, f (x c )) if ω > 2 (0, 0, h ′ , f (x c )) if 1 < ω < 2 (4.
12)

The explanation is the same than for formula (4.8).

In the next four chapters, we will study these transition maps. More precisely, we will study the transition map Υ ω,h,h ′ in Chapter 5, then the transition map Ψ ω,h,h ′ in Chapter 6, then the epoch transition map Φ ω,h,h ′ in Chapter 7 and finally the era transition map Φω,h and the double era transition map Φω,h in Chapter 8.

Dynamics in the neighbourhood of a point of the Kasner circle

The goal of this section is to give some sharp estimates on the transition map Υ ω,h s ,h u (see definition 4.21). Recall that Υ ω,h s ,h u describes the transition of the orbits of the Wainwright-Hsu vector field from the section S s ω,h s to the section S u ω,h u . Both sections S s ω,h s and S u ω,h u are close to the point P ω ∶= (0, 0, 0, ω) belonging to the Kasner circle in local coordinates. Actually, S s ω,h s should be thought as the "entrance gate" to the neighbourhood of P ω for the orbits, whereas S u ω,h u should be thought as the "exit gate". We will choose the parameters h s and h u so that the orbits starting in the section S s ω,h s hit the section S u ω,h u before they exit a small neighbourhood of P ω where the local vector field X ω is defined. Recall that the orbits of X ω are the same as those of X. Hence, we are left to investigate, for any ω ∈ ]1, +∞[, the dynamics generated by the local vector field X ω near the point P ω . The methods we use are generalizations and refinements of those used in the work of Liebscher & al. [Lie+11].

The following proposition is the main result of this section. For a technical reason explained below, we will often encounter the quantity ω-1 4 in the estimates. Hence, we introduce the notation

d(ω) def = ω -1 4
Proposition 5.1 (Control of the transition maps Υ ω,h s ,h u ). There exist two constants C > 0 and n ∈ N such that for every

ω ∈ ]1, +∞[, every 0 < h ≤ (Cω n ) -1 , every 0 < h ⊥ < min(h, d(ω)), for h s = (h, h ⊥ , min(h, d(ω))) and h u = (h, h, 2h), the transition map Υ ω,h s ,h u ∶ S s ω,h s → S u ω,
dist ∞ (Υ(x), A) = Υ(x) -Υ A (x) ⊥ ≤ h ω+2 ω+1 ⊥ h -1
(5.1) (Drift in the direction tangent to the Mixmaster attractor)

Υ(x) -Υ A (x) / / ≤ h ⊥ hCω n (5.2)
(Lipschitz estimate in the direction transverse to the Mixmaster attractor)

(Υ(x) -Υ(x)) -(Υ A (x) -Υ A (x)) ⊥ ≤ h 1 ω+1 ⊥ h -1 ∥x -x∥ ∞ (5.3)
(Lipschitz estimate in the direction tangent to the Mixmaster attractor)

(Υ(x) -Υ(x)) -(Υ A (x) -Υ A (x)) / / ≤ Cω n h ∥x -x∥ ⊥ + Cω n h ⊥ ∥x -x∥ / /
(5.4) Remark 5.2 (Purpose of Proposition 5.1). Recall that Υ describes the behaviour of all the orbits of the local vector field X ω near the Kasner circle K and the mixmaster attractor A, while Υ A describes the behaviour of the heteroclinic chains in A. Also recall that we have explicit formulas for the map Υ A (see Remark 4.23). The purpose of Proposition 5.1 is to compare the dynamics of Υ to the one of Υ A .

Remark 5.3 (Explanation of the different estimates). To fix the ideas, consider an orbit τ ↦ x(τ ) of the local vector field X ω traveling between the sections S

s 1 ω,h s and S u ω,h u . Denote by x in = (x in u , h, x in s 2 , x in c ) ∈ S
s ω,h s its initial condition. Estimate (5.1) means that the distance between the orbit x and the Mixmaster attractor is contracted during its travel. Moreover, it shows that this contraction degenerates when ω → +∞, that is, when x travels very close to a Taub point. Estimate (5.2) means that the more x in is close to the Mixmaster attractor, the more the orbit x does not deviate, in the direction tangent to the Mixmaster attractor, from the type II orbit passing through the point (0, h, 0, x in c ). Estimates (5.3) and (5.4) prove that Υ -Υ A is Lipschitz, and provide an explicit a Lipschitz constant for this map. As a summary, one can remember the following fact. There is a competition between two factors for the above estimates:

• The more the points are close to the Mixmaster attractor, the more the estimates are precise.

• The more the points are close to a Taub point, the less the estimates are precise.

Remark 5.4. Note that Proposition 5.1 holds for h small enough and for h ⊥ smaller than h. In the next chapters, we will apply Proposition 5.1 with h ⊥ much much smaller than h. Hence, it is extremely important that h ⊥ appears on the right hand side of the estimates (5.1)... (5.4).

Remark 5.5. The estimate (5.2) could be rewritten in a simpler way as

|Υ(x) c -x c | ≤ h ⊥ hCω n .
We did not make this choice to make it clear here that we compare Υ and Υ A . Same remark goes for (5.4).

Remark 5.6.

If x ∈ S u ω,h u , then for all y ∈ S u ω,h u ∩ A dist ∞ (x, A) = ∥x -y∥ ⊥ Now remark that if x ∈ S s ω,h s , then Υ A (x) ∈ S u ω,h u ∩ A. This is the reason why dist ∞ (Υ(x), A) = Υ(x) -Υ A (x)
⊥ Remark 5.7 (Technical detail). The quantity ω-1 4 in the upper bound on the size of the sections is purely technical. It is closely related to the fact that the coordinates are not defined for x c = 1. Basically, we need to make sure that the orbits do not start too close to this frontier so that they intersect the section S u ω,h u before they possibly encounter this boundary and cease to exist. To prove Proposition 5.1, we divide the study in two parts. In section 5.1, we study the behaviour of one orbit of X ω . This will lead to estimates (5.1) and (5.2). In section 5.2, we compare the behaviour of two orbits. This will lead to estimates (5.3) and (5.4).

Following the notations of Proposition 3.2, we will denote by x = (x u , x s 1 , x s 2 , x c ) the coordinates of any point x ∈ U ξ ⊂ R 4 . Recall from Proposition 3.8 that the differential equations associated with X ω have the following form

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x ′ u = µ u (ω)x u x ′ s 1 = -μ ω,s 1 (x c ) x s 1 + X u,s 1 ω,s 1 (x)x u x s 1 + X s 1 ,s 1 ω,s 1 (x)x 2 s 1 + X s 2 ,s 1 ω,s 1 (x)x s 2 x s 1 x ′ s 2 = -μ ω,s 2 (x c ) x s 2 + X u,s 2 ω,s 2 (x)x u x s 2 + X s 1 ,s 2 ω,s 2 (x)x s 1 x s 2 + X s 2 ,s 2 ω,s 2 (x)x 2 s 2 x ′ c = X u,s 1 ω,c (x)x u x s 1 + X u,s 2 ω,c (x)x u x s 2
(5.5)

Remark 5.8. Let ω ∈ ]1, +∞[ and τ ↦ x(τ ) be an orbit of the local vector field X ω . Denote by x in its initial condition. Using (5.5), one can see that the coordinate x u is strictly increasing. It follows that the section S u ω,h u is intersected by the orbit x at most once. If this is the case, the time of first (and unique) intersection of the orbit x with the section S u ω,h u is

τ out (x, ω, h) def = 1 µ u (ω) ln h x in u (5.6)
where h u = (h, h ⊥ , h / / ). We should insist on the fact that we are talking about the local vector field X ω and its orbits. The "real" orbits for the Wainwright-Hsu vector field X can intersect the section S u ω,h u infinitely many times. The local vector field X ω being a renormalization of X in a neighbourhood U of the point P ω on the Kasner circle, an orbit of X ω is exactly a connected component of an orbit of X intersected with U (modulo the local coordinate system ξ).

To conclude this introduction, we state the first ingredient for the shadowing theorem (see Section 10).

Proposition 5.9 (Shadowing near the Kasner circle). There exist two constants C ≥ 1 and n ∈ N such that the properties below hold for every

ω ∈ ]1, +∞[, every i ∈ {1, 2}, every 0 < h ≤ (Cω n ) -1 , every 0 < h ⊥ ≤ min( 1 3 h, d(ω)), for h s = (h, h ⊥ , h ⊥ ) and for h u = (h, h, 2h). Let τ ↦ x(τ )
τ * = τ * ω,h (x in ) def = 1 µ u (ω) + 4 5 µ s 1 (ω) ln 4 5 µ s 1 (ω)h µ u (ω)x in u Then 1. 0 ≤ τ * ≤ τ out where τ out = τ out (x, ω, h).

The point x(τ *

) is very close to the point P ω on the Kasner circle. More precisely,

∥x(τ * ) -P ω ∥ ∞ ≤ 8h 1 3 ⊥ (5.7) S s i ω,h s S u ω,h u x s i x u x c ξ(K 0 ) •x(0) • x(τ out ) •x(τ * ) • P s i ω,h • P ω •P u ω,h

The orbit segment joining

⊥

(5.8b)

Remark 5.10. Once again, it is crucial that h ⊥ appears on the right hand side of the estimates (5.7)... (5.8b) since, in the next chapters, we will apply Proposition 5.9 with h ⊥ much much smaller than h.

Control of one orbit

Our goal is, for any given ω ∈ ]1, +∞[, to study the behaviour of the orbits of the local vector field X ω which travel between two given sections S s ω,h s and S u ω,h u . We first study the behaviour of the orbits that are assumed to stay in a small neighbourhood of P ω , where the dynamics is "almost linear". Lemma 5.16 shows that, for such an orbit τ ↦ x(τ ), the coordinate x u is exponentially increasing (if it is not identically zero), the coordinates x s 1 and x s 2 are exponentially decreasing (if they are not identically zero) and the variation of the coordinate x c is small. Then, we prove that for h s and h u well chosen, any orbit crossing S s ω,h s will eventually cross S u ω,h u and stays in a small neighbourhood of P ω during its travel between those two sections. To do this, we need two preliminary tools:

• A control of the eigenvalues -μ ω,s 1 (x c ) and -μ ω,s 2 (x c ) of DX ω (0, 0, 0, x c ). This is done in corollary 5.13.

• An estimate on the quantity

χ = x u ∥x s 1 ,s 2 ∥ 1 = x u (x s 1 + x s 2 )
used to bound the variation of the coordinate x c . Lemma 5.14 shows that this map is exponentially decreasing along the orbits of the local vector field X ω which travel sufficiently close to the point P ω .

Lemma 5.11 (Control of the eigenvalues µ

u , µ s 1 et µ s 2 ). Let ω ∈ ]1, +∞[ and α ∈]0, 1[. For every ω ′ ∈ ]1, +∞[ such that ω ′ -ω ≤ 1 -α 3ω we have α ≤ µ u (ω ′ ) µ u (ω) , µ s 1 (ω ′ ) µ s 1 (ω) , µ s 2 (ω ′ ) µ s 2 (ω) ≤ α -1
(5.9)

Proof. According to (2.14a), dµ u dω (ω ′ ) = 6 1 -ω ′2 1 + ω ′ + ω ′2 2 ≤ 6 Let ω ′ ∈ ]1, +∞[ such that ω ′ -ω ≤ 1 -α 3ω According to the mean value theorem, µ u (ω ′ ) -µ u (ω) ≤ 6 1 -α 3ω ≤ 6 (1 -α)ω 1 + ω + ω 2 = (1 -α)µ u (ω)
Observe that 1 -α ≤ α -1 -1. Hence, (5.9) holds true for µ u . The computations are analogous for µ s 1 and µ s 2 .

Recall from Proposition 3.8 that μω,

s i ω ′ = µ u (ω) µ u (ω ′ ) µ s i (ω ′ )
is an eigenvalue of DX ω (0, 0, 0, ω ′ ).

Proposition 5.12 (Control of the eigenvalue μω,s i ). Let ω ∈ ]1, +∞[ and α ∈]0, 1[. For any i ∈ {1, 2} and any ω

′ ∈ ]1, +∞[ such that ω ′ -ω ≤ 1 -α 6ω , the eigenvalue μω,s i ω ′ of DX ω (0, 0, 0, ω ′ ) satisfies α ≤ μω,s i ω ′ µ s i (ω) ≤ α -1
(5.10)

Proof. Using the formula μω,s i ω ′ µ s i (ω) = µ u (ω) µ u (ω ′ ) µ s i (ω ′ ) µ s i (ω)
and the straightforward inequality

1 -α 6 ≤ 1 -α 3
we get the estimate (5.10) from Lemma 5.11 applied twice with α instead of α.

Corollary 5.13 below is a refinement of Proposition 5.12 easier to use in the proof of Lemma 5.14.

Corollary 5.13 (Control of the eigenvalue μω,s i , second version).

Let ω ∈ ]1, +∞[ and α ∈]0, 1[. For every ω ′ ∈ ]1, +∞[ such that ω ′ -ω ≤ 1 -α 24ω 2 , for i = 1, 2, the eigenvalue μω,s i ω ′ of DX ω (0, 0, 0, ω ′ ) satisfies α ′ ≤ μω,s i ω ′ µ s i (ω) ≤ α ′-1
(5.11)

where

α ′ = 1-α 2 µ u (ω) + 1+α 2 µ s 1 (ω) µ s 1 (ω) Proof. Using (2.14), it is straightforward to check that 1 -α 24ω 2 ≤ 1 -α 12(1 + ω)ω = 1 -α ′ 6ω for every ω ∈ ]1, +∞[ and every α ∈]0, 1[.
Hence, (5.11) follows immediately from Proposition 5.12.

Lemma 5.14 (Control of χ).

There exist two constants C > 0 and n ∈ N such that for every ω ∈ ]1, +∞[, every 3/4 < α < 1, every t ≥ 0 and every orbit τ ↦ x(τ ) of the local vector field X ω satisfying

max x u (τ ), x s 1 (τ ), x s 2 (τ ), |x c (τ ) -ω| ≤ 1 -α Cω n for every τ ∈ [0, t] the function χ(τ ) = x u (τ )(x s 1 (τ ) + x s 2 (τ )) satisfies χ(t) ≤ e -α µ s1 (ω)-µ u (ω) t χ(0) (5.12) Remark 5.15. Recall that µ s 1 (ω) ≥ µ u (ω).
Hence, χ is exponentially decreasing along the orbits of the local vector field X ω .

Proof. Basically, the proof amounts to obtain a differential inequation on χ and then to use a Gronwall estimate. For every C > 0 and every n ∈ N, we denote by E C,n the set of all (ω, α, t, x) such that ω ∈ ]1, +∞[, 3/4 < α < 1, t ≥ 0 and τ ↦ x(τ ) is an orbit of the local vector field X ω satisfying max(x u (τ ),

x s 1 (τ ), x s 2 (τ ), |x c (τ ) -ω|) ≤ 1 -α Cω n for every τ ∈ [0, t] (5.13)
Let C > 0 and n ∈ N be large enough such that for every ω ∈ ]1, +∞[, X ω is well defined on the open ball B ω,C,n (see definition 3.6). Let (ω, α, t, x) ∈ E C,n . We compute the derivative of χ ∶ τ ↦ x u (τ )(x s 1 (τ ) + x s 2 (τ )) by replacing the derivatives of x u , x s 1 and x s 2 by their respective expressions according to (5.5). The time τ ∈ [0, t] is implicit in the following computations.

χ ′ = µ u (ω)x u (x s 1 + x s 2 ) + x u -μ ω,s 1 (x c ) x s 1 + X u,s 1 ω,s 1 (x)x u x s 1 + X s 1 ,s 1 ω,s 1 (x)x 2 s 1 + X s 2 ,s 1 ω,s 1 (x)x s 2 x s 1 + x u -μ ω,s 2 (x c ) x s 2 + X s 1 ,s 2 ω,s 2 (x)x s 1 x s 2 + X u,s 2 ω,s 2 (x)x u x s 2 + X s 2 ,s 2 ω,s 2 (x)x 2 s 2
According to the estimate on the non-linear terms (3.13), the inequality (5.13) and the fact that μω,s 1 ≤ μω,s 2 , there exist C 0 ≥ 24 and n 0 ≥ 2 such that for every C ≥ C 0 , every n ≥ n 0 and every (ω, α, t, x) ∈ E C,n , we have

χ ′ ≤ (µ u (ω) -μω,s 1 (x c )) + C 0 ω n 0 max(x u , x s 1 , x s 2 ) χ (5.14)
According to the estimate (5.11) on the eigenvalues and the inequality (5.13), for every C ≥ C 0 , every n ≥ n 0 and every (ω, α, t, x) ∈ E C,n , we have the following control on μω,s 1 :

-μω,s 1 (x c ) ≤ -α ′ µ s 1 (ω) (5.15)
where

α ′ = (1 -1+α 2 )µ u (ω) + 1+α 2 µ s 1 (ω) µ s 1 (ω)
Plugging (5.15) into (5.14), it follows that for every C ≥ C 0 , every n ≥ n 0 and every (ω, α, t, x) ∈ E C,n , we have

χ ′ ≤ (µ u (ω) -α ′ µ s 1 (ω)) + C 0 ω n 0 max(x u , x s 1 , x s 2 ) χ ≤ 1 + α 2 (µ u (ω) -µ s 1 (ω)) + C 0 ω n 0 max(x u , x s 1 , x s 2 ) χ (5.16)
According to (2.14),

µ s 1 (ω) -µ u (ω) ≥ 2 ω 2
(5.17) It follows by (5.16) and (5.17) that for every C ≥ C 0 , every n ≥ n 0 + 2 and every (ω, α, t, x) ∈ E C,n , we have

χ ′ ≤ α(µ u (ω) -µ s 1 (ω))χ
which is the desired differential inequality. Indeed, one just needs to apply the standard Gronwall's lemma to obtain (5.12).

Lemma 5.16 (Control of one orbit close to P ω ). There exist two constants C > 0 and n ∈ N such that for every ω ∈ ]1, +∞[, every 3/4 < α < 1, every time t ≥ 0 and every orbit τ ↦ x(τ ) of the local vector field X ω satisfying

max(x u (τ ), x s 1 (τ ), x s 2 (τ ), |x c (τ ) -ω|) ≤ 1 -α Cω n for every τ ∈ [0, t]
we have the following estimates:

(Formulae in the unstable direction)

x u (t) = e µ u (ω)t x u (0) (5.18a)
(Estimate in the stable direction) For every i ∈ {1, 2},

x s i (t) ≤ e -αµ s1 (ω)t x s i (0) (5.18b)
(Estimate in the central direction)

|x c (t) -x c (0)| ≤ Cω n x u (0)(x s 1 (0) + x s 2 (0)) 1 -e -α µ s1 (ω)-µ u (ω) t
(5.18c)

Proof. The three controls are proven independently. The first one is a direct consequence of the evolution equation of x u in (5.5).

For every C > 0 and every n ∈ N, we denote by E C,n the set of all (ω, α, t, x) such that ω ∈ ]1, +∞[, 3/4 < α < 1, t ≥ 0 and τ ↦ x(τ ) is an orbit of the local vector field X ω satisfying max(x u (τ ),

x s 1 (τ ), x s 2 (τ ), |x c (τ ) -ω|) ≤ 1 -α Cω n for every τ ∈ [0, t] (5.19)
Control of the coordinate x s 1 (the case of x s 2 is analogous). We compute the derivative of x s 1 using (5.5). The time τ is implicit in the following computations. According to (3.13) and (5.19), there exist C 0 ≥ 24 and n 0 ≥ 2 such that for every C ≥ C 0 , every n ≥ n 0 and every (ω, α, t, x) ∈ E C,n , we have

x ′ s 1 ≤ -μ ω,s 1 (x c ) + C 0 ω n 0 max(x u , x s 1 , x s 2 ) x s 1 (5.20)
According to the estimate (5.11) on the eigenvalues and the inequality (5.19), for every C ≥ C 0 , every n ≥ n 0 and every (ω, α, t, x) ∈ E C,n , we have the following control on μω,s 1 :

-μω,s 1 (x c ) ≤ -α ′ µ s 1 (ω) ≤ - 1 + α 2 µ s 1 (ω) (5.21)
where

α ′ = (1 -1+α 2 )µ u (ω) + 1+α 2 µ s 1 (ω) µ s 1 (ω)
It follows from (5.20) and (5.21) that, for every C ≥ C 0 , every n ≥ n 0 and every (ω, α, t, x) ∈ E C,n , we have

x ′ s 1 ≤ - 1 + α 2 µ s 1 (ω) + C 0 ω n 0 max(x u , x s 1 , x s 2 ) x s 1 (5.22)
Using (2.14), we get

µ s 1 (ω) ≥ 4 ω 2 (5.23)
According to (5.22) and (5.23), for every C ≥ C 0 , every n ≥ n 0 + 2 and every (ω, α, t, x) ∈ E C,n , we have x ′ s 1 ≤ -αµ s 1 (ω)x s 1 Hence, Gronwall's lemma gives the desired control (5.18b) on x s 1 (t).

Control of the coordinate x c . We compute the derivative of x c using (5.5). The time τ is implicit in the following computations. According to (3.13) and (5.19), there exist C 1 ≥ C 0 and n 1 ≥ n 0 such that for every C ≥ C 1 , every n ≥ n 1 and every (ω, α, t, x) ∈ E C,n , we have

x ′ c ≤ C 1 ω n 1 χ (5.24)
where χ(τ ) = x u (τ )(x s 1 (τ ) + x s 2 (τ )). According to (5.24) and Lemma 5.14, there exist C 2 ≥ C 1 and n 2 ≥ n 1 such that for every C ≥ C 2 , every n ≥ n 2 and every (ω, α, t, x) ∈ E C,n , we have

|x c (t) -x c (0)| ≤ C 1 ω n 1 t 0 e -α(µ s1 (ω)-µ u (ω))τ χ(0) dτ so |x c (t) -x c (0)| ≤ C 1 ω n 1 χ(0) 1 -e -α(µ s1 (ω)-µ u (ω))t α(µ s 1 (ω) -µ u (ω)) (5.25)
Using (2.14), we get

µ s 1 (ω) -µ u (ω) ≥ 2 ω 2 (5.26)
Recall that α > 3/4. It follows by (5.25) and (5.26) that for every C ≥ max(C 2 , 8 3 C 1 ), every n ≥ max(n 2 , n 1 + 2) and every (ω, α, t, x) ∈ E C,n , we have

|x c (t) -x c (0)| ≤ Cω n χ(0) 1 -e -α µ s1 (ω)-µ u (ω) t
which concludes the proof.

We are now ready to formulate, for any given ω ∈ ]1, +∞[, two statements about the orbits τ ↦ x(τ ) of the local vector field X ω starting in the section S s ω,h s . Proposition 5.17 deals with the generic orbits whose initial condition x in verifies x in u ≠ 0 (generic case) while Proposition 5.20 deals with the exceptional orbits for which x in u = 0. More precisely, Proposition 5.17 below gives an explicit interval where the orbit x is well defined. On this interval, x will satisfy the estimates of Lemma 5.16. Recall that

d(ω) = ω-1 4 and τ out (x, ω, h) = 1 µ u (ω) ln h x in u (see remark 5.8).
Proposition 5.17 (Behaviour of generic orbits). There exist two constants C > 0 and n ∈ N such that for every ω ∈ ]1, +∞[, every 0 < h ≤ (Cω n ) -1 and for h s = (h, min (h, d(ω)) , min (h, d(ω))), the following properties holds. Let τ ↦ x(τ ) be an orbit of the local vector field X ω . If its initial condition

x in = (x in u , x in s 1 , x in s 2 , x in c ) ∶= x(0) belongs to the section S s ω,h s and satisfies x in u ≠ 0 then x is defined (at least) on 0, τ out (x, ω, h) . Moreover, if 3/4 < α < 1 and h ≤ (1 -α)(Cω n ) -1 , then for every τ ∈ [0, τ out (x, ω, h)],
we have the following estimates:

(Control in the unstable direction)

x u (τ ) = e µ u (ω)τ x in u
(5.27a) (Control in the stable direction) For i ∈ {1, 2},

x s i (τ ) ≤ e -αµ s1 (ω)τ x in s i (5.27b)
(Control in the central direction)

x c (τ ) -x in c ≤ Cω n x in u max x in s 1 , x in s 2 1 -e -α(µ s1 (ω)-µ u (ω))τ
(5.27c)

Proof. For every C > 0 and every n ∈ N, we denote by E C,n the set of all (ω, α, h, x) (ω, α, h, x) ∈ E C 1 ,n . We are going to show that the orbit x is well defined on [0, τ out (x, ω, h)] and that for every τ ∈ [0, τ out (x, ω, h)], we have max(

such that ω ∈ ]1, +∞[, 3/4 < α < 1, 0 < h ≤ (1 -α)(Cω n ) -1 and τ ↦ x(τ ) is an orbit of the local vector field X ω such that x in ∶= x(0) ∈ S
x u , x s 1 , x s 2 , |x c -ω|) ≤ 1-α C 0 ω n . Let us denote by ]τ -, τ + [ the maximal existence interval of x (with τ -< 0 and τ + > 0). For every τ ∈]τ -, τ + [, let N ⊥ (τ ) def = max(x u (τ ), x s 1 (τ ), x s 2 (τ )) N / / (τ ) def = |x c (τ ) -ω| Remark that N ⊥ (0) ≤ min(h, d(ω)) < 1-α C 0 ω n and N / / (0) ≤ min(h, d(ω)) < min( 1-α C 0 ω n , ω-1 2 ). Let us denote by τ max the supremum of all time t 0 ∈ [0, τ + [ such that for every τ ∈ [0, t 0 ], N ⊥ (τ ) ≤ 1-α C 0 ω n and N / / (τ ) ≤ min( 1-α C 0 ω n , ω-1 2 ). By definition, for every τ ∈ [0, τ max [, we have N ⊥ (τ ) ≤ 1-α C 0 ω n and N / / (τ ) ≤ min( 1-α C 0 ω n , ω-1 2 ).
It follows that x must be defined at τ = τ max since it cannot blow up. Now assume that τ max ≤ τ out (x, ω, h). By continuity, for every τ ∈ [0, τ max ], we have

N ⊥ (τ ) ≤ 1-α C 0 ω n and N / / (τ ) ≤ min( 1-α C 0 ω n , ω-1 2 ).
As a consequence, we can use Lemma 5.16 on [0, τ max ] to get

x u (τ max ) ≤ e µ u (ω)τ out (x,ω,h) x in u = h ≤ 1 -α C 1 ω n < 1 -α C 0 ω n , x s 1 (τ max ), x s 2 (τ max ) ≤ h < 1 -α C 0 ω n ,
and

|x c (τ max ) -ω| ≤ x c (τ max ) -x in c + x in c -ω ≤ 2C 0 ω n h min(h, d(ω)) + min(h, d(ω)) < min 1 -α C 0 ω n , ω -1 2
which contradicts the maximality of τ max by continuity. Hence, τ max > τ out (x, ω, h). This proves that the orbit x is well defined on [0, τ out (x, ω, h)] and that for every

τ ∈ [0, τ out (x, ω, h)], N ⊥ (τ ) ≤ 1-α C 0 ω n and N / / (τ ) ≤ min( 1-α C 0 ω n , ω-1 2 ).
As a consequence, we can use Lemma 5.16 on [0, τ out (x, ω, h)], which proves (5.27a), (5.27b) and (5.27c). Now remark that x is well defined on the interval [0, τ out (x, ω, h)], which is independant of α. Hence, we should find a condition that is also independant of α, as stated in Proposition 5.17. Let

C = 5C 1 . If ω ∈ ]1, +∞[, 0 < h ≤ (Cω n )
-1 and x is an orbit of the local vector field X ω such that x in ∶= x(0) ∈ S s ω,h s where h s = (h, min(h, d(ω)), min(h, d(ω))) and x in u ≠ 0, then (ω, 4/5, h, x) ∈ E C 1 ,n and we can apply the above reasoning to x. This proves that x is well defined on 0, τ out (x, ω, h) and this concludes the proof.

Corollary 5.18 below complements Proposition 5.17. It shows that if S s ω,h s and S u ω,h u are two sections "close enough" to P ω and if S s ω,h s is "sufficiently small", then any generic orbit of the local vector field X ω starting in S s ω,h s will eventually pass through S u ω,h u before leaving the neighbourhood of P ω where X ω is well defined. Moreover, it gives precise estimates about the position of the orbit in the section S u ω,h u .

Corollary 5.18 (Estimates in the section S u ω,h u ). There exist two constants C > 0 and n ∈ N such that for every

ω ∈ ]1, +∞[, every 0 < h ≤ (Cω n ) -1 , every 0 < h ⊥ < min(h, d(ω)), for h s = (h, h ⊥ , min (h, d(ω))) and h u = (h, h, 2h
), the following properties hold true. Let τ ↦ x(τ ) be an orbit of the local vector field X ω . If its initial condition x in ∶= x(0) belongs to the section S s ω,h s and satisfies x in u ≠ 0, then x intersects the section S u ω,h u exactly at the time τ = τ out (x, ω, h). Moreover, if

3/4 < α < 1 and h ≤ (1 -α)(Cω n ) -1 , then
(Distance to the Mixmaster attractor)

Υ(x in ) -Υ A (x in ) ⊥ ≤ (h ⊥ ) 1+ α ω h -α ω (5.28a)
(Drift in the direction tangent to the Mixmaster attractor)

Υ(x in ) -Υ A (x in ) / / ≤ h ⊥ hCω n (5.28b)
Remark 5.19. Choosing α = max ω ω+1 , 4 5 , estimate (5.28a) will lead to estimate (5.1) and estimate (5.28b) will lead to estimate (5.2).

Proof. Let C 0 > 0 and n 0 ∈ N be large enough such that we can apply Proposition 5.17 with these two constants. Let C 1 = 2C 0 and n

1 = n 0 + 1. Fix ω ∈ ]1, +∞[, 3/4 < α < 1, 0 < h ≤ (1 -α)(C 1 ω n 1 ) -1 , 0 < h ⊥ < min(h, d(ω)
) and an orbit τ ↦ x(τ ) of the local vector field X ω whose initial condition x in ∶= x(0) belongs to the section S s ω,h s where h s = (h, h ⊥ , min(h, d(ω))) and such that x in u ≠ 0. According to Proposition 5.17, x is well defined on 0, τ out (x, ω, h) .

Using (5.27a), we get

x u (τ out (x, ω, h)) = e µ u (ω)τ out (x,ω,h) x in u = h According to (5.27b),
x s 1 (τ out (x, ω, h)), x s 2 (τ out (x, ω, h)) ≤ h and according to (5.27c),

x c (τ out (x, ω, h)) -ω ≤ 2h It follows that x(τ out (x, ω, h)) ∈ S u ω,h u where h u = (h, h, 2h). Recall from remark 5.8 that τ out (x, ω, h)
is the unique time of intersection. Hence, Υ(x

in ) = x(τ out (x, ω, h)) is well defined. Remark that x in u ≤ h ⊥ , max x in s 1 , x in s 2 ≤ h and Υ A (x in ) c = x in c so (5.28b) is a direct consequence of (5.27c) applied with τ = τ out (x, ω, h). Let α ′ = (1-α)µ u (ω)+αµ s1 (ω) µ s1 (ω)
. Using (2.14), one can remark that

α ′ µ s 1 (ω) µ u (ω) -1 = α µ s 1 (ω) -µ u (ω) µ u (ω) = α ω and 1 -α ′ = (1 -α) µ s 1 (ω) -µ u (ω) µ s 1 (ω) = 1 -α 1 + ω ≥ 1 -α 2ω Hence, h ≤ 1 -α C 1 ω n 1 = 1 -α 2ω 1 C 0 ω n 0 ≤ 1 -α ′ C 0 ω n 0 Since Υ(x in ) -Υ A (x in ) ⊥ = max x s 1 (τ out (x, ω, h)), x s 2 (τ out (x, ω, h)) ,
(5.28a) follows from (5.27b) applied with τ out (x, ω, h) instead of τ and with α ′ instead of α.

Now remark that the fact that x intersects the section S u ω,h u is independant of α. Hence, we should find a condition that is also independant of α, as stated in corollary 5.18.

Let C = 5C 1 , ω ∈ ]1, +∞[, 0 < h ≤ (Cω n 1 )
-1 and 0 < h ⊥ ≤ min(h, d(ω)). Let x be an orbit of the local vector field X ω whose initial condition x in ∶= x(0) belongs to the section S s ω,h s where h s = (h, h ⊥ , min(h, d(ω))) and such that x in u ≠ 0. Remark that h ≤ 1/5(C 1 ω n 1 ) -1 . Hence, we can apply the above reasoning to x with α = 4/5. This proves that x intersects the section S u ω,h u (where h u = (h, h, 2h)) and concludes the proof.

We now deal with the exceptional orbits τ ↦ x(τ ) whose initial condition x in verifies x in u = 0. Recall that the local coordinate system ξ is constructed in such a way that the stable manifold of a point (0, 0, 0, xc ) for the local vector field X ω has for equation "x u = 0, x c = xc " (see (3.3b)). Hence, any exceptional orbit converges to a point of {0 R 3 } × ]1, +∞[ (which is the Kasner interval in local coordinates).

Proposition 5.20 (Behaviour of exceptional orbits).

There exist two constants C > 0 and n ∈ N such that for every

ω ∈ ]1, +∞[, every 3/4 < α < 1, every 0 < h ≤ (1 -α)(Cω n ) -1 , for h s = (h, h, h)
and for every orbit τ ↦ x(τ ) of the local vector field X ω whose initial condition x in ∶= x(0) belongs to the section S s ω,h s and such that x in u = 0, x is well defined on [0, +∞[ and stays forever in the stable manifold of the point (0, 0, 0, x in c ), i.e. for every τ ≥ 0, x u (τ ) = 0 and x c (τ ) = x in c . Moreover, the orbit x converges exponentially fast to (0, 0, 0, x in c ). More precisely, for i = 1, 2 and for every τ ≥ 0,

x s i (τ ) ≤ e -αµ s1 (ω)τ x in s i
(5.29)

Proof. Using the equations (5.5), this is a straightforward consequence of Lemma 5.16.

Next corollary shows that if the section S s ω,h s is small enough, any orbit of the local vector field X ω will intersect it at most once. This is useful for two reasons. Firstly, it allows us to define a time of intersection without ambiguity (see definition 5.22). Secondly, it implies that the time length between two consecutive intersections of an orbit t ↦ x(t) of the Wainwright-Hsu vector field X with the section S s ω,h s cannot be arbitrary small, i.e. admits a uniform positive lower bound (see Lemma 6.10).

Again, we insist on the fact that corollary 5.21 below is about the local vector field X ω and its orbits. The "real" orbits for the Wainwright-Hsu vector field X can intersect the section S s ω,h s infinitely many times. The local vector field X ω being a renormalization of X in the neighbourhood of the point P ω on the Kasner circle, an orbit of X ω is exactly a connected component of an orbit of X (modulo the local coordinate system ξ).

Corollary 5.21 (Unique intersection with S s ω,h s ).

There exist two constants C > 0 and n ∈ N such that for every

ω ∈ ]1, +∞[, every 0 < h ≤ (Cω n ) -1 , for h s = (h, min (h, d(ω)) , min (h, d(ω))
) and for every orbit τ ↦ x(τ ) of the local vector field X ω whose initial condition x(0) belongs to the section S s ω,h s , x does not intersect S s ω,h s again in the future nor in the past.

Proof. Let C 0 > 0 and n ∈ N be large enough such that we can apply Proposition 5.17 and Proposition 5.20 with these two constants. Let C = 10C 0 and α

= 4 5 . Let ω ∈ ]1, +∞[, 0 < h ≤ (Cω n ) -1 , h s = (h, min(h, d(ω)), min(h, d(ω))
) and τ ↦ x(τ ) be an orbit of the local vector field X ω whose initial condition x in ∶= x(0) belongs to the section S s ω,h s . Let us denote by ]τ -, τ + [ the maximal existence interval of x. By symmetry, it is enough to prove that x does not intersect again S s ω,h s in the future. First, assume that x in u ≠ 0. Using Proposition 5.17 with C 0 and α, we can apply (5.27b) to get that for every τ ∈ 0, τ out (x, ω, h) and every i ∈ {1, 2},

x s i (τ ) < h so x(τ ) ∉ S s ω,h s .
The first equation of the system (5.5) implies that, for every τ ∈ τ out (x, ω, h), τ + , the coordinate x u (τ ) is bigger than h, hence x(τ ) ∉ S s ω,h s . We are left to deal with the case where x in u = 0. Using (5.29), we get that for every τ > 0 and every i ∈ {1, 2}, x s i (τ ) < h. Hence, x(τ ) ∉ S s ω,h s . This concludes the proof.

We can now give a proof of the proposition on the shadowing of a heteroclinic chain, stated in the introduction of the present section.

Proof of Proposition 5.9. To simplify the proof, let us treat the case i = 1. Let C 0 ≥ 1 and n ∈ N be large enough such that we can apply Proposition 5.17 and corollary 5.18 with these two constants.

Let C 1 = 5C 0 and α = 4 5 . Remark that C

-1

1 = (1 -α)C -1 0 . Let ω ∈ ]1, +∞[, 0 < h ≤ (C 1 ω n ) -1 , 0 < h ⊥ ≤ min( 1 3 h, d(ω))
µu(ω)+αµs 1 (ω) ln αµs 1 (ω)h µu(ω)x in u ≤ e 2 3 ln αµs 1 (ω)h µu (ω)x in u ≤ 4h 2 3 (x in u ) -2 3 (5.31) and e -αµ s1 (ω)τ * = e - αµs 1 (ω) µu (ω)+αµs 1 (ω) ln αµs 1 (ω)h µu (ω)x in u ≤ e -1 3 ln αµs 1 (ω)h µu (ω)x in u ≤ 2h -1 3 (x in u ) 1 3
(5.32)

Plugging (5.31) and (5.32) into (5.30), we get that (5.7) holds true. Recall that P s 1 ω,h , P ω = {(0, z, 0, ω) | 0 < z ≤ h} According to (5.27a), (5.27b), (5.27c) and (5.31), we have, for every τ ∈ 0, τ * , 0 ≤ x s 1 (τ ) ≤ h and

∥x(τ ) -(0, x s 1 (τ ), 0, ω)∥ ∞ ≤ 4h 2 3 h 1 3 ⊥ + 3h ⊥ ≤ 7h 1 3

⊥

(5.33) Moreover, using (5.7), we have, for every z ∈ 0,

x s 1 (τ * ) , ∥x(τ * ) -(0, z, 0, ω)∥ ∞ ≤ ∥x(τ * ) -P ω ∥ ∞ ≤ 6h 2 3 h 1 3 ⊥ + 2h ⊥ ≤ 8h 1 3 ⊥ (5.34)
Finally, for every z ∈ x s 1 (τ * ), h = x s 1 (0) , there exists τ ∈ 0, τ * such that z = x s 1 (τ ) and we can use (5.33) and (5.34) to conclude that

d H x 0, τ * , P s 1 ω,h , P ω ≤ 8h 1 3
⊥ Analogously, we recall that P ω , P u ω,h = {(z, 0, 0, ω) | 0 < z ≤ h} and we get by a straightforward computation that

d H x τ * , τ out , P ω , P u ω,h ≤ 8h 1 3 ⊥
Hence, (5.8a) and (5.8b) hold true. The proof is similar in the case i = 2 (the only difference is that µ s 1 (ω) has to be replaced by µ s 2 (ω) in (5.30) and in the left-hand side of (5.32); the estimates remain true since e

-αµ s2 (ω)τ * ≤ e -αµ s1 (ω)τ * ).

Comparison of two orbits

In this section, we are going to compare two orbits of the local vector field X ω which simultaneously intersect a section S u ω,h u . This will lead to the Lipschitz estimates (5.3) and (5.4) on Υ. Until the end of this section, we fix C 0 ≥ 2 and n 0 ∈ N large enough such that we can apply Proposition 5.17, corollary 5.18 and corollary 5.21 with these two constants. In particular, for every

C ≥ C 0 > 0, every n ≥ n 0 ∈ N, every 0 < h ≤ (Cω n ) -1 , for h s = (h, min(h, d(ω)), min(h, d(ω))), for h u = (h,
. Let ω ∈ ]1, +∞[, 0 < h ≤ (Cω n ) -1 , h s = (h, min(h, d(ω)), min(h, d(ω))), h u = (h, h, 2h
) and i ∈ {1, 2}. Let τ ↦ x(τ ) and τ ↦ x(τ ) be two orbits of the local vector field X ω . We say that (x, x) is a pair of S

s i ω,h s , S u ω,h u -synchronized orbits if 1. x(0) ∈ S s i ω,h s . 2. x intersects S s i ω,h s before x, i.e τ in (x, ω, h s ) < 0.

x and x intersect S

u ω,h u at the same time, i.e.

τ out (x, ω, h) = τ out (x, ω, h).
If this is the case, we define

x in def = x(τ in (x, ω, h s )) ∈ S s i ω,h s xin def = x(τ in (x, ω, h s )) ∈ S s i ω,h s x out def = x(τ out (x, ω, h)) ∈ S u ω,h u xout def = x(τ out (x, ω, h)) ∈ S u ω,h u
See also figure 16.

Remark 5.24. Let (x, x) be a pair of S

s i ω,h s , S
u ω,h u -synchronized orbits. Since x and x both intersect S u ω,h u , it follows that x in u > 0 and xin u > 0. With these notations, it is straightforward to check that

τ in (x, ω, h s ) = 0 τ in (x, ω, h s ) = - 1 µ u (ω) ln x in u xin u ≤ 0 τ out (x, ω, h) = τ out (x, ω, h) = 1 µ u (ω) ln h x in u Remark 5.25. If (x, x) is a pair of S s i ω,h s , S
u ω,h u -synchronized orbits, then for every τ ∈ R such that x(τ ) and x(τ ) are well defined, we have x u (τ ) = xu (τ ) (see figure 16).

Remark 5.26. Up to reparametrization, any pair of orbits which both intersect the section S

s i ω,h s is a pair of S s i ω,h s , S
u ω,h u -synchronized orbits. More precisely, let τ ↦ x(τ ) and τ ↦ x(τ ) be two orbits of the local vector field X ω which both intersect the section S s ω,h s . Up to a translation in time of x, one can assume that x(0) ∈ S s ω,h s . According to corollary 5.18, x and x both intersect the section S u ω,h u . Up to a translation in time of x, one can assume that x and x intersect simultaneously the section S u ω,h u . Up to symmetry, one can assume that x intersects the section S s ω,h s before x. With these conventions, (x, x) is a pair of S s ω,h s , S u ω,h u -synchronized orbits. Given a pair (x, x) of S s i ω,h s , S u ω,h u -synchronized orbits, the following lemma provides some estimates concerning the orbit x at the time t = 0. Lemma 5.27. There exist two constants C > 0 and n ∈ N such that for every

ω ∈ ]1, +∞[, every 0 < h ≤ (Cω n ) -1 , for h s = (h, min(h, d(ω)), min(h, d(ω))), h u = (h, h, 2h
) and for every pair (x, x) of S s ω,h s , S u ω,h u -synchronized orbits, we have the following estimates: (Estimate in the stable direction) For every i ∈ {1, 2},

xs i (0) - xin s i ≤ Cωh 1 x in u x in u - xin u (5.36)
(Estimate in the central direction)

xc (0) - xin c ≤ Cω n h xin u x in u x in u - xin u
(5.37)

x u = x in u S s ω,h s S u ω,h u x s 1 , x s 2 x u x c K • xin • x(0) • xout • x in • x out Figure 16: A pair of S s ω,h s , S u ω,h u -synchronized orbits.
Proof. The estimates (5.36) and (5.37) will be proven independently.

For every C > 0 and every n ∈ N, we denote by E C,n the set of all (ω, h, x, x) such that ω ∈ ]1, +∞[, 0 < h ≤ (Cω n ) -1 and (x, x) is a pair of synchronized orbits of the local vector field X ω (with respect to the sections S s ω,h s and S u ω,h u , where h s = (h, min(h, d(ω)), min(h, d(ω))) and h u = (h, h, 2h)). Fix 3/4 < α < 1.

Estimate of the coordinate xc

. Recall that τ in (x, ω, h s ) = -1 µ u (ω) ln x in u xin u
. Applying (5.27c) on the time interval τ in (x, ω, h s ), 0 (using a translation in time), we get that for every C ≥ C 0 , every n ≥ n 0 and every (ω, h, x, x) ∈ E C,n , we have

xc (0) - xin c = xc (0) -xc - 1 µ u (ω) ln x in u xin u ≤ C 0 ω n 0 xin u h ⎛ ⎜ ⎜ ⎜ ⎝ 1 - xin u x in u α µs 1 (ω)-µu (ω) µu (ω) ⎞ ⎟ ⎟ ⎟ ⎠
Moreover, using (2.14), we get that α

µ s1 (ω)-µ u (ω) µ u (ω) = α ω < 1. Recall that 0 < xin u x in u ≤ 1.
Hence, estimate (5.37) is a consequence of the above inequality.

Estimate of the coordinate xs

i (i ∈ {1, 2}). Let C ≥ C 0 , n ≥ n 0 and (ω, h, x, x) ∈ E C,n .
According to (5.5), xs i is a solution of the following first order linear differential equation of variable y:

y ′ = -μ ω,s i (x c )y + X s i (x)x s i where X s i (x) def = X u,s i s i (x)x u + X s 1 ,s i s i (x)x s 1 + X s 2 ,s i s i (x)x s 2
Using variation of parameters, we get an implicit expression of xs i (0), which can be written as follows:

xs i (0) - xin s i = A 1 + A 2
where

A 1 = e ∫ 0 τ in (x,ω,h s ) -μ ω,s i (x c (w)) dw -1 xin s i
and

A 2 = 0 τ in (x,ω,h s ) e ∫ 0 w -μ ω,s i (x c (σ)) dσ X s i (x(w))x s i (w) dw
Estimate of |A 1 |. Applying (5.27c) once again on the time interval τ in (x, ω, h s ), 0 , we get that for every τ ∈ τ in (x, ω, h s ), 0 ,

|x c (τ ) -ω| ≤ xc (τ ) - xin c + h ≤ 2h (5.38)
Let us fix some constants C 1 ≥ min(C 0 , 12 1-α ) and n 1 ≥ n 0 . For every C ≥ C 1 , every n ≥ n 1 and every 0 < h ≤ (Cω n )

-1 , we have 2h ≤ 1-α 6ω . It follows from (5.38) and (5.10) that for every C ≥ C 1 , every n ≥ n 1 and every (ω, h, x, x) ∈ E C,n , we have

e ∫ 0 τ in (x,ω,h s ) -μ ω,s i (x c (w)) dw -1 ≤ 1 -e α -1 µ s2 (ω)τ in (x,ω,h s ) ≤ 1 - xin u x in u µs 2 (ω) αµu(ω)
Hence,

|A 1 | ≤ h ⎛ ⎜ ⎜ ⎜ ⎝ 1 - xin u x in u µs 2 (ω) αµu (ω) ⎞ ⎟ ⎟ ⎟ ⎠
Moreover, we have the elementary fact for every 0 < z < 1 and every

v > 0, 1 -z v ≤ max(1, v)(1 -z) (5.39)
Let us apply (5.39) with z = Hence, according to (3.13) and Proposition 5.17, there exist C 2 ≥ C 1 and n 2 ≥ n 1 such that for every C ≥ C 2 , every n ≥ n 2 and every (ω, h, x, x) ∈ E C,n , we have

xin u x in u and v = µ s2 (ω) αµ u (ω) = α -1 (1 + ω) ≥ 1. It gives: |A 1 | ≤ hα -1 (1 + ω) 1 - xin u x in u ≤ 3ωh
|A 2 | ≤ C 2 ω n 2 h 2 0 τ in (x,ω,h s ) e αµ s1 (ω)w dw ≤ C 2 ω n 2 h 2 αµ s 1 (ω) ⎛ ⎜ ⎜ ⎜ ⎝ 1 - xin u x in u αµs 1 (ω) µu (ω) ⎞ ⎟ ⎟ ⎟ ⎠ Let us apply (5.39) with z = xin u x in u and v = αµ s1 (ω) µ u (ω) = α 1+ω ω .
One can remark that v < 1 for every ω large enough and µ s 1 (ω) ∼ ω→+∞ 6 ω . It follows that there exist C 3 ≥ C 2 such that for every C ≥ C 3 , every n ≥ n 3 ∶= n 2 + 1 and every (ω, h, x, x) ∈ E C,n , we have We are now going to prove the main technical result of this section. The following proposition gives Lipschitz estimates on the distance between two synchronized orbits when intersecting the section S u ω,h u . We prove that the Lipschitz constant mostly depends on the distance between their initial conditions in the section S s ω,h s and the Mixmaster attractor.

|A 2 | ≤ C 3 ω n 3 h 2 1 x in u x in u - xin u ≤ h 1 x in u x in u - xin u ( 5
Proposition 5.28 (Lipschitz estimates in the section S u ω,h u ). There exist two constants C > 0 and n ∈ N such that for every

ω ∈ ]1, +∞[, every 3/4 < α < 1, every 0 < h ≤ (1 -α)(Cω n ) -1 , every 0 < h ⊥ ≤ min(h, d(ω)), for h s = (h, h ⊥ , min(h, d(ω))), h u = (h, h, 2h
) and for every pair (x, x) of S s ω,h s , S u ω,h u -synchronized orbits, we have the following estimates:

(Lipschitz estimate in the direction transverse to the Mixmaster attractor)

x out - xout ⊥ ≤ (h ⊥ ) α ω Cω n h -α ω x in - xin ∞ (5.42)
(Lipschitz estimate in the direction tangent to the Mixmaster attractor)

x out - xout -x in - xin / / ≤ Cω n h x in - xin ⊥ + Cω n h ⊥ x in - xin / /
(5.43)

Proof. We can assume that x in ≠ xin . Otherwise, the orbits τ ↦ x(τ ) and τ ↦ x(τ ) coincide and the estimates in Proposition 5.28 are trivial.

Notation. For every C > 0 and every n ∈ N, we denote by

E C,n the set of all (ω, α, h, h ⊥ , x, x) such that ω ∈ ]1, +∞[, 3/4 < α < 1, 0 < h ≤ (1 -α)(Cω n ) -1 , 0 < h ⊥ ≤ min(h, d(ω)) and (x, x) is a pair of S s ω,h s , S u ω,h u -synchronized orbits, where h s = (h, h ⊥ , min(h, d(ω))) and h u = (h, h, 2h)).
Let C 0 > 0 and n 0 ∈ N be large enough such that we can apply Proposition 3.8 and Lemma 5.27 with these two constants. For every C ≥ C 0 , every n ≥ n 0 and every (ω, α, h, x, x) ∈ E C,n , let

τ out def = 1 µ u (ω) ln h x in u = τ out (x, ω, h) = τ out (x, ω, h) d c (τ ) def = τ 0 dx c dτ (z) - dx c dτ (z) dz for every τ ∈ 0, τ out d s (τ ) def = |x s 1 (τ ) -xs 1 (τ )| + |x s 2 (τ ) -xs 2 (τ )| for every τ ∈ 0, τ out α 0 def = (1 -α)µ u (ω) + αµ s 1 (ω) µ s 1 (ω) α ′ 0 def = 1-α 0 2 µ u (ω) + 1+α 0 2 µ s 1 (ω) µ s 1 (ω) First, remark that x out - xout ⊥ ≤ d s (τ out ). Secondly, remark that 1 -α 0 = 1 -α 1 + ω (5.44)
and

α 0 µ s 1 (ω) µ u (ω) -1 = α µ s 1 (ω) µ u (ω) -1 = α ω (5.45)
Idea of the proof. We are looking for upper bounds of d c (τ out ) and d s (τ out ). The idea is to obtain cross estimates on both d c (τ ) and d s (τ ), and then to progress step by step towards some estimates that are independent from each other.

Step 1: estimate of d s (0). According to (5.36), for every C ≥ C 0 , every n ≥ n 0 and every (ω, α, h, x, x) ∈ E C,n , we have

d s (0) ≤ 2C 0 ωh 1 x in u x in u - xin u + x in s 1 - xin s 1 + x in s 2 - xin s 2
(5.46)

Step 2: estimate of |x c (τ ) -xc (τ )| for τ ∈ [0, τ out ]. According to (5.37), for every C ≥ C 0 , every n ≥ n 0 and every (ω, α, h, x, x) ∈ E C,n , we have

|x c (τ ) -xc (τ )| ≤ d c (τ ) + |x c (0) -xc (0)| ≤ d c (τ ) + xc (0) - xin c + x in c - xin c ≤ d c (τ ) + C 0 ω n 0 h xin u x in u x in u - xin u + x in c - xin c Moreover, C 0 ω n 0 h ≤ 1 and xin u x in u ≤ 1 so |x c (τ ) -xc (τ )| ≤ d c (τ ) + x in - xin ⊥ + x in - xin / /
(5.47)

Step 3: an estimate of d s (τ ) depending on d c (τ ). From now on, τ will often be implicit in the estimates. The following estimates are valid for every τ ∈ [0, τ out ]. By definition, we have

dd s dτ = xs 1 -x s 1 |x s 1 -xs 1 | dx s 1 dτ - dx s 1 dτ + xs 2 -x s 2 |x s 2 -xs 2 | dx s 2 dτ - dx s 2 dτ
According to (5.5),

dx s 1 dτ - dx s 1 dτ = -μ ω,s 1 (x c )(x s 1 -x s 1 ) + (μ ω,s 1 (x c ) -μω,s 1 (x c ))x s 1 + X ω,s 1 (x)(x s 1 -x s 1 ) + (X ω,s 1 (x) -X ω,s 1 (x))x s 1 (5.48)
where

X ω,s 1 (x) = X u,s 1 ω,s 1 (x)x u + X s 1 ,s 1 ω,s 1 (x)x s 1 + X s 2 ,s 1 ω,s 1 (x)x s 2 . According to (5.27c), there exist C 1 ≥ C 0 and n 1 ≥ n 0 such that for every C ≥ C 1 , every n ≥ n 1 and every (ω, α, h, x, x) ∈ E C,n , we have |x c -ω| ≤ 1 -α 24(1 + ω)ω 2
(5.49) Using (5.49) with (5.44) and (5.11), we get

-μω,s 1 (x c ) ≤ -α ′ 0 µ s 1 (ω) ≤ - 1 + α 0 2 µ s 1 (ω) (5.50)
According to the expression of μω,s 1 (see (3.12)) and formulas (2.14), for every C ≥ C 1 , every n ≥ n 1 and every (ω, α, h, x, x) ∈ E C,n , we have

|μ ω,s 1 (x c ) -μω,s 1 (x c )| ≤ 6 |x c -xc | (5.51)
According to (3.13), for every C ≥ C 1 , every n ≥ n 1 and every (ω, α, h, x, x) ∈ E C,n , we have

|X ω,s 1 (x)| |x s 1 -xs 1 | ≤ C 0 ω n 0 h |x s 1 -xs 1 | |X ω,s 1 (x) -X ω,s 1 (x)| x s 1 ≤ C 0 ω n 0 (∥x -x∥ ⊥ + |x c -xc |) x s 1 (5.52)
We can estimate in the same way the terms that appear in the expression of

dx s2 dτ - dx s2
dτ . It follows from (5.48), (5.50), (5.51) and (5.52) (and similar estimates for s 2 instead of s 1 ) that, for every C ≥ C 1 , every n ≥ n 1 and every (ω, α, h, x, x) ∈ E C,n , we have

dd s dτ ≤ - 1 + α 0 2 µ s 1 (ω)d s + 6 |x c -xc | (x s 1 + x s 2 ) + C 0 ω n 0 hd s + C 0 ω n 0 (∥x -x∥ ⊥ + |x c -xc |) (x s 1 + x s 2 )
For every τ ∈ [0, τ out ], we have x u (τ ) = xu (τ ). Hence, ∥x -x∥ ⊥ ≤ d s . Using (x s 1 + x s 2 ) ≤ 2h, it follows that, there exists C 2 = max(C 1 , C 0 + 6) and n 2 = n 1 so that, for every C ≥ C 2 , every n ≥ n 2 and every (ω, α, h, x, x) ∈ E C,n , we have

dd s dτ ≤ - 1 + α 0 2 µ s 1 (ω) + 3C 2 ω n 2 h d s + 2C 2 ω n 2 |x c -xc | (x s 1 + x s 2 )
Using (2.14), we get

µ s 1 (ω) ≥ 2 ω (5.53)
According to (5.44) and (5.53), there exist C 3 ≥ 2C 2 and n 3 ≥ n 2 such that for every C ≥ C 3 , every n ≥ n 3 and every (ω, α, h, x, x) ∈ E C,n , we have

3C 2 ω n 2 h ≤ 1 -α 0 2 µ s 1 (ω)
Hence, for every C ≥ C 3 , every n ≥ n 3 and every (ω, α, h, x, x) ∈ E C,n , we have

dd s dτ ≤ -α 0 µ s 1 (ω)d s + C 3 ω n 3 |x c -xc | (x s 1 + x s 2 ) (5.54)
According to (5.44), there exist C 4 ≥ 2C 3 and n 4 ≥ n 3 such that for every C ≥ C 4 , every n ≥ n 4 and every (ω, α, h, x, x) ∈ E C,n , we can apply the estimate (5.27b) to (ω, α 0 , h, x) and obtain the following estimate of (x s 1 + x s 2 ):

(x s 1 (τ ) + x s 2 (τ )) ≤ e -α 0 µ s1 (ω)τ (x s 1 (0) + x s 2 (0)) (5.55)
Plugging (5.47) and (5.55) into (5.54) and using the fact that (x s 1 (0) + x s 2 (0)) ≤ 2h, we get that for every C ≥ C 4 , every n ≥ n 4 and every

(ω, α, h, x, x) ∈ E C,n , dd s dτ (τ ) ≤ -α 0 µ s 1 (ω)d s (τ ) + C 4 ω n 3 he -α 0 µ s1 (ω)τ (d c (τ ) + x in - xin ⊥ + x in - xin / / )
which can be rewritten in the form

e α 0 µ s1 (ω)τ dd s dτ (τ ) + α 0 µ s 1 (ω)e α 0 µ s1 (ω)τ d s (τ ) ≤ C 4 ω n 3 h d c (τ ) + x in - xin ⊥ + x in - xin / /
We recognize the derivative of e α 0 µ s1 (ω)τ d s (τ ) in the left side of the above inequality. By integrating between 0 and τ , we find:

d s (τ ) ≤ e -α 0 µ s1 (ω)τ d s (0) + C 4 ω n 3 he -α 0 µ s1 (ω)τ τ 0 d c (z) dz + τ x in - xin ⊥ + x in - xin / /
(5.56)

Step 4: an integral inequation for d c (τ ). According to (5.5) and the fact that x u (τ ) = xu (τ ) for every τ ∈ [0, τ out ], we have

dd c dτ = x u |X u,s 1 ω,c (x)(x s 1 -x s 1 ) + (X u,s 1 ω,c (x) -X u,s 1 ω,c (x))x s 1 +X u,s 2 ω,c (x)(x s 2 -x s 2 ) + (X u,s 2 ω,c (x) -X u,s 2 ω,c (x))x s 2 |
According to the estimate (3.13) on the non linear terms, there exist C 5 ≥ C 4 and n 5 ≥ n 4 such that for every C ≥ C 5 , every n ≥ n 5 and every (ω, α, h, x, x) ∈ E C,n , we have

dd c dτ ≤ C 5 ω n 5 x u (d s + |x c -xc | (x s 1 + x s 2 ))
(5.57) Plugging (5.47) and (5.55) into (5.57), using the formula x u (τ ) = e µ u (ω)τ x in u and the estimate

(x s 1 (0) + x s 2 (0)) ≤ 2h
we get that, for every C ≥ C 5 , every n ≥ n 5 and every (ω, α, h, x, x) ∈ E C,n :

dd c dτ (τ ) ≤ C 5 ω n 5 e µ u (ω)τ x in u d s (τ ) + 2C 5 ω n 5 he (µ u (ω)-α 0 µ s1 (ω))τ x in u d c (τ ) + x in - xin ⊥ + x in - xin / /
(5.58) Plugging (5.56) into (5.58), there exist C 6 ≥ C 5 and n 6 ≥ n 5 such that for every C ≥ C 6 , every n ≥ n 6 and every (ω, α, h, x, x) ∈ E C,n , we have

dd c dτ (τ ) ≤ C 6 ω n 6 e (µ u (ω)-α 0 µ s1 (ω))τ x in u d s (0) +h d c (τ ) + τ 0 d c (θ) dθ + (1 + τ ) x in - xin ⊥ + x in - xin / /
For any C ≥ C 6 , any n ≥ n 6 and any (ω, α, h, x, x) ∈ E C,n , let

T (x, x) def = sup t ≥ 0 | ∀θ ∈ [0, t] , d c (θ) ≤ x in - xin ⊥ + x in - xin / / > 0 T (x, x) def = min T (x, x), τ out
By definition of T (x, x), for every C ≥ C 6 , every n ≥ n 6 , every (ω, α, h, x, x) ∈ E C,n and every τ ∈ [0, T (x, x)], we have

dd c dτ (τ ) ≤ C 6 ω n 6 e (µ u (ω)-α 0 µ s1 (ω))τ x in u d s (0) + 2h(1 + τ ) x in - xin ⊥ + x in - xin / /
and by using formula (5.45) under the form µ u (ω) -

α 0 µ s 1 (ω) = -α(µ s 1 (ω) -µ u (ω)), we get dd c dτ (τ ) ≤ C 6 ω n 6 e -α(µ s1 (ω)-µ u (ω))τ x in u d s (0) + 2h(1 + τ ) x in - xin ⊥ + x in - xin / /
(5.59)

Using the fact that d c (0) = 0, integration of the inequality (5.59) between 0 and τ gives

d c (τ ) ≤ γ(τ ) (5.60)
where

γ(τ ) = 2C 6 ω n 6 x in u d s (0) + h x in - xin ⊥ + x in - xin / /
τ 0 e -α(µ s1 (ω)-µ u (ω))z dz

+ 2C 6 ω n 6 x in u h x in - xin ⊥ + x in - xin / / τ 0 ze -α(µ s1 (ω)-µ u (ω))z dz.
Here, the proof is essentially complete. Indeed, (5.60) is an explicit estimate on d c (τ ) and by plugging it in (5.56), we obtain an explicit estimate on d s (τ ). We are left to find an upper bound on the explicit function γ.

Step 5: Estimates of γ(τ ) for τ ∈ [0, T (x, x)]. Using (2.14), we get

µ s 1 (ω) -µ u (ω) ≥ 2 ω 2
Hence, for every C ≥ C 6 , every n ≥ n 6 , every (ω, α, h, x, x) ∈ E C,n and every τ ∈ [0, T (x, x)], we have

τ 0 e -α(µ s1 (ω)-µ u (ω))z dz ≤ 1 α(µ s 1 (ω) -µ u (ω)) ≤ 2 3 ω 2 τ 0 ze -α(µ s1 (ω)-µ u (ω))z dz ≤ 1 α 2 (µ s 1 (ω) -µ u (ω)) 2 ≤ 4 9 ω 4
(5.61)

According to (5.61), there exist C 8 ≥ C 6 and n 8 ≥ n 6 such that for every C ≥ C 8 , every n ≥ n 8 , every (ω, α, h, x, x) ∈ E C,n and every τ ∈ [0, T (x, x)], we have

γ(τ ) ≤ C 8 ω n 8 x in u d s (0) + h x in - xin ⊥ + x in - xin / /
(5.62) Plugging (5.46) into (5.62), it follows that there exist C 9 ≥ C 8 and n 9 ≥ n 8 such that for every C ≥ C 9 , every n ≥ n 9 , every (ω, α, h, x, x) ∈ E C,n and every τ ∈ [0, T (x, x)], we have

γ(τ ) ≤ C 9 ω n 9 h x in u - xin u + C 9 ω n 9 x in u x in s 1 - xin s 1 + x in s 2 - xin s 2 + x in c - xin c (5.63)
Step 6: estimate of d c and proof of (5.43). Plugging the estimates obtained in the preceding step into (5.60), it follows that for every C ≥ C 9 , every n ≥ n 9 , every (ω, α, h, x, x) ∈ E C,n and every τ ∈ [0, T (x, x)], we have

d c (τ ) ≤ C 9 ω n 9 h x in u - xin u + C 9 ω n 9 x in u x in s 1 - xin s 1 + x in s 2 - xin s 2 + x in c - xin c (5.
d c (T (x, x)) ≤ 1 2 x in - xin ⊥ + x in - xin / /
Since d c is well defined and continuous (at least) on 0, τ out , the above inequality contradicts the maximality of T (x, x). It follows that T (x, x) ≥ τ out (5.66)

By definition of d c , we have

(x out c -x out c ) -(x in c -x in c ) ≤ d c (τ out ) + xc (0) - xin c
(5.67) Plugging (5.64) into (5.67) and using (5.37), it follows that there exists C 11 ≥ C 10 and n 11 ≥ n 10 such that for every C ≥ C 11 , every n ≥ n 11 and every (ω, α, h, x, x) ∈ E C,n , we have

x out - xout -x in - xin / / ≤ C 11 ω n 11 h x in - xin ⊥ + C 11 ω n 11 h ⊥ x in - xin / /
Hence, (5.43) holds true.

Step 7: estimate of d s and proof of (5.42). According to (5.66) and (5.56), for every C ≥ C 11 , every n ≥ n 11 , every (ω, α, h, x, x) ∈ E C,n and every τ ∈ [0, τ out ], we have

d s (τ ) ≤ e -α 0 µ s1 (ω)τ d s (0) + 2C 4 ω n 3 hτ e -α 0 µ s1 (ω)τ x in - xin ⊥ + x in - xin / /
(5.68) Plugging (5.46) into (5.68), it follows that there exist C 12 ≥ C 11 and n 12 ≥ n 11 such that for every C ≥ C 12 , every n ≥ n 12 , every (ω, α, h, x, x) ∈ E C,n and every τ ∈ [0, τ out ], we have

d s (τ ) ≤ C 12 ω n 12 he -α 0 µ s1 (ω)τ x in - xin ∞ 1 x in u + τ (5.69)
It remains to evaluate this inequality for τ = τ out . According to formula (5.45), we have

x out - xout ⊥ ≤ C 12 ω n 12 h x in u h α 0 µs 1 (ω) µu (ω) x in - xin ∞ 1 x in u + 1 µ u (ω) ln h x in u ≤ C 12 ω n 12 x in u h α ω x in - xin ∞ 1 - h µ u (ω) x in u h ln x in u h
Moreover, z ↦ z ln z is bounded on [0, 1] and µ u (ω) ∼ ω→+∞ 6/ω (see (2.14)) so there exist C 13 ≥ C 12 and n 13 ≥ n 12 such that for every C ≥ C 13 , every n ≥ n 13 and every (ω, α, h, x, x) ∈ E C,n , we have

x out - xout ⊥ ≤ (h ⊥ ) α ω C 13 ω n 13 h -α ω x in - xin ∞ (5.70)
Hence, (5.42) holds true.

Control of the transition maps Υ ω,h s ,h u

Recall that d(ω) = ω -1 4

Proof of Proposition 5.1. Let C 0 > 0 and n 0 ∈ N be large enough such that we can apply corollary 5.18 and Proposition 5.28 with these two constants.

Proof of (5.1) and (5.2). There is nothing to prove when x u = 0 since in that case the left hands of the inequalities vanish (see (4.10)). For every C > 0 and every n ∈ N, we denote by E C,n the set of all (ω, h, h

⊥ , t ↦ x(t)) such that ω ∈ ]1, +∞[, 0 < h ≤ (Cω n ) -1 , 0 < h ⊥ < min(h,
def = max ω ω + 1 , 4 5
Observe that

1 -α(ω) = min 1 ω + 1 , 1 5 ≥ 1 5ω Set C 1 = 5C 0 and n 1 = n 0 + 1. Let C ≥ C 1 , n ≥ n 1 and (ω, h, h ⊥ , x) ∈ E C,n . Observe that h ≤ 1 Cω n ≤ 1 -α(ω) C 0 ω n 0
It follows that we can apply corollary 5.18 to (ω, α(ω), h, h ⊥ , t ↦ x(t)). This yields

Υ(x in ) -Υ A (x in ) ⊥ ≤ (h ⊥ ) 1+ α(ω) ω h - α(ω) ω Υ(x in ) -Υ A (x in ) / / ≤ h ⊥ hC 0 ω n 0
Moreover, we have

α(ω) ω < 1, 1 + α(ω) ω ≥ ω + 2 ω + 1 , and 0 < h ⊥ < 1. Hence, (h ⊥ ) 1+ α(ω) ω h - α(ω) ω ≤ h ω+2 ω+1 ⊥ h -1
This concludes the proof of (5.1) and (5.2).

Continuity of Υ. Recall that for every z ∈ S

s ω,h s such that z u = 0, we have Υ(z) = (h, 0, 0, z c ) (see (4.10)). According to (5.1) and (5.2), for every z ∈ S s ω,h s such that z u = 0, we have

lim x→z Υ(x) = (h, 0, 0, z c ) = Υ(z) so Υ is continuous at z.
Proof of (5.3) and (5.4). For every C > 0 and every n ∈ N, we denote by F C,N the set of all For every C ≥ C 1 , every n ≥ n 1 and every (ω, h, h ⊥ , x, x) ∈ F C,N , we have

(ω, h, h ⊥ , t ↦ (x, x)(t)) such that ω ∈ ]1, +∞[, 0 < h ≤ (Cω n ) -1 , 0 < h ⊥ < min(h, d(ω)) and t ↦ (x, x)(t) is a pair of S s ω,h s , S
h ≤ (Cω n ) -1 ≤ 1 -α(ω) C 0 ω n 0
Hence, we can apply Proposition 5.28 to (ω, α(ω), h, h ⊥ , x), which yields

Υ(x in ) -Υ(x in ) ⊥ ≤ (h ⊥ ) α(ω) ω C 0 ω n 0 h - α(ω) ω x in - xin ∞ Υ(x in ) -Υ(x in ) -(x in - xin ) / / ≤ C 0 ω n 0 h x in - xin ⊥ + C 0 ω n 0 h ⊥ x in - xin / /
One can remark that there exists 4/5 < d < 1 such that for every ω ∈ ]1, +∞[,

α(ω) ω ≤ d. Moreover, for every C 2 ≥ C 1 and every n 2 ≥ n 1 such that C 1-d 2 ≥ C 0 and n 2 (1 -d) ≥ n 0 every ω ∈ ]1, +∞[ and every 0 < h ≤ (C 2 ω n 2 ) -1 , we have C 0 ω n 0 h - α(ω) ω ≤ h -1 . Hence, for every C ≥ C 2 , every n ≥ n 2 and every (ω, h, h ⊥ , x, x) ∈ F C,N , we have Υ(x in ) -Υ(x in ) ⊥ ≤ h 1 ω+1 ⊥ h -1 ∥x -x∥ ∞ Υ(x in ) -Υ(x in ) -(x in - xin ) / / ≤ C 0 ω n 0 h x in - xin ⊥ + C 0 ω n 0 h ⊥ x in - xin / / Since Υ(x in ) -Υ(x in ) ⊥ = Υ(x in ) -Υ(x in ) -Υ A (x in ) -Υ A (x in ) ⊥ Υ(x in ) -Υ(x in ) -(x in - xin ) / / = Υ(x in ) -Υ(x in ) -Υ A (x in ) -Υ A (x in ) / /
this concludes the proof of (5.3) and (5.4).

Dynamics in the neighbourhood of a type II orbit

The goal of this section is to give some estimates on the transition map Ψ ω,h u ,h s (see definition 4.24). We will show that this map is "very close" to the Kasner map f . Recall that Ψ ω,h u ,h s describes the behaviour of the orbits of the Wainwright-Hsu vector field X in the neighbourhood of the type II orbit O P ω →P f (ω) . More precisely, Ψ ω,h u ,h s is the transition map from the section S u ω,h u (which intersects O P ω →P f (ω) close to its "initial point" P ω ) to the section S s f (ω),h s (which intersects O P ω →P f (ω) close to its "final point" P f (ω) ) Observe that the situation is quite different from the one of Section 5. We are no more studying the local dynamics of a vector field in the vicinity of a singular point, but rather the large scale dynamics of a non-linear vector field. As a consequence, the estimates proven here for the map Ψ ω,h u ,h s will be far less precise than the ones obtained in Proposition 5.1 for the map Υ ω,h s ,h u .

Define, for any

ω ∈ ]1, +∞[ \ {2}, i(ω) def = 1 if ω > 2 2 if 1 < ω < 2 (6.1) Recall that for any ω ∈ ]1, +∞[ \ {2}, the type II orbit O P ω →P f (ω) is tangent to the direction ∂ x s i(ω)
at the final point P f (ω) . As a consequence, by continuity of the flow, if the section S u ω,h u is sufficiently small, the orbits starting in S u ω,h u will intersect the section S s f (ω),h s for the first time in S s i(ω) f (ω),h s . For a technical reason explained below, we will often encounter the quantity min 1, (ω -2)

2 in the estimates. Hence, we introduce the notation

m(ω) def = min 1, (ω -2) 2
Recall that Proj A is the projection on the Mixmaster attractor (see definition 4.1) and recall that Ψ A ω,h u ,h s = Ψ ω,h u ,h s • Proj A . Moreover, the map Ψ A ω,h u ,h s admits an explicit expression (see (4.12)). We can now give a formal statement of the main results of this section.

Proposition 6.1 (Control of the transition map Ψ ω,h u ,h s ). There exist two constants C1 ≥ 1 and ñ1 ∈ N such that the properties below hold for

ω ∈ ]1, +∞[ \ {2}, 0 < h u ≤ ( C1 ω ñ1 ) -1 , 0 < h s ≤ ( C1 f (ω) ñ1 ) -1 , h = min h u , h s , 0 < h ⊥ ≤ h C1 ω m(ω), h u = h u , h ⊥ , h C1 ω m(ω) and h s = h s , h s , h s . The transition map Ψ ω,h u ,h s ∶ S u ω,h u → S s f (ω),
dist ∞ (Ψ(y), A) = Ψ(y) -Ψ A (y) ⊥ ≤ h ⊥ h -C1 ω (6.2)
(Control of the drift tangential to the Mixmaster attractor)

Ψ(y) -Ψ A (y) / / ≤ h ⊥ h -C1 ω (6.3)
(Lipschitz control in the direction transverse to the Mixmaster attractor)

(Ψ(y) -Ψ(ỹ)) -Ψ A (y) -Ψ A (ỹ) ⊥ ≤ ∥y -ỹ∥ ⊥ + h ⊥ ∥y -ỹ∥ / / h -C1 ω (6.4)
(Lipschitz control in the direction tangent to the Mixmaster attractor)

(Ψ(y) -Ψ(ỹ)) -Ψ A (y) -Ψ A (ỹ) / / ≤ ∥y -ỹ∥ ⊥ + h ⊥ ∥y -ỹ∥ / / h -C1 ω (6.5)
Remark 6.2. Proposition 6.1 describes the behaviour of the orbits of the Wainwright-Hsu vector field X traveling from a section S u ω,h u to a section S s f (ω),h s . The vector field is non-linear and the traveling time is very long (it tends to infinity as h → 0 or ω → +∞). As a consequence, to ensure that an orbit starting in S u ω,h u will cut the section S s f (ω),h s , the size of the section S u ω,h u must be very small. This is why, in Proposition 6.1, the size

h C1 ω m(ω)
of the section S u ω,h u is "extremely small" compared to the parameters h u and h s , especially when ω is very large, i.e. when the type II orbit O P ω →P f (ω) is "close" to the Taub point.

Remark 6.3. The quantity m(ω) appears in the upper bound of the size of the section S u ω,h u for some purely technical reasons. If ω = 2, the type II orbit O u ω arrives at the point P f (ω) of Kasner parameter f (ω) = 1. However, the local coordinate system ξ = (x u , x s 1 , x s 2 , x c ) is not defined in the neighbourhood of this point. For this reason, we do not want the section S u ω,h u to cross the hyperplane ω = 2.

The second result of this section will be used in Section 10 to prove that certain orbits shadow a heteroclinic chain. Proposition 6.4 (Shadowing of a type II orbit). Let C1 and ñ1 be the constants defined in Proposition 6.1. For every > 0, there exists η > 0 such that for

ω ∈ ]1, +∞[ \ {2}, 0 < h u ≤ ( C1 ω ñ1 ) -1 , 0 < h s ≤ ( C1 f (ω) ñ1 ) -1 , h = min h u , h s , h u = h u , ηh C1 ω m(ω), ηh C1 ω m(ω) and h s = h s , h s , h s ,
the Hausdorff distance between two (minimal) orbit segments joining the section S u ω,h u and the section S s f (ω),h s (in that order) is less than . We now define a hitting time with the section S s f (ω),h s for the orbits in B + .

Definition 6.5 (Hitting time). Let

ω ∈ ]1, +∞[ \ {2}, h s > 0 and h s = (h s , h s , h s ). Assume that S s f (ω)
,h s is included in the range of the local coordinates ξ, so that the geometrical section S s f (ω),h s is well defined. For every q ∈ B + , we define

τ ω,h s (q) def = inf t > 0 | X t (q) ∈ S s f (ω),h s ∈ ]0, +∞]
Remark 6.6. With the notation of Proposition 6.1, for q ∈ S u ω,h u , τ ω,h s (q) is the traveling time between q and its image by the transition map Ψ ω,h u ,h s . In particular, τ ω,h s (P u ω,h u ) is the traveling time of the type II orbit O P ω →P f (ω) between the sections S u ω,h u and S s f (ω),h s .

Organization of the proof of Proposition 6.1. The main difficulty is to find some estimates on the traveling time τ ω,h s . Once we will have proven these estimates on τ ω,h s , we will easily deduce the estimates on the transition map Ψ ω,h u ,h s using Gronwall's lemma. To study τ ω,h s , we proceed as follows:

1. We first obtain an estimate on τ ω,h s (P u ω,h u ) using directly the Wainwright-Hsu equations (1.3c). This is possible because τ ω,h s (P u ω,h u ) is the traveling time of the type II orbit O P ω →P f (ω) between the sections S u ω,h u and S s f (ω),h s and this orbit is explicit.

2. Then we construct a flow box in the neighbourhood of the point P s i(ω) f (ω),h s and we bound the flow box coordinates. Recall that P

s i(ω) f (ω),h s = X τ ω,h s (P u ω,h u ) (P u ω,h u ) = ξ -1 • Ψ ω,h u ,h s • ξ(P u ω,h u ).
3. Finally, we use a formula for Ψ ω,h u ,h s depending on X , the traveling time τ ω,h s (P u ω,h u ) and the flow box to get the desired estimates on Ψ ω,h u ,h s .

Traveling time of type II orbits

Recall that τ ω,h s (P u ω,h u ) is the traveling time of the type II orbit O P ω →P f (ω) from the section S u ω,h u to the section S s f (ω),h s . Proposition 6.7 (Estimates on the traveling time of type II orbits). There exist two constants C2 > 0 and ñ2 ∈ N such that for every

ω ∈ ]1, +∞[, every 0 < h u ≤ ( C2 ω ñ2 ) -1 , every 0 < h s ≤ ( C2 f (ω) ñ2 ) -1
and for h = min h u , h s , the traveling time satisfies

ω C2 ≤ τ ω,h s (P u ω,h u ) ≤ C2 ω ln 1 h (6.6)
Proof. According to Proposition 3.2, there exist C 0 > 0 and n 0 ≥ 1 such that for any ω ∈ ]1, +∞[, the range U ξ of the local coordinate system contains the ball B ω,C 0 ,n 0 . We can and we will assume that

C 0 ≥ 2000. Let ω ∈ ]1, +∞[, 0 < h u ≤ (C 0 ω n 0 ) -1 , 0 < h s ≤ (C 0 f (ω) n 0 ) -1 and h = min h u ,
O P ω →P f (ω) has six lifts in B + .
Two of these lifts are such that N 1 > 0, N 2 = 0 and N 3 = 0. We choose one, denoted by t ↦ O(t) = (N 1 (t), 0, 0, Σ 1 (t), Σ 2 (t), Σ 3 (t))

Using a time translation, we can and we will assume that O(0) is a lift of

P u ω,h u . This property is equivalent to N 1 (0) = h u and N ′ 1 (0) > 0. With this parametrization, τ ω,h s (P u ω,h u ) is the unique time T verifying N 1 (T ) = h s and N ′ 1 (T ) < 0. Moreover, O(τ ω,h s (P u ω,h u )) is a lift of P s i(ω)
f (ω),h s . See figure 17. Denote by P ω the lift of P ω such that O starts at P ω , i.e. lim t→-∞ O(t) = P ω . Recall that near the point P ω , N 1 = x u , while near the point F (P ω ), N 1 = x s i(ω) .

Recall the evolution equations

N ′ 1 = -(q + 2Σ 1 )N 1 (6.7) Σ ′ 1 = 1 6 N 2 1 (Σ 1 + 4) (6.8) where q = 1 3 Σ 2 1 + Σ 2 2 + Σ 2 3 .
To control the traveling time, one must control the quantities q + 2Σ 1 and N 2 1 . Next lemma shows that these two quantities cannot be simultaneously "too small". Claim 1. For every point

(N 1 , 0, 0, Σ 1 , Σ 2 , Σ 3 ) in the type II orbit O, either N 1 > 1 1000ω or |q + 2Σ 1 | ≥ 1 ω . Proof of claim 1. Let (N 1 , 0, 0, Σ 1 , Σ 2 , Σ 3 ) be a point in the type II orbit O.
Let M = (0, 0, 0, Σ 1 , Σ 2 , Σ 3 ) be its projection onto the (Σ 1 , Σ 2 , Σ 3 )-plane. Denote by d the Euclidean distance on R 6 . The proof essentially follows from the formula

q + 2Σ 1 = 1 3 d(M, Q 1 ) 2 -2 (6.9) t N 1 (t) 0 - h u - h s T end - 1 1000ω t 2 t 1 Figure 17: Graph of t ↦ N 1 (t). T end = τ ω,h s (P u ω,h u ).
which proves that q + 2Σ 1 varies as a squared distance. At the point P ω , the quantity -(q + 2Σ 1 ) coincides with the unstable eigenvalue of the Wainwright-Hsu vector field µ u (ω). So, using (2.14a) and (6.9), one gets 1 3

d(P ω , Q 1 ) 2 -2 ≥ 2 ω (6.10)
Analogously, at the point F (P ω ), the quantity -(q +2Σ 1 ) coincides with the eigenvalue µ s 1 (F (P ω )) if f (ω) = ω -1, and with the eigenvalue µ s 2 (F (P ω )) if f (ω) = 1 ω (see Remark 2.9). So, using (2.14b), (2.14c) and (6.9), 1 3

d(F (P ω ), Q 1 ) 2 -2 ≥ 2 ω (6.11)
Recall the constraint equation (2.4):

6 -3q = 1 2 N 2 1 (6.12)
and observe that 3q is the square of the distance between the point M and the center of the Kasner circle and 6 is the square of the radius of the Kasner circle. The constraint equation (6.12) implies that, if N 1 is small, then M is very close to the Kasner circle. Since M belongs to the projection of the type II orbit O, M must be close to one of the two end points P ω and F (P ω ). More precisely, one easily checks that if N 1 ≤ 1 1000ω , then min (d (M, P ω ) , d (M, F (P ω ))) ≤ 1 100ω (6.13) Using (6.10), (6.11) and (6.13), we get that if

N 1 ≤ 1 1000ω , then 1 3 d(M, Q 1 ) 2 -2 ≥ 1 ω .
The claim follows from (6.9). We know that N 1 is increasing and then decreasing along the type II orbit. Moreover, recall that N 1 (0) = h u and N 1 (τ ω,h s (P

u ω,h u )) = h s (see figure 17). Hence, |N 1 (0)| < 1 1000ω and |N 1 (τ ω,h s (P u ω,h u ))| < 1 1000ω . It follows that there exist 0 < t 1 < t 2 < τ ω,h s (P u ω,h u ) such that 1. On [0, t 1 ], N 1 is increasing and N 1 (t) ≤ 1 1000ω . 2. On ]t 1 , t 2 [, N 1 (t) > 1 1000ω . 3. On t 2 , τ ω,h s (P u ω,h u ) , N 1 is decreasing and N 1 (t) ≤ 1
1000ω . Upper bound for t 1 and τ ω,h s (P u ω,h u ) -t 2 . Using the evolution equation (6.7) and claim 1 on [0, t 1 ], we get that for every t ∈ [0,

t 1 ], N ′ 1 (t) ≥ 1 ω N 1 (t)
. By integrating this inequality between 0 and t 1 , we get Claim 2. For every point (N 1 , 0, 0, Σ 1 , Σ 2 , Σ 3 ) in the type II orbit O, we have

t 1 ≤ ω ln 1 h u ≤
N 1 ≤ 100 ω (6.16)
Proof of claim 2. Let (N 1 , 0, 0, Σ 1 , Σ 2 , Σ 3 ) be a point in the type II orbit O.

Let M = (0, 0, 0, Σ 1 , Σ 2 , Σ 3 ) be its projection onto the (Σ 1 , Σ 2 , Σ 3 )-plane. The projection of O onto the (Σ 1 , Σ 2 , Σ 3 )-plane is explicitly known: it is the chord whose end points are P ω and F (P ω ). Using the coordinates of P ω and F (P ω ), one can get that d(P ω , F (P ω )) ≤ 18 2 ω . Hence, d(M, K ) ≤ 100 ω 2 . Recall that 3q is the square of the distance between the point M and the center of the Kasner circle and 6 is the square of the radius of the Kasner circle. It follows that 3q ≥ 6 -1000 ω 2 and, using the constraint equation (6.12), we get N 1 ≤ 100 ω . This concludes the proof of claim 2. We are left to find some lower and upper bounds for the variation of Σ 1 on ]t 1 , t 2 [. According to the constraint equation (6.12), q(t 1 ) = q(t 2 ). According to claim 1, (q + 2Σ 1 ) (t 2 ) ≥ 1 ω and (q

+ 2Σ 1 ) (t 1 ) ≤ -1 ω . Hence, Σ 1 (t 2 ) -Σ 1 (t 1 ) ≥ 1 ω (6.17)
Moreover, Σ 1 is increasing along the type II orbit and its variation Σ 1 (t 2 ) -Σ 1 (t 1 ) is smaller than its variation between P ω and F (P ω ). Using (2.14), we get

Σ 1 (t 2 ) -Σ 1 (t 1 ) ≤ 12 ω (6.18)
Using the estimate (6.16), the fact that 2 ≤ Σ 1 + 4 ≤ 6 and the evolution equation (6.8), we get that for every t ∈ ]t 1 , t 2 [,

1 10 7 ω 2 ≤ Σ ′ 1 (t) ≤ 10 4 ω 2 (6.19)
Integrating (6.19) between t 1 and t 2 , estimates (6.17) and (6.18) give ω 10 4 ≤ t 2 -t 1 ≤ 10 9 ω (6.20)

Estimates (6.14), (6.15) and (6.20) give the desired control on τ ω,h s (P u ω,h u ). This concludes the proof with C2 ∶= max C 0 , 10 10 and ñ2 ∶= n 0 .

Construction of a flow box

Given ω ∈ ]1, +∞[, i ∈ {1, 2} and a small constant h > 0, we are going to construct a flow box in a neighbourhood of the point P s i ω,h . The usual flow box theorem states that, since P s i ω,h is a non singular point for X , there exists a neighbourhood of P s i ω,h (called a "flow box") and a local coordinate system on this neighbourhood such that the integral curves of the vector field X are parallel straight lines in this local coordinate system.

The following lemma, in addition to givng a precise statement of the flow box theorem in our context, gives estimates concerning the size of the flow box and the C 2 -norm of the local coordinate system.

To study the map map Ψ ω,h u ,h s , we will apply this lemma at f (ω) instead of ω.

Lemma 6.8 (Construction of a flow box).

There exist two constants C3 ≥ C2 and ñ3 ≥ ñ2 such that for every ω ∈ ]1, +∞[, every i ∈ {1, 2}, every 0 < h ≤ ( C3 ω ñ3 ) -1 , for

r box = min h 2 ( C3 ω ñ3 ) -1 , ω -1 2 h box = (h, r box , r box )
there exist a neighbourhood V ω,h of P

s i ω,h in B + and a C 2 -diffeomorphism θ ω,h ∶ V ω,h → -r box , r box × 0, r box 2 × -r box , r box ⊂ R 4
with the following properties. If we denote by (x 1 , x 2 , x 3 , x 4 ) the coordinates on the space R 4 where θ ω,h takes its values, then 1. θ ω,h P s i ω,h = (0, 0, 0, 0).

V

∥Dθ ω,h ∥ ∞ ≤ C3 ω ñ3 h 2 D 2 θ ω,h ∞ ≤ C3 ω ñ3 h 6 (6.21) 5. The C 2 -norm of θ -1
ω,h admits an upper bound which is polynomial in ω. More precisely:

θ -1 ω,h C 2 ≤ C3 ω ñ3 (6.22) 6. For every 0 < r ′ ≤ r box , the set θ -1 ω,h -r ′ , r ′ × 0, r ′ 2 × -r ′ , r ′ contains the ball B P s i ω,h , r ′ h 2 ( C3 ω ñ3 ) -1 open in U ξ and θ -1 ω,h {0} × 0, r ′ 2 × -r ′ , r ′ = S s i ω,(h,r ′ ,r ′ ) ⊂ V ω,h (6.23)
Remark 6.9. Items 2 and 3 imply that for every y ∈ V ω,h , -x 1 (θ ω,h (y)) is the unique time t ∈ -r box , r box such that X t (y) ∈ S s i ω,h box . In particular, for a flow box around the point P

s i(ω) f (ω),h , if -x 1 (θ f (ω),h (y)) > 0, then τ ω,h (y) = -x 1 (θ f (ω),h (y)).
In order to make the proof of Lemma 6.8 easier to read, we extract here an independant result that will be used in the course of the proof. Roughly speaking, this result states that the orbits of the Wainwright-Hsu vector field X crossing a section S s i ω,h do not cross it again "too fast". Lemma 6.10 (No loop in small time). There exist two constants C > 0 and n ∈ N such that the following property holds for ω ∈ ]1, +∞[ and 0 < h ≤ (Cω n ) -1 . Let t ↦ q(t) be an orbit of the Wainwright-Hsu vector field X whose initial condition q(0) belongs to the section S s ω,h where h = h, h 2 , h 2 . Then, q is well defined (at least) on the time interval 0, ln 2 12 and does not cross the section S s ω,h for t ∈ 0, ln 2 12 . Proof. Let C 0 > 0 and n 0 ∈ N be large enough such that we can apply Proposition 5.17, Proposition 5.20 and corollary 5.21 with these two constants. Let ω ∈ ]1, +∞[, 0 < h ≤ (C 0 ω n 0 ) -1 and t ↦ q(t) be an orbit of the vector field X whose initial condition q(0) belongs to the section S s ω,h

where h = h, h 2 , h 2 . Let y in ∶= ξ(q(0)) ∈ S
s ω,h and denote by t ↦ y(t) the orbit of the vector field X = ξ * X with initial condition y(0) = y in . Remark that y = ξ • q whenever y is well defined.

Case y in u = 0. In that case, the orbit y converges exponentially fast to the point (0, 0, 0, y c ) and according to (5.29), for every t > 0 and every i ∈ {1, 2}, y i (t) < h/2. This implies that for every t > 0, y(t) ∉ S s ω,h . Hence, q does not cross the section S s ω,h for t > 0. Case y in u > 0. Denote by t ↦ x(t) the orbit of the renormalized local vector field X ω = γ ω .X (see definition 3.10) with initial condition x(0) = y in . Remark that x is a reparametrization of the orbit y.

According to Proposition 5.17, x is at least defined for t ∈ [0, τ loc ] where

τ loc = 1 µ u (ω) ln h y in u ≥ ln 2 6
Using the estimate (3.27) concerning the renormalization function γ ω , we get that the orbit y is at least defined for t ∈ 0, ln 2 12 and there exists a C 1 -map s ∶ 0, ln 2 12 → 0, ln 2 6 such that s(0) = 0 and for every t ∈ [0, ln 2 12 ], y(t) = x(s(t)). Moreover, x intersects the section S s ω,h at most one time (see corollary 5.21) so y intersects the section S s ω,h at most one time on 0, ln 2 12 . It follows that q intersects the section S s ω,h at most one time on 0, ln 2 12 . This concludes the proof. Proof of Lemma 6.8. To fix the ideas, we will only treat the case i = 2, that is, we will construct a flow box around the point P s 2 ω,h = ξ -1 (0, 0, h, ω). For every C > 0 and every n ∈ N, we denote by E C,n the set of all (ω, h) such that ω ∈ ]1, +∞[ and 0 < h ≤ (Cω n ) -1 . Let C 0 ≥ 100 and n 0 ∈ N be large enough such that we can apply Proposition 3.2 and Lemma 6.10 with these two constants.

We will use several times in this proof that the vector field X is bounded on every compact subset of B for the C 2 -norm. In particular, even if it means taking C 0 larger, we can assume that ∥X ∥ C 2 ≤ C 0 on a compact set containing all the orbits playing a role in this proof.

For every C ≥ C 0 , every n ≥ n 0 and every (ω, h) ∈ E C,n , let

r def = min h 2 (Cω n ) -1 , ω -1 2 h def = (h, r, r) D def = [0, r] 2 × [-r, r] let χ∶ D → S s 2 ω,h (x u , x s 1 , x c ) ↦ ξ -1 (x u , x s 1 , h, x c + ω) and let ϕ∶ [-r, r] × D → U ξ (t, z) ↦ X t (χ(z))
where X t denotes the flow of the Wainwright-Hsu vector field X . The map χ is a bijective C 2 parametrization of the section S s 2 ω,h such that χ(0) = P s 2 ω,h . The map ϕ is a C 2 map such that, for every z ∈ D, t ↦ ϕ(t, z) is a (local) parametrization of the orbit of the Wainwright-Hsu vector field X passing through the point χ(z) ∈ S s 2 ω,h at t = 0. Note that the domain of ϕ depends on C, n and (ω, h). Roughly speaking, the map θ ω,h will be obtained as the inverse of ϕ.

Claim 1. For every C ≥ C 0 , every n ≥ n 0 and every

(ω, h) ∈ E C,n , ϕ is injective on [-r, r] × D. Proof of claim 1. Let C ≥ C 0 , n ≥ n 0 and (ω, h) ∈ E C,n . Let (t, z), (t ′ , z ′ ) ∈ [-r, r] × D and assume that ϕ(t, z) = ϕ(t ′ , z ′ )
. By symmetry, one can assume that t ≤ t ′ . We have χ(z) = X t ′ -t (χ(z ′ )) and since r < ln 2 24 , we have 0 ≤ t ′ -t ≤ ln 2 12 . According to Lemma 6.10, we necessarily have t = t ′ . It follows that χ(z) = χ(z ′ ) and since χ is injective, we have z = z ′ . It follows that ϕ is injective.

Claim 2. There exist C 1 ≥ C 0 and n 1 ≥ n 0 such that for every C ≥ C 1 , every n ≥ n 1 and every

(ω, h) ∈ E C,n , ∥ϕ∥ C 2 ≤ C 1 ω n 1 . Proof of claim 2. Let C ≥ C 0 , n ≥ n 0 and (ω, h) ∈ E C,n .
The first and second derivatives of the flow (t, y) ↦ X t (y) are controlled by the C 2 -norm of X (which is bounded by C 0 ) and the size of the time interval on which we study the flow. This time interval is [-r, r] so its size is bounded independantly of (ω, h). Moreover, according to the estimate (3.5) about the adapted system of local coordinates ξ, we have ∥χ∥ C 2 ≤ C 0 ω n 0 . Since ϕ(t, z) = X t (χ(z)), this leads to the desired result.

Claim 3. There exist C 2 ≥ C 1 and n 2 ≥ n 1 such that for every C ≥ C 2 , every n ≥ n 2 and every (ω, h) ∈ E C,n , the derivative Dϕ(0) is invertible and (Dϕ(0))

-1 ≤ C 2 ω n2 h 2 Proof of claim 3. Let C ≥ C 1 , n ≥ n 1 and (ω, h) ∈ E C,n . Observe that Dϕ(0) = X (P s 2 ω,h ) | ∂ξ -1 ∂x u P s 2 ω,h | ∂ξ -1 ∂x s1 P s 2 ω,h | ∂ξ -1 ∂x c P s 2 ω,h
and

X (P s 2 ω,h ) = Dξ -1 (P s 2 ω,h ).X P s 2 ω,h
Recall from the formula (3.6) that X P s 2 ω,h is collinear to the vector ∂ ∂x s2

. It follows that

X (P s 2 ω,h ) = a ∂ξ -1 ∂x s 2 P s 2 ω,h (6.24)
for a certain a ∈ R and |det Dϕ(0)| = |a| det Dξ -1 (P

s 2 ω,h ) (6.25)
According to (3.5), there exist

C ′ 1 ≥ C 1 and n ′ 1 ≥ n 1 such that for every C ≥ C ′ 1 , n ≥ n ′ 1 , for every (ω, h) ∈ E C,n , det Dξ -1 (P s 2 ω,h ) ≥ 1 C ′ 1 ω n ′ 1 (6.26)
and ∂ξ

-1 ∂x s 2 P s 2 ω,h ≤ C ′ 1 ω n ′ 1 (6.27)
According to (3.1a) and the expression of the vector field X induced by (1.3c), the Σ s 2 -coordinate of the vector X (P

s 2 ω,h ) is Σ s 2 X (P s 2 ω,h ) = 1 6 N s 2 P s 2 ω,h 2 Σ s 2 P s 2 ω,h + 4 = 1 6 h 2 Σ s 2 P s 2 ω,h + 4 ≥ 1 3 h 2 so X (P s 2 ω,h ) ≥ h 2 3 (6.28)
Using (6.24), (6.25), (6.26), (6.27) and (6.28), we find that for every

C ≥ C ′ 1 , every n ≥ n ′ 1 and every (ω, h) ∈ E C,n , we have |det Dϕ(0)| ≥ X (P s 2 ω,h ) ∂ξ -1 ∂x s2 P s 2 ω,h det Dξ -1 (P s 2 ω,h ) ≥ h 2 3 C ′ 1 ω n ′ 1 2 (6.29)
In particular, Dϕ(0) is invertible. Denote by t Co(A) the adjugate of a square matrix A. Using (6.29), the standard formula (Dϕ(0))

-1 = 1 det Dϕ(0) t Co(Dϕ(0))
and claim 2, it follows that there exist C 2 ≥ C 1 and n 2 ≥ n 1 such that for every C ≥ C 2 , every n ≥ n 2 and every (ω, h) ∈ E C,n , Dϕ(0) is invertible and (Dϕ(0)) -1 ≤ C 2 ω n2 h 2 . The next claim relies on a standard argument for the local inversion theorem. Denote

V ω,h def = ϕ ([-r, r] × D)
Claim 4. There exist C 3 ≥ C 2 and n 3 ≥ n 2 such that for every C ≥ C 3 , every n ≥ n 3 and every

(ω, h) ∈ E C,n , ϕ is a C 2 -diffeomorphism from [-r, r] × D onto V ω,h and Dϕ -1 ∞ ≤ C 3 ω n3 h 2 . Proof of claim 4. Let C ≥ C 2 , n ≥ n 2 and (ω, h) ∈ E C,n . Let u = Dϕ(0) and η = Dϕ(0)-Dϕ. We have Dϕ = u(Id -u -1 η).
According to the claim 2 and the mean value theorem,

∥η∥ ∞ ≤ C 1 ω n 1 r. According to the claim 3, u -1 ≤ C 2 ω n2 h 2 . It follows that, for every C ≥ C 3 ∶= 2C 1 C 2 , every n ≥ n 3 ∶= n 1 +n 2 and every (ω, h) ∈ E C,n , u -1 η ∞ ≤ 1 2 . Hence, for every C ≥ C 3 , every n ≥ n 3 and every (ω, h) ∈ E C,n , Dϕ is invertible on [-r, r] × D and Dϕ -1 ∞ ≤ 2C 2 ω n2 h 2 ≤ C 3 ω n3 h 2 .
Recall that claim 1 implies that ϕ is one-to-one. So, according to the global inversion theorem,

ϕ is a C 2 -diffeomorphism from [-r, r] × D to V ω,h .
Let us denote by θ the inverse of ϕ. By construction, it is clear that θ P s 2 ω,h = (0, 0, 0, 0). Next claim is also a standard computation for the local inversion theorem.

Claim 5. There exist C 4 ≥ C 3 and n 4 ≥ n 3 such that for every C ≥ C 4 , every n ≥ n 4 and every

(ω, h) ∈ E C,n , ∥Dθ∥ ∞ ≤ C 4 ω n4 h 2 and D 2 θ ∞ ≤ C 4 ω n4 h 6 . Proof of claim 5. Let C ≥ C 3 , n ≥ n 3 and (ω, h) ∈ E C,

Claim 6. For every

C ≥ C 4 , every n ≥ n 4 , every (ω, h) ∈ E C,n and every 0 < r ′ ≤ r, θ -1 -r ′ , r ′ × 0, r ′ 2 × -r ′ , r ′ contains the open ball B P s 2 ω,h , r ′ h 2 ( C3 ω ñ3 ) -1 in U ξ and θ -1 ω,h {0} × 0, r ′ 2 × -r ′ , r ′ = S s 2 ω,(h,r ′ ,r ′ ) ⊂ V ω,h Proof of claim 6. Let C ≥ C 4 , n ≥ n 4 , (ω, h) ∈ E C,n and 0 < r ′ ≤ r.
Let us denote by R the supremum of every δ > 0 such that

B(P s 2 ω,h , δ) ⊂ θ -1 -r ′ , r ′ × 0, r ′ 2 × -r ′ , r ′ Recall that θ P s 2
ω,h = (0, 0, 0, 0). Using the mean value theorem and claim 5, we get that

r ′ ≤ ∥Dθ∥ ∞ R ≤ C 4 ω n 4 h 2 R Hence, R ≥ r ′ h 2 C 4 ω n 4 Moreover, θ -1 ω,h {0} × 0, r ′ 2 × -r ′ , r ′ = ϕ {0} × 0, r ′ 2 × -r ′ , r ′ = χ 0, r ′ 2 × -r ′ , r ′ = ξ -1 0, r ′ 2 × {h} × -r ′ , r ′ = S s 2 ω,(h,r ′ ,r ′ ) ⊂ V ω,h
This concludes the proof of claim 6.

As a particular case with r ′ = r, it follows from claim 6 that θ ω,h maps the section S

s 2 ω,h to {0} × [0, r] 2 × [-r, r]. Moreover, by definition of ϕ, ∂ϕ ∂t (t, z) = X (ϕ(t, z)) so Dθ(ϕ(t, z))X (ϕ(t, z)) = ∂ ∂x 1
. Hence, θ ω,h straightens the vector field X onto the vector field ∂ ∂x 1

. This shows that Lemma 6.8 holds with C3 ∶= C 4 and ñ3 ∶= n 4 .

Hitting time

Lemma 6.11 (Hitting time). There exist two constants C4 ≥ C3 and ñ4 ≥ ñ3 such that the properties below hold for

ω ∈ ]1, +∞[ \ {2}, 0 < h u ≤ ( C4 ω ñ4 ) -1 , 0 < h s ≤ ( C4 f (ω) ñ4 ) -1 , 0 < η ≤ 1, h = min h u , h s , h u = h u , ηh C4 ω m(ω), ηh C4 ω m(ω) and h s = h s , h s , h s .
1. For every q ∈ S u ω,h u , the forward X -orbit of q intersects the section S s f (ω),h s and its first intersection point belongs to S

s i(ω) f (ω),h s . Moreover, τ ω,h s (q) = τ ω,h s (P u ω,h u ) -x 1 θ f (ω),h s X τ ω,h s (P u ω,h u ) (q) (6.30)
2. For every q ∈ S u ω,h u and every t ∈ [0, 2τ ω,h s (q)], we have

d B X t (q), X t (P u ω,h u ) ≤ η (6.31)
Proof. Setting. For every C > 0 and every n ∈ N, we denote by E C,n the set of all (ω, h u , h s , η) such

that ω ∈ ]1, +∞[ \ {2}, 0 < h u ≤ (Cω n ) -1 , 0 < h s ≤ (Cf (ω) n )
-1 and 0 < η ≤ 1. Let C 0 ≥ C3 and n 0 ≥ ñ3 be large enough such that we can apply Proposition 3.2, Proposition 6.7 and Lemma 6.8 with these two constants.

For every C ≥ C 0 , every n ≥ n 0 and every

(ω, h u , h s , η) ∈ E C,n , let h = min h u , h s , h u = h u , ηh Cω m(ω), ηh Cω m(ω) , h s = h s , h s , h s and define the map g ∶ S u ω,h u → R by the formula g(q) = τ ω,h s (P u ω,h u ) -x 1 θ f (ω),h s X τ ω,h s (P u ω,h u ) (q)
Remark that g(q) is well defined if and only if X τ ω,h s (P u ω,h u ) (q) belongs to the flow box V f (ω),h s . According to remark 6.9, if g(q) is well defined then

X g(q) (q) ∈ S s f (ω),h box ⊂ S s f (ω),h s where r box = min (h s ) 2 C3 f (ω) ñ3 , f (ω) -1 2 h box = (h s , r box , r box )
We are going to prove that 1. If C and n are large enough, then, for every q ∈ S u ω,h u , g(q) is well defined and g(q) > 0 (claim 1).

2. If C and n are large enough, then, for every q ∈ S u ω,h u , g(q) is the first time such that the forward X -orbit of q intersects the section S s f (ω),h s (claims 2 and 3). More precisely, first we prove that g(q) is the first time such that the forward X -orbit of q intersects a small section S

s i(ω) f (ω),
h (defined below) and then we extend this result to our initial section S s f (ω),h s .

As an immediate consequence of these results, we will get that g = τ ω,h s on S u ω,h u . Inequality (6.31) will be proved along the way. The main arguments are the logarithmic upper bound (6.6) of τ ω,h s (P u ω,h u ), Gronwall's lemma, and the lower bound on the size of the flow box V f (ω),h s .

Using (2.15), it is straightforward to check that for every ω ∈ ]1, +∞[, we have

f (ω) -1 ≥ |ω -2|
Hence, there exist C 1 ≥ C 0 and n 1 ≥ n 0 such that for every C ≥ C 1 , every n ≥ n 1 and every (ω, h u , h s , η) ∈ E C,n , we have

C 0 f (ω) n 0 (h s ) 4 C 1 f (ω) n 1 m(ω) 1 2 < 1 2 r box (h s ) 2 C3 f (ω) ñ3 (6.32) For every C ≥ C 1 , every n ≥ n 1 and every (ω, h u , h s , η) ∈ E C,n , let r = (h s ) 4 C 1 f (ω) n 1 m(ω) 1 2 h = h s , r, r Claim 1. There exist C4 ≥ C 1 and ñ4 ≥ n 1 such that for all C ≥ C4 , n ≥ ñ4 , (ω, h u , h s , 1) ∈ E C,n
and q ∈ S u ω,h u , g(q) is well defined, g(q) > 0 and X g(q) (q) ∈ S

s i(ω) f (ω), h ⊂ S s i(ω) f (ω),h s .
Proof of claim 1. Let us consider a compact subset C of the phase space B, so that C contains a rr boxneighbourhood of the Mixmaster attractor. We denote by A an upper bound of the norm of the Wainwright-Hsu vector field X on C. For every point q ∈ C, we denote by

τ C (q) ∶= inf{t > 0, X t (q) ∉ C} ∈ [0, +∞]. × [-r, r] ⊂ V f (ω),h s
Hence, g is well defined on S u ω,h u . Moreover, using (6.23), we get that X g(q) (q) ∈ S

s i(ω) f (ω), h ⊂ S s f (ω)
,h s . Now, remark that according to the lower bound (6.6) on τ ω,h s (P u ω,h u ),

x 1 θ f (ω),h s X τ ω,h s (P u ω,h u ) (q) < r < 1 2 r box ≤ 1 2 C3 ≤ 1 2 C2 ≤ ω 2 C2 ≤ 1 2 τ ω,h s (P u ω,h u ) (6.38)
It follows that g(q) > 0. Hence, the forward X -orbit of q intersects the section S s f (ω),h s . This concludes the proof of claim 1.

Let us fix C ≥ C4 , n ≥ ñ4 , (ω, h u , h s , 1) ∈ E C,n and q ∈ S u ω,h u until the end of this proof.

Claim 2. g(q)

is the time of first intersection of the forward X -orbit of q with the section S

s i(ω) f (ω), h.
Proof of claim 2. Let us denote by t min ∈ [0, g(q)] the time of first intersection of the forward X -orbit of q with the section S

s i(ω) f (ω),
h. We have t min = g(q) if and only if

t min -τ ω,h s (P u ω,h u ) = -x 1 θ f (ω),h s X τ ω,h s (P u ω,h u ) (q) Moreover X t min -τ ω,h s (P u ω,h u ) (X τ ω,h s (P u ω,h u ) (q)) = X t min (q) ∈ S s i(ω) f (ω), h ⊂ S s i(ω) f (ω),h box and -x 1 θ f (ω),h s X τ ω,h s (P u ω,h u ) (q)
is, according to remark 6.9, the unique time t ∈ -r box , r box such that X t (X

τ ω,h s (P u ω,h u ) (q)) ∈ S s i(ω) f (ω),h box . Hence, it is sufficient to prove that |t min -τ ω,h s (P u ω,h u )| < r box (6.39)
According to (6.38), we have

t min ≤ g(q) ≤ τ ω,h s (P u ω,h u ) + x 1 θ f (ω),h s X τ ω,h s (P u ω,h u ) (q) ≤ 2τ ω,h s (P u ω,h u ) (6.40)
Using (6.37), (6.32), the estimate (3.5b) on the local coordinate system ξ and the mean value theorem, we get

d B X t min (P u ω,h u ), P s i(ω) f (ω),h s ≤ d B X t min (q), P s i(ω) f (ω),h s + d B X t min (q), X t min (P u ω,h u ) ≤ C 0 f (ω) n 0 r + 1 2 rr box C 0 f (ω) n 0 < 1 2 r box (h s ) 2 C3 f (ω) ñ3 + 1 2 r box (h s ) 4 m(ω) 1 2 C 0 f (ω) n 0 C 1 f (ω) n 1 < 1 2 r box (h s ) 2 C3 f (ω) ñ3 + 1 2 r box (h s ) 2 C3 f (ω) ñ3 < r box (h s ) 2 C3 f (ω) ñ3
Hence, using point 6 of Lemma 6.8, X t min (P u ω,h u ) ∈ V f (ω),h s . Moreover, the type II orbit O P ω →P f (ω) passes through the section S

s i(ω) f (ω),
h exactly one time so, according to remark 6.9,

-x 1 θ f (ω),h s X t min (P u ω,h u ) is the unique time t ∈ R such that X t (X t min (P u ω,h u )) ∈ S s i(ω) f (ω),
h and it satisfies

x 1 θ f (ω),h s X t min (P u ω,h u ) < r box (6.41) Since X τ ω,h s (P u ω,h u )-t min (X t min (P u ω,h u )) = X τ ω,h s (P u ω,h u ) (P u ω,h u )) = P s i(ω) f (ω),h s ∈ S s i(ω) f (ω), h it follows that τ ω,h s (P u ω,h u ) -t min = -x 1 θ f (ω),h s X t min (P u ω,h u ) (6.42)
Hence, (6.39) is a consequence of (6.41) and (6.42). This concludes the proof of claim 2.

We now extend claim 2 to the full section S s f (ω),h s . Claim 3. g(q) is the time of first intersection of the forward X -orbit of q with the section

S s f (ω),h s . Proof of claim 3. Let j(ω) = 2 if i(ω) = 1 and j(ω) = 1 if i(ω) = 2. By definition, τ ω,h s (q) ≤ g(q).
Assume that τ ω,h s (q) < g(q). This implies that either X τ ω,h s (q) (q) ∈ S

s j(ω) f (ω),h s or X τ ω,h s (q) (q) ∈ S s i(ω) f (ω),h s \ S s i(ω) f (ω),
h, otherwise it would contradict claim 2. According to (6.40), we can use (6.37) to get

d B X τ ω,h s (q) (q), X τ ω,h s (q) (P u ω,h u ) < 1 2 rr box C 0 f (ω) n 0 (6.43)
According to the estimate (3.5b) on the local coordinate system ξ and the mean value theorem, we have

ξ X τ ω,h s (q) (q) -ξ X τ ω,h s (q) (P u ω,h u ) ∞ ≤ C 0 f (ω) n 0 d B X τ ω,h s (q) (q), X τ ω,h s (q) (P u ω,h u ) (6.44)
We are now going to treat the two cases differently.

Case X τ ω,h s (q) (q) ∈ S s j(ω) f (ω),h s . Remark that the orbit of the Wainwright-Hsu vector field X starting at P u ω,h u is a type II orbit passing through the section S

s i(ω) f (ω),h s . Hence, h s ≤ x s j(ω) X τ ω,h s (q) (q) -x s j(ω) X τ ω,h s (q) (P u ω,h u ) ≤ ξ X τ ω,h s (q) (q) -ξ X τ ω,h s (q) (P u ω,h u ) ∞ (6.
45) It follows from (6.43), (6.44) and (6.45) that h s ≤ 1 2 rr box , which is absurd.

Case X τ ω,h s (q) (q) ∈ S s i(ω) f (ω),h s \ S s i(ω) f (ω),
h. In that case, we have ξ X τ ω,h s (q) (q) -P

s i(ω) f (ω),h s ∞ > r and x s i(ω) ξ X τ ω,h s (q) (q) = x s i(ω) P s i(ω) f (ω),h s = h s
Moreover, since the orbit of the Wainwright-Hsu vector field X starting at P u ω,h u is a type II orbit passing through the section S

s i(ω) f (ω),h s at the point P s i(ω) f (ω),h s , it follows that ξ X τ ω,h s (q) (P u ω,h u ) and P s i(ω)
f (ω),h s have the same coordinates except for the coordinate x s i(ω) . Hence,

ξ X τ ω,h s (q) (q) -ξ X τ ω,h s (q) (P u ω,h u ) ∞ > r (6.46)
It follows from (6.43), (6.44) and (6.46) that r ≤ 1 2 rr box , which is absurd. This concludes the proof of claim 3.

It follows that τ ω,h s (q) = g(q). To finish the proof, remark that (6.31) is a consequence of estimates (6.36) and (6.38).

Control of the transition map Ψ ω,h u ,h s

With the context and notations of Lemma 6.11, item 1 of Lemma 6.11 implies that the map Ψ ω,h u ,h s is well defined at every point of the section S u ω,h u and is C 2 . Recall that for every

y ∈ S u ω,h u , Ψ A ω,h u ,h s (y u , y s 1 , y s 2 , y c ) = Ψ ω,h u ,h s • Proj A (y u , y s 1 , y s 2 , y c ) = Ψ ω,h u ,h s (y u , 0, 0, y c )
Using standard Hadamard's lemma, we get that there exists a C

1 map Ψ ∆ ω,h u ,h s from S u ω,h u into the space of (4 × 2) real valued matrices such that for every y ∈ S u ω,h u , Ψ ω,h u ,h s (y u , y s 1 , y s 2 , y c ) = Ψ A ω,h u ,h s (y u , y s 1 , y s 2 , y c ) + Ψ ∆ ω,h u ,h s (y u , y s 1 , y s 2 , y c ).(y s 1 , y s 2 ) (6.47)
One can think about the map Ψ ∆ ω,h u ,h s as a tool to measure the "deviation" of the transition map Ψ ω,h u ,h s from the map Ψ A ω,h u ,h s . Since the map Ψ A ω,h u ,h s is essentially the Kasner map f , it amounts to study the deviation of generic orbits from type II orbits. Next lemma gives some estimates on Ψ ∆ ω,h u ,h s . Lemma 6.12 (Control of Ψ ∆ ω,h u ,h s ). There exist two constants C > 0 and n ∈ N such that for every

ω ∈ ]1, +∞[ \ {2}, every 0 < h u ≤ (Cω n ) -1 , every 0 < h s ≤ (Cf (ω) n ) -1 , for h = min h u , h s , h u = h u , h Cω m(ω), h Cω m(ω) and h s = h s , h s , h s , we have Ψ ∆ ω,h u ,h s C 1 ≤ h -Cω (6.48)
Proof. For every C > 0 and every n ∈ N, we denote by E C,n the set of all (ω, h u , h s ) such that

ω ∈ ]1, +∞[ \ {2}, 0 < h u ≤ (Cω n ) -1 and 0 < h s ≤ (Cf (ω) n ) -1
. For every C ≥ C4 , every n ≥ ñ4 and every (ω, h u , h s ) ∈ E C,n , define h, h u and h s as in Lemma 6.12.

According to the standard Hadamard's lemma, estimates on the k-th derivative of Ψ ∆ ω,h u ,h s follow from estimates on the (k + 1)-th derivative of Ψ ω,h u ,h s . By definition of the transition map Ψ ω,h u ,h s and the hitting time τ ω,h s , for every q ∈ S u ω,h u ,

ξ -1 • Ψ ω,h u ,h s • ξ(q) = X τ ω,h s (q) (q)
Hence, estimates on Ψ ω,h u ,h s are consequences of estimates on the local coordinate system ξ, the flow of the Wainwright-Hsu vector field and the hitting time τ ω,h s . According to Proposition 6.7, τ ω,h s (P u ω,h u ) ≤ -C2 ω ln h. Moreover, X is bounded on every compact. Hence, Gronwall's lemma implies that there exist C 1 ≥ C 0 and n 1 ≥ n 0 such that for every

C ≥ C 1 , every n ≥ n 1 , every (ω, h u , h s ) ∈ E C,n , every q ∈ S u ω,h u and every t ∈ [0, 2τ ω,h s (P u ω,h u )], we have D t,q X t (q) ≤ h -Cω and D 2 t,q X t (q) ≤ h -Cω (6.49)
According to the expression of the hitting time (6.30), the estimate (6.21) on the derivative of the flow box coordinates and the preceding control on the flow of the Wainwright-Hsu vector field X , there exist C 2 ≥ C 1 and n 2 ≥ n 1 such that for every C ≥ C 2 , every n ≥ n 2 , every (ω, h u , h s ) ∈ E C,n and every q ∈ S u ω,h u , we have

∥Dτ ω,h s (q)∥ ≤ h -Cω and D 2 τ ω,h s (q) ≤ h -Cω (6.50)
Using (6.49) and (6.50), we get some estimates on the first and second derivatives of Ψ ω,h u ,h s : there exist C 3 ≥ C 2 and n 3 ≥ n 2 such that for every

C ≥ C 3 , every n ≥ n 3 , every (ω, h u , h s ) ∈ E C,n and every q ∈ S u ω,h u , we have D(ξ -1 • Ψ ω,h u ,h s • ξ)(q) ≤ h -Cω and D 2 (ξ -1 • Ψ ω,h u ,h s • ξ)(q) ≤ h -Cω (6.51)
The estimates (6.51) together with estimates (3.5) on the local coordinate system ξ yield some estimates on the first and second derivatives of Ψ ω,h u ,h s . These estimates give the desired estimates on Ψ ∆ ω,h u ,h s .

At this point, Proposition 6.1 on the transition map Ψ ω,h u ,h s must be seen as a straightforward consequence of Lemma 6.12.

Proof of Proposition 6.1. Let C 0 ≥ C4 and n 0 ≥ ñ4 be large enough such that we can apply Lemma 6.12 with these two constants. For every C > 0 and n ∈ N, we denote by

E C,N the set of all (ω, h u , h s , h ⊥ , y, ỹ) such that ω ∈ ]1, +∞[ \ {2}, 0 < h u ≤ (Cω n ) -1 , 0 < h s ≤ (Cf (ω) n ) -1 , 0 < h ⊥ ≤ h Cω m(ω) and y, ỹ ∈ S u ω,h u where h = min h u , h s and h u = h u , h ⊥ , h Cω m(ω) . For every C ≥ C 0 , every n ≥ n 0 and every (ω, h u , h s , h ⊥ , y, ỹ) ∈ E C,N , we use the notations h s = h s , h s , h s , Ψ = Ψ ω,h u ,h s , Ψ A = Ψ A ω,h u ,h s and Ψ ∆ = Ψ ∆ ω,h u ,h s . Let C ≥ C 0 , n ≥ n 0 and (ω, h u , h s , h ⊥ , y, ỹ) ∈ E C,N .
According to Lemma 6.11, Ψ is well defined. According to (6.47), we have

Ψ(y) -Ψ A (y) = Ψ ∆ (y)y s 1 ,s 2
where y s 1 ,s 2 = (y s 1 , y s 2 ). Hence, using (6.48), we get

Ψ(y) -Ψ A (y) / / ≤ Ψ ∆ (y)y s 1 ,s 2 / / ≤ Ψ ∆ C 1 ∥y s 1 ,s 2 ∥ ∞ ≤ h -Cω h ⊥
This proves estimate (6.3). Estimate (6.2) is proven analogously. According to (6.47), we have

(Ψ(y) -Ψ(ỹ)) -Ψ A (y) -Ψ A (ỹ) = Ψ ∆ (y) -Ψ ∆ (ỹ) y s 1 ,s 2 + Ψ ∆ (ỹ) y s 1 ,s 2 -ỹs 1 ,s 2 Moreover, Ψ ∆ (y) -Ψ ∆ (ỹ) y s 1 ,s 2 + Ψ ∆ (ỹ) y s 1 ,s 2 -ỹs 1 ,s 2 ∞ ≤ Ψ ∆ C 1 ∥y -ỹ∥ ∞ h ⊥ + ∥y s 1 ,s 2 -ỹs 1 ,s 2 ∥ ∞ ≤ Ψ ∆ C 1 (∥y -ỹ∥ ∞ h ⊥ + ∥y -ỹ∥ ⊥ ) ≤ 2h
-Cω ∥y -ỹ∥ / / h ⊥ + ∥y -ỹ∥ ⊥ using (6.48). There exist C 1 ≥ C 0 and n 1 ≥ n 0 such that for every (ω, h

u , h s , h ⊥ , y, ỹ) ∈ E C,N , we have 2h -C 0 ω ≤ h -C 1 ω
This proves estimates (6.4) and (6.5). This shows that Proposition 6.1 holds true with C1 ∶= C 1 and ñ1 ∶= n 1 .

We finish this section with a short proof of Proposition 6.4.

Proof of Proposition 6.4. Using the notations of Proposition 6.4, this is a straightforward consequence of the Gronwall's estimate (6.31) and the fact that τ ω,h s (q) is uniformly arbitrary close to τ ω,h s (P u ω,h u ) when η is taken small enough.

Dynamics along an epoch

The goal of this section is to give some estimates on the epoch transition map Φ ω,h ω ,h f (ω) (see definition 4.17). Recall that this map describes the behaviour of the orbits of the Wainwright-Hsu vector field between the sections S s ω,h ω and S s f (ω),h f (ω) . In other words, it describes the behaviour of the orbits between the moment they arrive in the neighbourhood of the point P ω and the moment they arrive in the neighbourhood of the point P f (ω) .

Our first task will be to prove that we can write the epoch transition map

Φ ω,h ω ,h f (ω) as a compo- sition Φ ω,h ω ,h f (ω) = Ψ ω,h u ,h f (ω) • Υ ω,h ω ,h u (7.1)
of the transition maps Ψ ω,h u ,h f (ω) and Υ ω,h ω ,h u studied in the two preceding sections. This amounts to prove that, for h ω , h u and h f (ω) well chosen, any orbit starting in the section S s ω,h ω will pass through the section S u ω,h u before hitting the section S s f (ω),h f (ω) . Once the relation (7.1) will be proven, we will be able to combine the estimates proven in the two preceding sections and deduce from them some estimates on the map Φ ω,h ω ,h f (ω) . More precisely, we will show that this map is a strong contraction in the direction transversal to the Mixmaster attractor while it is very close to the Kasner map f in the direction tangential to the Mixmaster attractor. The key point is the fact that the super-linear contraction of Υ ω,h ω ,h u in the direction transversal to the Mixmaster attractor dominates everything else.

From now on, we will systematically use the continued fraction expansion of the Kasner parameter ω. This will make our results easier to formulate and to read. Recall that we denote by [k 0 ; k 1 , k 2 , k 3 , . . . ] the unique (infinite) continued fraction

k 0 + 1 k 1 + 1 k 2 + 1 k 3 + . . . Moreover, we denote by [k 0 (ω); k 1 (ω), k 2 (ω), . . . ] the continued fraction expansion of a real number ω ∈ ]0, +∞[ \ Q. Also, recall that m(ω) = min 1, (ω -2) 2 , i(ω) = 1 if ω > 2 2 if 1 < ω < 2
Recall that Proj A is the projection on the Mixmaster attractor (see definition 4.1) and Φ

A ω,h ω ,h f (ω) = Φ ω,h ω ,h f (ω) • Proj A .
Now, let us introduce some constants that will be used to quantify the dilatation properties of the Kasner map. Define, for

ω ∈ ]1, +∞[ \ Q, K f (ω) def = 36 25 if 1 < ω < 5 3 1 if ω > 5 3 (7.2) Lip f (ω) def = 16k 1 (ω) 2 if 1 < ω < 2 1 if ω > 2 (7.3) and Lip f ′ (ω) def = 128k 1 (ω) 3 if 1 < ω < 2 0 if ω > 2 (7.4)
We will prove that, on the one hand, K f (ω) is a local expansion constant for the Kasner map and, on the other hand, Lip f (ω) and Lip f ′ (ω) are some local Lipschitz constants for the Kasner map and its derivative in the neighbourhood of ω. Proposition 7.1 is the main result of this section, it shows that the decisive parameter to control the epoch transition map is the size h ⊥ of the section S s ω,h ω in the direction transverse to the Mixmaster attractor. Its proof does not require new ideas, it is just the concatenation of Proposition 5.1 and Proposition 6.1.

Proposition 7.1 (Control of the epoch transition map).

There exist two constants C5 ≥ 1 and ñ5 ∈ N such that the properties below hold for

ω ∈ ]1, +∞[ \ Q, 0 < h ω ≤ ( C5 ω ñ5 ) -1 , 0 < h f (ω) ≤ ( C5 f (ω) ñ5 ) -1 , h = min h ω , h f (ω) , 0 < h ⊥ ≤ h C5 k 0 (ω) 3 m(ω), h ω = h ω , h ⊥ , h C5 k 0 (ω) m(ω) and h f (ω) = (h f (ω) , h f (ω) , h f (ω) ). If k 0 (ω) = k 1 (ω) = 1, assume that h ω = h f (ω) . The epoch transition map Φ ω,h ω ,h f (ω) ∶ S s ω,h ω → S s f (ω),h f (ω)
is well defined and takes its values in S s i(ω) f (ω),h f (ω) . Moreover, for every x, x ∈ S s ω,h ω , we have the following estimates, where

Φ ∶= Φ ω,h ω ,h f (ω) and Φ A ∶= Φ A ω,h ω ,h f (ω) :
(Control of the distance to the Mixmaster attractor)

dist ∞ (Φ(x), A) = Φ(x) -Φ A (x) ⊥ ≤ h k0(ω)+4 k0(ω)+3 ⊥ (7.5)
(Control of the drift tangential to the Mixmaster attractor)

Φ(x) -Φ A (x) / / ≤ 2h ⊥ Lip f (ω) (7.6)
(Contraction in the direction transverse to the Mixmaster attractor)

∥Φ(x) -Φ(x)∥ ⊥ ≤ h 1 k0 (ω)+3 ⊥ ∥x -x∥ ∞ (7.7)
(Lipschitz control in the direction tangential to the Mixmaster attractor)

(Φ(x) -Φ(x)) -(Φ A (x) -Φ A (x)) / / ≤ h 1 k0 (ω)+3 ⊥ ∥x -x∥ ∞ + Lip f (ω) ∥x -x∥ ⊥ (7.8)
(Expansion in the direction tangential to the Mixmaster attractor)

∥Φ(x) -Φ(x)∥ / / ≥ K f (ω) ∥x -x∥ / / -h 1 k0(ω)+3 ⊥ ∥x -x∥ ∞ -C5 k 0 (ω) ñ5 h ω ∥x -x∥ ⊥ (7.9) (Global lipschitz constant) ∥Φ(x) -Φ(x)∥ ∞ ≤ 4 Lip f (ω) ∥x -x∥ ∞ (7.10)

Some estimates concerning the Kasner map

In this section, we explore two properties of the Kasner map f : the fact that it is locally expansive and the fact that it is locally Lipschitz. Those properties are direct consequences of the explicit formula (2.15), but we need to state some precise quantitative results. The proposition below states that K f (ω) (defined by formula (7.2)) is an expansivity constant in the neighbourhood of ω.

Proposition 7.2 (Local expansion constant for f ). For

ω ∈ ]1, +∞[ \ Q, the Kasner map f is K f (ω)-expansive on the interval ]ω -η, ω + η[ where η = min ω-1 2 , |ω-2| 2 .
Proof. We divide the proof in three cases: ω > 2, 5 3 < ω < 2 and 1 < ω < 5 3 . If ω > 2, then f = Id on ]ωη, ω + η[ according to (2.15). Using (2.15), remark that for every

x ∈ ]1, 2[, f ′ (x) = - 1 (x -1) 2 and f is monotonous on ]1, 2[. Let y, ỹ ∈ ]ω -η, ω + η[. If 5 3 < ω < 2, then |f (y) -f (ỹ)| ≥ min x∈]ω-η,ω+η[ f ′ (x) |y -ỹ| ≥ f ′ (2) |y -ỹ| ≥ K f (ω) |y -ỹ| If 1 < ω < 5 3 , then |f (y) -f (ỹ)| ≥ min x∈]ω-η,ω+η[ f ′ (x) |y -ỹ| ≥ f ′ 11 6 |y -ỹ| ≥ K f (ω) |y -ỹ|
Hence, Proposition 7.2 has been proved for all ω.

The next proposition states that Lip f (ω) and Lip f ′ (ω) (defined by fromulas (7.3) and (7.4)) are local Lipschitz constants for the Kasner map f and its derivative f ′ in the neighbourhood of ω.

Proposition 7.3 (Local Lipschitz constant for f and f ′ ). For ω ∈ ]1, +∞[ \ Q, the Kasner map f is Lip f (ω)-Lipschitz and its derivative f ′ is Lip f ′ (ω)-Lipschitz on the interval ]ω -η, ω + η[ where η = min ω-1 2 , |ω-2| 2 . Proof. If ω > 2, then f = Id on ]ω -η, ω + η[ according to (2.15). If 1 < ω < 2, then (2.15) implies that max x∈]ω-η,ω+η[ f ′ (x) ≤ f ′ ω + 1 2 ≤ 4 (ω -1) 2 ≤ 16k 1 (ω) 2 and max x∈]ω-η,ω+η[ f ′′ (x) ≤ f ′′ ω + 1 2 ≤ 16 (ω -1) 3 ≤ 128k 1 (ω) 3
The statement follows immediately from these inequalities and the mean value theorem.

Travels along an epoch

In this section, we state a proposition that gives some conditions under which we can write Φ ω = Ψ ω • Υ ω . Equivalently, we give some conditions on h ω , h u and h f (ω) under which every orbit starting in the section S s ω,h ω will pass through the section S u ω,h u before hitting the section S s f (ω),h f (ω) . There are essentially two cases depending on ω = [k 0 ; k 1 , k 2 , . . . ] ∈ ]1, +∞[ \ Q: the first case is when k 0 = k 1 = 1 and the second case is when either k 0 ≥ 2 or (k 0 = 1 and k 1 ≥ 2). For the first case, we use the contraction in the direction transversal to the Mixmaster attractor. For the second case, we use the gap between the sections S s ω and S s f (ω) in the direction tangential to the Mixmaster attractor. The first case is special, in the sense that we need to choose more carefully the parameters for the sections than in the second case.

Lemma 7.4. There exist two constants C > 0 and n ∈ N such that the properties below hold true for

ω ∈ ]1, +∞[ \ Q, 0 < h ω ≤ (Cω n ) -1 , 0 < h f (ω) ≤ (Cf (ω) n ) -1 , h = min h ω , h f (ω) , h ω = h ω , h Ck 0 (ω) m(ω), h Ck 0 (ω) m(ω) , h u = h ω , h C1 ω m(ω), h C1 ω m(ω) and h f (ω) = (h f (ω) , h f (ω) , h f (ω) ). If k 0 (ω) = k 1 (ω) = 1, assume that h ω = h f (ω) . The epoch transition map Φ ω,h ω ,h f (ω) ∶ S s ω,h ω → S s f (ω),h f (ω)
is well defined and takes its values in S

s i(ω) f (ω),h f (ω) . The map Ψ ω,h u ,h f (ω) • Υ ω,h ω ,h u is also well defined on the section S s ω,h ω . Moreover, Φ ω,h ω ,h f (ω) = Ψ ω,h u ,h f (ω) • Υ ω,h ω ,h u
Proof. For every C > 0 and every n ∈ N, we denote by E C,n the set of all (ω,

h ω , h f (ω) ) such that ω = [k 0 ; k 1 , k 2 , . . . ] ∈ ]1, +∞[ \ Q, 0 < h ω ≤ (Cω n ) -1 , 0 < h f (ω) ≤ (Cf (ω) n ) -1 such that h ω = h f (ω) if k 0 (ω) = k 1 (ω) = 1.

We also define h, h ω , h

u , h f (ω) as in Lemma 7.4. Let C 0 ≥ 100 and n 0 ∈ N * be large enough such that we can apply Proposition 5.1, Proposition 5.17 and Proposition 6.1 with these two constants. Take C 1 ≥ C 0 such that for every C ≥ C 1 , every n ≥ n 0 and every (ω, h ω , h f (ω) ) ∈ E C,n , we have

h Ck 0 ω m(ω) ω+2 ω+1 h -1 ω ≤ h C1 ω ω m(ω) (7.11a) h Ck 0 ω m(ω) h ω C 0 ω n 0 + 1 ≤ h C1 ω ω m(ω) (7.11b) Let C ≥ C 0 , n ≥ n 0 and (ω, h ω , h f (ω) ) ∈ E C,
f (ω),h f (ω) . Observe that h Ck 0 (ω) m(ω) is always smaller than d(ω) = ω-1 4 (where d(ω) is a parameter introduced for Proposition 5.1). Indeed, if ω ∈]1, 2[, then f (ω) = 1 ω-1 , hence h Ck 0 (ω) m(ω) ≤ h f (ω) ≤ d(ω) = ω-1 4 since C ≥ 4 . And if ω > 2, then h Ck 0 (ω) m(ω) ≤ 1 4 ≤ d(ω). It follows that we can use Proposition 5.1 to get that Υ ω,h ω ,(h ω ,h ω 2h ω ) is well defined. Hence, Υ ω,h ω ,h u is well defined if Υ ω,h ω ,(h ω ,h ω 2h ω ) S s ω,h ω ⊂ S u ω,h u (7.12) Let x ∈ S s ω,h ω .
According to (5.1) and (7.11a), we have

Υ ω,h ω ,(h ω ,h ω 2h ω ) (x) -Υ A ω,h ω ,(h ω ,h ω 2h ω ) (x) ⊥ ≤ h Ck 0 ω m(ω) ω+2 ω+1 h -1 ω ≤ h C1 ω ω m(ω)
According to (5.2), we have

∥Υ ω,h ω ,(h ω ,h ω 2h ω ) (x) -x∥ / / ≤ h Ck 0 ω m(ω)h ω C 0 ω n 0
so, using (7.11b), we get

∥Υ ω,h ω ,(h ω ,h ω 2h ω ) (x) -P u ω,h ω ∥ / / ≤ ∥Υ ω,h ω ,(h ω ,h ω 2h ω ) (x) -x∥ / / + ∥x -P u ω,h ω ∥ / / ≤ h Ck 0 ω m(ω)h ω C 0 ω n 0 + h Ck 0 ω m(ω) ≤ h C1 ω ω m(ω)
It follows that for every C ≥ C 1 , every n ≥ n 0 and every (ω, h ω , h f (ω) ) ∈ E C,n , (7.12) holds true. Hence, the maps Υ ω,h ω ,h u and Ψ ω,h u ,h f (ω) • Υ ω,h ω ,h u are well defined on the section

S s ω,h ω . Moreover, the map Ψ ω,h u ,h f (ω) • Υ ω,h ω ,h u takes its values in S s i(ω) f (ω),h f (ω) . This implies that the epoch transition map Φ ω,h ω ,h f (ω) is well defined. We are left to prove that Φ ω,h ω ,h f (ω) = Ψ ω,h u ,h f (ω) • Υ ω,h ω ,h u . Let x ∈ S s ω,h ω .
First case: x u = 0. According to (4.8), (4.10) and (4.12),

Φ ω,h ω ,h f (ω) (x) = Ψ ω,h u ,h f (ω) • Υ ω,h ω ,h u (x)
Second case: x u ≠ 0. By definition of Ψ ω,h u ,h f (ω) , we only need to prove that the orbit t ↦ y(t) of the locally renormalized Wainwright-Hsu vector field X ω starting from x does not intersect the section S s f (ω),h f (ω) before it intersects the section S u ω,h u . Assume that k 0 = k 1 = 1. It follows from (5.27b) that during its travel between S s ω,h ω and S u ω,h u , the orbit y satisfies y s 1 (t) < h ω and y s 2 (t) < h ω . Since h ω = h f (ω) , y(t) does not belong to the section

S s f (ω),h f (ω) .
Assume that k 0 = 1 and k 1 ≥ 2. It follows from (5.27c) that during its travel between S s ω,h ω and S u ω,h u , the orbit y satisfies |y c (t) -ω| ≤ 2h C f (ω) ≤ 1 8 . Hence, y c (t) ≤ 13 8 . Moreover, h f (ω) ≤ 1 8 and f (ω) ≥ 2 so any point z belonging to the section S s f (ω),h f (ω) must satisfy z c ≥ 2 -1 8 = 15 8 . Hence, y(t) does not belong to the section S s f (ω),h f (ω) . Assume that k 0 ≥ 2. Having in mind that in this case, f (ω) = ω -1, one can repeat the above argument.

This shows that Ψ ω,h u ,h f (ω) • Υ ω,h ω ,h u (x) is the first intersection point of y with the section

S s f (ω),h f (ω) . Hence Φ ω,h ω ,h f (ω) (x) = Ψ ω,h u ,h f (ω) • Υ ω,h ω ,h u (x).

Control of the epoch transition map Φ ω,h ω ,h f (ω)

In this section, we prove Proposition 7.1 using the decomposition Φ ω = Ψ ω • Υ ω (see Lemma 7.4), the estimates on Υ ω proven in Section 5 (see Proposition 5.1) and the estimates on Ψ ω proven in Section 6 (see Proposition 6.1).

Proof of Proposition 7.1. For every C > 0 and every n ∈ N, we denote by E C,n the set of all

(ω, h ω , h f (ω) , h ⊥ , x, x) such that ω = [k 0 ; k 1 , k 2 , . . . ] ∈ ]1, +∞[ \ Q, 0 < h ω ≤ (Cω n ) -1 , 0 < h f (ω) ≤ (Cf (ω) n ) -1 such that h ω = h f (ω) if k 0 (ω) = k 1 (ω) = 1, 0 < h ⊥ ≤ h Ck 3 0 m(ω) where h = min h ω , h f (ω) and x, x ∈ S s ω,h ω where h ω = (h ω , h ⊥ , h
Ck 0 m(ω)). Let C 0 > 0 and n 0 ∈ N be large enough such that we can apply Proposition 5.1, Proposition 6.1 and Lemma 7.4 with these two constants. For every C ≥ C 0 , every n ≥ n 0 and every (ω,

h ω , h f (ω) , h ⊥ , x, x) ∈ E C,n , define h u as in Lemma 7.4 and h f (ω) as in Proposition 7.1 and let Υ ∶= Υ ω,h ω ,h u , Ψ ∶= Ψ ω,h u ,h f (ω) and Φ ∶= Φ ω,h ω ,h f (ω) .
Step 1: estimates (7.5) and (7.6).

Let C ≥ C 0 , n ≥ n 0 and (ω, h ω , h f (ω) , h ⊥ , x, x) ∈ E C,n . Recall that Φ(x) -Φ A (x) ⊥ = dist ∞ (Φ(x), A) = ∥Φ(x) -y∥ ⊥ for any y ∈ S s i(ω) f (ω),h f (ω) ∩ A. Hence, Φ(x) -Φ A (x) ⊥ = Ψ • Υ(x) -Ψ • Υ A (x) ⊥ using Lemma 7.4 = Ψ • Υ(x) -Ψ A • Υ(x) ⊥ It follows that Φ(x) -Φ A (x) ⊥ ≤ Υ(x) -Υ A (x) ⊥ h -C1 ω using (6.2) ≤ h ω+2 ω+1 ⊥ h -1 ω h -C1 ω using (5.1) (7.13)
To simplify the estimate found above, let us fix C 1 ≥ C 0 such that for every C ≥ C 1 , every n ≥ n 0 and every (ω, h ω , h f (ω) , h ⊥ , x, x) ∈ E C,n , we have

h ω+2 ω+1 ⊥ h -1 ω h -C1 ω ≤ h k0 +4 k0 +3 ⊥ (7.14)
Plugging (7.14) into (7.13), we get that estimate (7.5) holds true. According to (5.2), there exists C 2 ≥ C 1 such that for every C ≥ C 2 , every n ≥ n 0 and every

(ω, h ω , h f (ω) , h ⊥ , x, x) ∈ E C,n , we have |Υ(x) c -ω| ≤ min ω -1 4 , |ω -2| 2 (7.15) Let C ≥ C 2 , n ≥ n 0 and (ω, h ω , h f (ω) , h ⊥ , x, x) ∈ E C,n . We have Φ(x) -Φ A (x) / / = Ψ • Υ(x) -Ψ A • Υ A (x) / /
using Lemma 7.4

≤ Ψ • Υ(x) -Ψ A • Υ(x) / / + Ψ A • Υ(x) -Ψ A • Υ A (x) / / ≤ Υ(x) -Υ A (x) ⊥ h -C1 ω + Lip f (ω) Υ(x) -Υ A (x)
/ / using (6.3), (7.15) and proposition 7.3

≤h ω+2 ω+1 ⊥ h -1 ω h -C1 ω + Lip f (ω)h ⊥ h ω C 0 ω n 0
using (5.1) and (5.2)

≤h k0+4 k0+3 ⊥ + h ⊥ Lip f (ω) using (7.14) ≤2h ⊥ Lip f (ω)
Hence, estimate (7.6) holds true.

Step 2: estimates (7.7), (7.9) and (7.10). Using estimates (5.1), (5.3) and (5.4) and taking C 3 large enough, we get that

∥Υ(x) -Υ(x)∥ ⊥ + Υ(x) -Υ A (x) ⊥ ∥Υ(x) -Υ(x)∥ / / h -C1 ω ≤ h 1 ω+1 ⊥ h -1 ω + h ω+2 ω+1 ⊥ h -1 ω 1 + C 0 ω n 0 h ω + C 0 ω n 0 h ⊥ h -C1 ω ∥x -x∥ ∞ ≤h 1 k0+2 ⊥ h -C 3 k 0 ∥x -x∥ ∞ (7.16)
Plugging (7.16) into (6.4), we get that there exists C 4 ≥ C 3 such that for every C ≥ C 4 , every n ≥ n 0 and every (ω, h ω , h f (ω) , h ⊥ , x, x) ∈ E C,n , we have

∥Φ(x) -Φ(x)∥ ⊥ ≤ h 1 k0 +2 ⊥ h -C 3 k 0 ∥x -x∥ ∞ ≤ h 1 k0+3 ⊥ ∥x -x∥ ∞
Hence, estimate (7.7) holds true.

Plugging (7.16) into (6.5), we get that

(Φ(x) -Φ(x)) -Ψ A • Υ(x) -Ψ A • Υ(x) / / ≤ h 1 k0+2 ⊥ h -C 3 k 0 ∥x -x∥ ∞ (7.17)
Recall that Ψ A is essentially the Kasner map (see remark 4.26), hence

∥Φ(x) -Φ(x)∥ / / ≥ Ψ A • Υ(x) -Ψ A • Υ(x) / / -(Φ(x) -Φ(x)) -Ψ A • Υ(x) -Ψ A • Υ(x) / / ≥K f (ω) ∥Υ(x) -Υ(x)∥ / / -h 1 k0 +2 ⊥ h -C 3 k 0 ∥x -x∥ ∞ using (7.15
), proposition 7.2 and (7.17)

It follows that

∥Φ(x) -Φ(x)∥ / / ≥K f (ω) ∥x -x∥ / / -K f (ω)C 0 ω n 0 h ω ∥x -x∥ ⊥ -K f (ω)C 0 ω n 0 h ⊥ ∥x -x∥ / / -h 1 k0+2 ⊥ h -C 3 k 0 ∥x -x∥ ∞ using (5.4) ≥K f (ω) ∥x -x∥ / / -C 5 k n 0 0 h ω ∥x -x∥ ⊥ -h 1 k0+2 ⊥ h -C 5 k 0 ∥x -x∥ ∞ for C 5 large enough
According to the above inequality, there exists C 6 ≥ max(C 4 , C 5 ) such that for every C ≥ C 6 , every n ≥ n 0 and every (ω,

h ω , h f (ω) , h ⊥ , x, x) ∈ E C,n , we have ∥Φ(x) -Φ(x)∥ / / ≥ K f (ω) ∥x -x∥ / / -C 5 k n 0 0 h ω ∥x -x∥ ⊥ -h 1 k0+3 ⊥ ∥x -x∥ ∞
Hence, estimate (7.9) holds true.

We have

∥Φ(x) -Φ(x)∥ / / ≤ Ψ A • Υ(x) -Ψ A • Υ(x) / / + (Φ(x) -Φ(x)) -Ψ A • Υ(x) -Ψ A • Υ(x) / / ≤ Lip f (ω) ∥Υ(x) -Υ(x)∥ / / + h 1 k0 +2 ⊥ h -C 3 k 0 ∥x -x∥ ∞
using (7.15), Proposition 7.3 on the Kasner map and (7.17)

≤ h 1 k0+2 ⊥ h -C 3 k 0 + 3 Lip f (ω) ∥x -x∥ ∞ using (5.4) ≤4 Lip f (ω) ∥x -x∥ ∞ for C ≥ C 4
It follows from the above inequality and (7.7) that estimate (7.10) holds true.

Step 3: estimate (7.8). Let C ≥ C 6 , n ≥ n 0 and (ω, h ω , h f (ω) , h ⊥ , x, x) ∈ E C,n . We have

(Φ(x) -Φ(x)) -(Φ A (x) -Φ A (x)) / / ≤ (Φ(x) -Φ(x)) -Ψ A • Υ(x) -Ψ A • Υ(x) / / + Ψ A • Υ(x) -Ψ A • Υ(x) -Ψ • Υ A (x) -Ψ • Υ A (x) / / (7.18)
The first term of the right hand side of (7.18) is controlled by (7.17). To control the second term of the right hand side of (7.18), let us define the map

λ ∶ x ↦ Ψ • Υ u (x) -Ψ • Υ A (x)
where Υ u = Proj A •Υ. Remark that the second term is equal to ∥λ(x) -λ(x)∥ / / so we are left to apply the mean value theorem to λ. Remark that λ is continuous on S s ω,h ω and smooth on Int S s ω,h ω (we do not know if it is smooth on the hyperplane {x u = 0}). Let us identify the tangent space

T x S s 1 ω,h ω = Vect ∂ ∂x u ⊕ Vect ∂ ∂x s2 ⊕ Vect ∂ ∂x c with R
3 (and analogously for T x S s 2 ω,h ω , permuting the roles of s 1 and s 2 ). Assume that x, x ∈ Int S s ω,h ω and x ≠ x. We will only prove estimate (7.8) in the case where ∥x -x∥ ⊥ ≤ ∥x -x∥ / / (this is the only case useful later on and the other case is similar).

Let v = (v 1 , v 2 , v 3 ) ∈ R 3 such that ∥(v 1 , v 2 )∥ ∞ ≤ α |v 3 |, where α = ∥x-x∥ ⊥ ∥x-x∥ / /
. We have

Dλ(x).v = DΨ(Υ u (x)) -DΨ(Υ A (x)) DΥ u (x).v + DΨ(Υ A (x)) DΥ u (x).v -DΥ A (x).v Recall that Υ u (x) = (h ω , 0, 0, Υ(x) c )
Using (4.12), (5.4) and Proposition 7.3 (with (7.15)), we get

DΨ(Υ A (x)) DΥ u (x).v -DΥ A (x).v / / ≤ Lip f (ω) DΥ u (x).v -DΥ A (x).v / / ≤ Lip f (ω) α + C 0 ω n 0 h ⊥ ∥v∥ ∞
Using (4.12), (5.2), (5.4) and Proposition 7.3 (with (7.15)), we get

DΨ(Υ u (x)) -DΨ(Υ A (x)) DΥ u (x).v / / ≤ Lip f ′ (ω) ∥Υ(x) -x∥ / / ∥DΥ u (x).v∥ / / ≤ Lip f ′ (ω)h ⊥ h ω C 0 ω n 0 1 + C 0 ω n 0 (h ω α + h ⊥ ) ∥v∥ ∞ There exist C 8 ≥ C 7 ≥ C 6 such that for every C ≥ C 8 , every n ≥ n 0 and every (ω, h ω , h f (ω) , h ⊥ , x, x) ∈ E C,n , we have Lip f (ω)C 0 ω n 0 ≤ h -C 7 k 0 Lip f ′ (ω)h ω C 0 ω n 0 1 + C 0 ω n 0 (h ω α + h ⊥ ) ≤ h -C 7 k 0 3h 1 k0+2 ⊥ h -C 7 k 0 ≤ h 1 k0 +3 ⊥
Applying the mean value theorem to the last coordinate of λ, it follows that

(Φ(x) -Φ(x)) -(Φ A (x) -Φ A (x)) / / ≤h 1 k0 +2 ⊥ h -C 3 k 0 ∥x -x∥ ∞ + Lip f (ω) ∥x -x∥ ⊥ + 2h ⊥ h -C 7 k 0 ∥x -x∥ ∞ ≤h 1 k0 +3 ⊥ ∥x -x∥ ∞ + Lip f (ω) ∥x -x∥ ⊥
Hence, estimate (7.8) holds true on Int S s ω,h ω and then on S s ω,h ω by continuity. To conclude, Proposition 7.1 holds true with C5 = C 8 and ñ5 = n 0 .

Dynamics along an era

The goal of this section is to give some estimates on the era return map Φh ∶ S h → S h and the double era return map Φh ∶ S h → S h (see definitions 4.8 and 4.9). Recall that Φh ∶ S h → S h is essentially the first return map of the orbits of the Wainwright-Hsu vector field on the global section S h and that Φh ∶ S h → S h is just the square of Φh .

Our first task will be to prove that, for any ω ∈ ]1, 2[ \ Q, we can write the era transition map Φω,h as a composition of k 1 (ω) epoch transition maps,

Φω,h = Φ k 1 (ω)-1 • ⋅ ⋅ ⋅ • Φ 0 (8.1)
where Φ j is the epoch transition map from a section S s f j (ω),h j at the entrance of a neighbourhood of P f j (ω) to a section S s f j+1 (ω),h j+1 at the entrance of a neighbourhood of P f j+1 (ω) . Once equality (8.1) will be established, we will be able to use the estimates proven in the preceding section on the epoch transition maps (see Proposition 7.1) to get some estimates on the era transition map Φω,h (see Proposition 8.4). The main technical difficulty will be to set up an induction on the length of the era. Analogously to the epoch transition maps, we will show that the era transition map Φω,h is a strong contraction in the direction transversal to the Mixmaster attractor while it is very close to the era Kasner map f (see (2.16)) in the direction tangential to the Mixmaster attractor.

We would like to prove that the era transition map Φω,h admits some hyperbolic properties. Unfortunately, it does not expand enough in the direction tangential to the Mixmaster attractor. Indeed, the era transition map is "close" to the era Kasner map f in the direction tangential to the Mixmaster attractor and f does not expand uniformly in the neighbourhood of ω = 2. Nevertheless, since f expands uniformly on every interval ]1, 2 -[ and since f (]2 -, 2[) ⊂ ]1, 2 -[ for small enough, it follows that f ∶= f • f expands uniformly on ]1, 2[. This is the reason why we introduce the double era transition map Φω,h .

Before we give the estimates on the double era transition map Φω,h , we need some definitions. Let

K f def = 36 25 (8.2)
Let us explain why K f is a local expansion constant for the double era Kasner map f . Recall that we denote by K f (ω) a local expansion constant for the Kasner map f in the neighbourhood of ω (see

Proposition 7.2). Let ω = [1; k 1 , k 2 , . . . ] ∈ ]1, 2[ \ Q and 1 ≤ j ≤ k 1 -1. Using formula (7.2), we get that K f (f j (ω)) = 1 Hence, k 1 -1 j=0 K f (f j (ω)) = K f (ω)
As a consequence, K f (ω) is also a local expansion constant for the era Kasner map f in the neighbourhood of ω. As a consequence, K f ( f (ω))K f (ω) is a local expansion constant for the double era Kasner map f in the neighbourhood of ω. Using formula (7.2), it is easy to check that

K f ( f (ω))K f (ω) ≥ K f (8.3)
We are going to prove that the double era transition map is, as the double era Kasner map, expansive in the direction tangent to the Mixmaster attractor with a slightly lesser constant, say

K c def = 1 + K f 2 (8.4)
Later on (see Section 9), we will show that there exists a cone field invariant by the double era transition map, say of width σ. This invariant cone field will allow us to define a graph transformation that maps σ-Lipschitz graphs to σ-Lipschitz graphs. Local stable manifolds (for the double era transition map) will be obtained as fixed points of the graph transformation. The condition for this graph transformation to be a contraction mapping is

K c 1 - σ2 > 1 (8.5)
Hence, we fix now a positive constant σ satisfying (8.5) and we will prove an expansion estimate for the double era transition map that is adapted to this particular constant. Remark that σ < 1/2. Definition 8.1.

For any ω ∈ ]1, 2[ \ Q, let us define s 2 (ω) def = k 1 (ω) 2 + k 2 (ω) 2 + k 3 (ω) 2 + k 4 (ω) 2 (8.6a) s 4 (ω) def = k 1 (ω) 4 + k 2 (ω) 4 + k 3 (ω) 4 + k 4 (ω) 4 (8.6b) Define, for any ω ∈ ]1, 2[ \ Q, î(ω) def = 1 if k 2 (ω) ≥ 2 2 if k 2 (ω) = 1 (8.7)
Proposition 8.2 (Double era transition map). There exist two constants C8 ≥ 1 and ĥ > 0 such that the properties below hold for every

ω ∈ ]1, 2[ \ Q and 0 < h ⊥ ≤ e -C8 s 4 (ω) . Let h = ĥ, h ⊥ , e -C8 s 2 (ω) .
The double era transition map Φω,h ∶ S s ω,h ⊂ Sĥ → Sĥ is well defined and takes its values in S sî (ω) ĥ

. Moreover, for every x, x ∈ S s ω,h , we have the following estimates, where Φ ∶= Φω,h and ΦA ∶= ΦA ω,h : (Control of the distance to the Mixmaster attractor)

dist ∞ Φ(x), A = Φ(x) - ΦA (x) ⊥ ≤ h 1+ k1 (ω) 4 + k2(ω) 4 ⊥ (8.8)
(Control of the drift tangential to the Mixmaster attractor)

Φ(x) - ΦA (x) / / ≤ 578h ⊥ k 1 (ω) 2 k 2 (ω) 2 (8.9)
(Contraction in the direction transverse to the Mixmaster attractor)

Φ(x) -Φ(x) ⊥ ≤ h k1(ω) 100 + k2 (ω) 100 ⊥ ∥x -x∥ ∞ (8.10)
(Lipschitz control in the direction tangential to the Mixmaster attractor)

Φ(x) -Φ(x) - ΦA (x) - ΦA (x) / / ≤ h 1 26k1(ω) ⊥ + h 1 26k2(ω) ⊥ ∥x -x∥ ∞ + 16 2 k 1 (ω) 2 k 2 (ω) 2 ∥x -x∥ ⊥ (8.11)
(Expansion in the direction tangent to the Mixmaster attractor)

Φ(x) -Φ(x) / / ≥ K f ∥x -x∥ / / - K f -K c 1 + 1 σ ∥x -x∥ ∞ (8.12)

Control of the era transition map Φω,h

The estimates on the double era transition map Φ stated in Proposition 8.2 cannot be proven directly. One first needs to prove estimates on the era transition map Φ. This is what we are going to do. Proposition 8.4 below shows that the decisive parameter to control the era transition map is the size h ⊥ of the section S s ω,h in the direction transverse to the Mixmaster attractor. Recall that for all

ω ∈ ]1, +∞[ \ Q, m(ω) = min 1, (ω -2) 2 and define for all ω ∈ ]1, 2[ \ Q, m(ω) def = min 0≤j≤k 1 (ω) m(f j (ω)) (8.13) Lemma 8.3. For all ω ∈ ]1, +∞[ \ Q, m(ω) ≥ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 (4k 2 (ω)) 2 if k 0 (ω) = 1 1 (2k 1 (ω)) 2 if k 0 (ω) = 2 1 if k 0 (ω) ≥ 3 (8.14) For all ω ∈ ]1, 2[ \ Q, m(ω) ≥ 1 (4k 2 (ω)k 3 (ω)) 2 (8.15)
Proof. Estimate (8.14) is an immediate consequence of the formula

ω = k 0 (ω) + 1 k 1 (ω) + 1 k 2 (ω) + 1 k 3 (ω) + . . . Let ω ∈ ]1, 2[ \ Q. Remark that k 0 (f j (ω)) ≥ 3 for 1 ≤ j ≤ k 1 -2 and k 0 (f k 1 (ω)-1 (ω)) = 2 k 1 (f k 1 (ω)-1 (ω)) = k 2 k 0 (f k 1 (ω) (ω)) = 1 k 2 (f k 1 (ω) (ω)) = k 3
Together with (8.14), this yields

m(ω) ≥ min 1, 1, . . . , 1, 1 (2k 2 (ω)) 2 , 1 (4k 3 (ω)) 2 ≥ 1 (4k 2 (ω)k 3 (ω)) 2
Hence, estimate (8.15) is proven.

Define, for any

ω ∈ ]1, 2[ \ Q, ī(ω) def = 1 if k 1 (ω) ≥ 2 2 if k 1 (ω) = 1 (8.16)
Recall that ñ5 is a constant fixed in Proposition 7.1.

Proposition 8.4 (Era transition map).

There exists a constant C6 ≥ C5 such that the properties below hold for

ω ∈ ]1, 2[ \ Q, 0 < h ≤ C-1 6 , h = hk 1 (ω) -ñ 5 , 0 < h ⊥ ≤ h C6 (k 1 (ω)+1) 3 m(ω) 2 . Let h = h, h ⊥ , h C6 (k 1 (ω)+1) m(ω) 2 k 2 (ω)k 3 (ω)
. The era transition map

Φω,h ∶ S s ω,h ⊂ S h → S h
is well defined and takes its values in S sī (ω) h

. Moreover, for every x, x ∈ S s ω,h , we have the following estimates, where Φ = Φω,h and ΦA = ΦA ω,h :

(Control of the distance to the Mixmaster attractor)

dist ∞ Φ(x), A = Φ(x) - ΦA (x) ⊥ ≤ h 1+ k1 (ω) 4 ⊥ (8.17)
(Control of the drift tangential to the Mixmaster attractor)

Φ(x) - ΦA (x) / / ≤ 34h ⊥ k 1 (ω) 2 (8.18)
(Contraction in the direction transverse to the Mixmaster attractor)

∥ Φ(x) -Φ(x)∥ ⊥ ≤ h k1(ω) 25 ⊥ ∥x -x∥ ∞ (8.19)
(Lipschitz control in the direction tangential to the Mixmaster attractor)

Φ(x) -Φ(x) - ΦA (x) - ΦA (x) / / ≤ h 1 k1(ω)+4 ⊥ ∥x -x∥ ∞ + 16k 1 (ω) 2 ∥x -x∥ ⊥ (8.20)
(Control of the expansion in the direction tangent to the Mixmaster attractor)

∥ Φ(x) -Φ(x)∥ / / ≥ K f (ω) ∥x -x∥ / / -h 1 k1 (ω)+4 ⊥ ∥x -x∥ ∞ -h C6 ∥x -x∥ ⊥ (8.21) (Global lipschitz constant) ∥ Φ(x) -Φ(x)∥ ∞ ≤ 4 k 1 (ω)+2 k 1 (ω) 2 ∥x -x∥ ∞ (8.22)
First, we will show that the map Φω,h can be expressed as a composition of several epoch transition maps. Once this is done, we will be left to apply recursively Proposition 7.1 to obtain the estimates on the era transition map.

From now on, assume that C4 ≥ 1, C5 ≥ 1000 C4 and ñ5 ≥ 1000. Fix ω ∈ ]1, 2[ \ Q and 0 < h ≤ 1 6 C5 2 ñ5 -1 . We now proceed to define the epoch transition maps that we will be using to decompose the era transition map. Let h ∶= hk 1 (ω)

-ñ 5 and

h j def = h if j = 0 or j = k 1 (ω) h if 1 ≤ j ≤ k 1 (ω) -1
Define, for 0 ≤ j ≤ k 1 (ω), the section parameters

h j,⊥ def = h C5 (k 1 (ω)-j+1) 3 m(ω) 2 and h j,// def = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ h C5 (k 1 (ω)-j+1) m(ω) 2 k 2 (ω)k 3 (ω) if j = 0 h C5 (k 1 (ω)-j+1) m(ω) 2 if 1 ≤ j ≤ k 1 (ω) and h j def = h j , h j,⊥ , h j,// , h ′ j def = h j , h j , h j
We will use the epoch transition maps

Φ j def = Φ f j (ω),h j ,h ′ j+1 ∶ S s f j (ω),h j → S s f j+1 (ω),h ′ j+1 , 0 ≤ j ≤ k 1 (ω) -1 Define, for 0 ≤ j ≤ k 1 (ω) -1, Φ * j def = Φ j • ⋅ ⋅ ⋅ • Φ 0 , Φ * A j def = Φ * j • Proj A Our goal is to prove that Φω,h 0 = Φ * k 1 (ω)-1
To simplify the notation, let

S s j def = S s f j (ω),h j , 0 ≤ j ≤ k 1 (ω)
Remark that the sections we consider become larger as j increases from 0 to k 1 (ω). The departure section S s 0 and the arrival section S s k 1 (ω) are at distance h from the Kasner circle. The intermediate sections S s j (0 < j < k 1 (ω)) are chosen much closer to the Kasner circle, at distance h ≪ h. Lemma 8.5. For every 0 ≤ j ≤ k 1 (ω) -1, the epoch transition map Φ j is well defined on the section

S s j . Proof. Remark that k 0 (f j (ω)) = k 1 (ω) -j + 1, for all 1 ≤ j ≤ k 1 (ω) and h j ≤ C5 f j (ω) ñ5 -1 , for all 0 ≤ j ≤ k 1 (ω) and h = min(h, h) ≤ min(h j , h j+1 ), for all 0 ≤ j ≤ k 1 (ω) -1 Moreover, recall that m(ω) = min 0≤j≤k 1 (ω) m(f j (ω)) so m(ω) ≤ m(f j (ω)), for all 0 ≤ j ≤ k 1 (ω)
Hence, Lemma 8.5 is a direct consequence of Proposition 7.1.

Lemma 8.6. For every 0

≤ j ≤ k 1 (ω) -1, Φ j S s j ⊂ S s j+1 . Proof. Let x ∈ S s 0 . According to (7.5), dist ∞ (Φ 0 (x), A) = Φ 0 (x) -Φ A 0 (x) ⊥ ≤ h 5 4 0,⊥ ≤ h 0,⊥ ≤ h 1,⊥
According to (7.6) and Proposition 7.3 on the local Lipschitz constant for the Kasner map,

Φ 0 (x) -P s 2 f (ω),h 1 / / ≤ Φ 0 (x) -Φ A 0 (x) / / + Φ A 0 (x) -P s i(ω) f (ω),h 1 / / ≤ 2h 0,⊥ Lip f (ω) + Lip f (ω)h 0,// ≤ 16k 1 (ω) 2 2 h C5 (k 1 (ω)+1) 3 + h C5 (k 1 (ω)+1) k 2 (ω)k 3 (ω) m(ω) 2 using Lip f (ω) = 16k 1 (ω) 2 ≤ h C5 k 1 (ω) m(ω) 2 × 48k 1 (ω) 2 h C5
Since C5 ≥ 1000 and ñ5 ≥ 1000, one can check that 48k 1 (ω)

2 h C5 ≤ 1. Hence, Φ 0 (x) -P s 2 f (ω),h 1 / / ≤ h 1,//
and we can conclude that Φ 0 S

s 0 ⊂ S s 1 . Now, fix 1 ≤ j ≤ k 1 (ω) -1 and x ∈ S s j . According to (7.5), dist ∞ Φ j (x), A = Φ j (x) -Φ A j (x) ⊥ ≤ h k1 (ω)-j+5 k1 (ω)-j+4 j,⊥ ≤ h j,⊥ ≤ h j+1,⊥
According to (7.6) and Proposition 7.3 on the local Lipschitz constant for the Kasner map,

Φ j (x) -P s 1 f j+1 (ω),h j+1 / / ≤ Φ j (x) -Φ A j (x) / / + Φ A j (x) -P s 1 f j+1 (ω),h j+1 / / ≤ 2h j,⊥ Lip f (f j (ω)) + Lip f (f j (ω))h j,// ≤ 2 h C5 (k 1 (ω)-j+1) 3 + h C5 (k 1 (ω)-j+1) m(ω) 2 using Lip f (f j (ω)) = 1 ≤ h j+1,// × 3 h C5 using m(ω) ≤ 1 ≤ h j+1,//
Hence, Φ j S s j ⊂ S s j+1 . This concludes the proof of Lemma 8.6.

Corollary 8.7. The map Φ * k 1 (ω)-1 is well defined on the whole section S s 0 and takes its values in the global section S

sī (ω) h . Proof. Recall that S h = S s 1 h ∪ S s 2 h where S s 1 h def = x = (x u , x s 1 , x s 2 , x c ) | x s 1 = h, 0 ≤ x u ≤ h, 0 ≤ x s 2 ≤ h, 1 < x c < 2
and analogously for S s 2 h . The fact that Φ * k 1 (ω)-1 is well defined on the section S s 0 is a direct consequence of Lemma 8.6. Moreover, Lemma 8.6 informs us that Φ * k 1 (ω)-1 takes its values in the section S

s k 1 (ω) = S s f k1 (ω) (ω),h k1 (ω) . Hence, for any x ∈ S s 0 , Φ * k 1 (ω)-1 (x) -Φ * A k 1 (ω)-1 (x) ⊥ ≤ h k 1 (ω),⊥ ≤ h and Φ * k 1 (ω)-1 (x) -P sī (ω) f (ω),h / / ≤ h k 1 (ω),// ≤ 1 2 m(ω) ≤ min f (ω) -1 2 , 2 -f (ω) 2 Since 1 < f (ω) < 2 and x c P sī (ω) f (ω),h = f (ω), the above inequality implies that 1 < x c Φ * k 1 (ω)-1 (x) < 2 Hence, Φ * k 1 (ω)-1 (x) ∈ S h . More precisely, Proposition 7.1 implies that Φ * k 1 (ω)-1 takes its values in S sī (ω) h .
Lemma 8.8. The era transition map Φω,h 0 is well defined on the whole section S s 0 and takes its values in S

sī (ω) h . Moreover, Φω,h 0 = Φ * k 1 (ω)-1 = Φ k 1 (ω)-1 • ⋅ ⋅ ⋅ • Φ 0 (8.23)
Proof. In this proof, we will denote k 1 = k 1 (ω) and we will assume that k 1 ≥ 2. Indeed, if k 1 = 1, the era transition map coincides with the epoch transition map and Lemma 8.8 is a straightforward consequence of Proposition 7.1.

Claim 1. The era transition map Φω,h 0 is well defined on

S s 0 ∩ B IX and Φω,h 0 (x) = Φ * k 1 -1 (x) for every x ∈ S s 0 ∩ B IX .
Proof of claim 1. Let x ∈ S s 0 ∩B IX and q be the orbit of the Wainwright-Hsu vector field X with initial condition q(0) = ξ -1 (x). Let q 0 = ξ -1 (x) and q j = ξ -1 (Φ * j-1 (x)) for any 1 ≤ j ≤ k 1 . Since x ∈ B IX , we have x u ≠ 0, x s 1 ≠ 0 and x s 2 ≠ 0. It follows by induction that for any 0 ≤ j ≤ k 1 -1, we have q j u ≠ 0 and q j+1 is the first intersection point of the orbit of the Wainwright-Hsu vector field X starting at q j with the section S s j+1 ∶= ξ -1 S s j+1 . Hence, q k 1 is a point belonging both to the orbit starting at q 0 = ξ -1 (x) and to the global section S h . This proves that the era transition map Φω,h 0 is well defined on S s 0 ∩ B IX . We are now going to prove that Φω,h 0

(x) = Φ * k 1 -1 (x), i.e. that ξ -1 (Φ * k 1 -1 (x))
is the first intersection point of the orbit q with the section S h . Let t s 0 = 0 and let t u 0 be the first time t > 0 such that q(t) ∈ S u ω,h ′ 0 . By induction, define, for every 1

≤ j ≤ k 1 -1, t s j = min t > t u j-1 | q(t) ∈ S s j t u j = min t > t s j | q(t) ∈ S u f j (ω),h ′ j t s k 1 = min t > t u k 1 -1 | q(t) = ξ -1 (Φ * k 1 -1 (x))
With these notations, we are left to prove that for any t ∈ 0, t s k 1 , q(t) does not belong to S h . The general idea is simple: either q(t) is close to a type II orbit that is far away from the section S h or q(t) is close to the Kasner circle and we can use the local estimates of Section 5.

Case t ∈ t s j , t u j , 0 ≤ j ≤ k 1 -1. According to Proposition 5.17 and corollary 5.18, q s 1 and q s 2 are exponentially decreasing. Hence, q s 1 (t) < q s 1 (t s j ) ≤ h and q s 2 (t) < q s 1 (t s j ) ≤ h. This implies that q(t) does not belong to the section S h . , q u is strictly increasing and q u (t u 0 ) = h so q(t) does not belong to the section S h .

Case t ∈ t

Let us denote by p the (type II) orbit with initial condition p(t

u 0 ) = P u ω,h . Let η = max ∥ξ(q(t u 0 )) -P u ω,h ∥ ⊥ , ∥ξ(q(t u 0 )) -P u ω,h ∥ / / h C4 ω m(ω)
Using (5.1) and (5.2), we get

η ≤ max (h 0,⊥ ) ω+2 ω+1 h -1 , h 0,⊥ h C5 ω ñ5 h C4 ω m(ω) ≤ h 0,⊥ h C4 ω m(ω) using h ≤ 1 6 C5 2 ñ5 -1 and (h 0,⊥ ) 1 ω+1 h -1 ≤ 1 ≤ h C5 (k 1 +1) 3 m(ω) 2 h C4 ω m(ω) ≤ h100 m(ω) ≤ 1 C5 f (ω) ñ5 ω ñ5 h 1000 m(ω) ≤ 1
Hence, we can apply (6.31) with h u = h and h s = h to get that, for every . As a consequence, estimate (8.24) together with the estimate (3.5b) on the local coordinate system ξ imply that

t ∈ t u 0 , t s 1 , d B (q(t), p(t)) ≤ η ≤ 1 C5 f (ω) ñ5 ω ñ5 h 1000 m(ω) (8.
q s 1 (t out 0 ) < 1 2 C5 ω ñ5 , q s 2 (t out 0 ) < 1 2 C5 ω ñ5 , q c (t out 0 ) -ω < min 1 2 C5 ω ñ5 , ω -1 2
Recall that the orbit ξ •q leaves the open ball B ω,2 C5 ,ñ 5 at time t = t out 0 . Hence, q u (t out 0 ) = 1 2 C5 ω ñ5 ≥ 3h. Using (8.24) together with (3.5b) once again, we get that p u (t out 0 ) ≥ 2h. It follows that for every t ∈ t out 0 , t s 1 , one of the three following properties hold:

1. p u (t) ≥ 2h (roughly, before p leaves U ξ ) 2. p(t) ∉ U ξ 3. p c (t) = f (ω) (roughly, after p re-enters U ξ )
Let t ∈ t out 0 , t s 1 and assume that q(t) ∈ S h . Estimate (8.24) implies that p(t) belongs to the domain U ξ of the local coordinates system ξ. If p u (t) ≥ 2h, then we get q u (t) > h using (8.24) with (3.5b). If p c (t) = f (ω), then we get

|q c (t) -f (ω)| ≤ f (ω)
-2 2 using (8.24) with (3.5b) once again. Since f (ω) > 2, it follows that q c (t) > 2. In both cases, q(t) ∉ S h so this is absurd.

Case t ∈ t u j , t s j+1 , 1 ≤ j ≤ k 1 -2.
Let us denote by p the (type II) orbit with initial condition p(t

u j ) = P u f j (ω), h. Recall that k 1 -j + 1 ≤ f j (ω) = [k 1 -j + 1; k 2 , . . . ] ≤ k 1 -j + 2
We now prove a similar estimate to (8.24), using the same arguments. For every t ∈ t u j , t s j+1 , we have

d B (q(t), p(t)) ≤ max ξ(q(t u j )) -P u f j (ω), h ⊥ , ξ(q(t u j )) -P u f j (ω), h / / h C4 f j (ω) m(f j (ω)) using (6.31) with h u = h and h s = h ≤ max (h j,⊥ ) f j (ω)+2 f j (ω)+1 h-1 , h j,⊥ h C5 f j (ω) ñ5 h C4 f j (ω) m(f j (ω))
using (5.1) and (5.2)

≤ h j,⊥ h C4 f j (ω) m(f j (ω)) ≤ h C5 (k 1 -j+1) 3 m(ω) 2 h C4 f j (ω) m(f j (ω)) ≤ 1 C5 f j (ω) ñ5 f j+1 (ω) ñ5 h 1000 m(ω) (8.25) Let t ∈ t u j , t s j+1
and assume that q(t) ∈ S h . It follows from (8.25) and the estimate (3.5b) on the local coordinate system ξ that p(t) ∈ U ξ . Since p is a type II orbit, we have either p c (t) = f j (ω) or p c (t) = f j+1 (ω). Moreover, estimate (8.25) together with the estimate (3.5b) on the local coordinate system implies that

|q c (t) -p c (t)| ≤ f j+1 (ω) -2 2 Since 2 < f j+1 (ω) < f j (ω)
, it follows that q c (t) > 2. This contradicts the fact that q(t) belongs to the section S h .

Case t ∈ t u k 1 -1 , t s k 1 . Assume that there exists t ∈ t u k 1 -1 , t s k 1 such that q(t) ∈ S h . Then it must satisfy q(t) ∈ S h \ S s f (ω),h ′ k1(ω) because ξ -1 (Φ * k 1 -1 (x))
is the first intersection point of the orbit starting at q(t u k 1 -1 ) with the section

S s f (ω),h ′ k1 (ω)
. But estimate (8.25) with j = k 1 -1 is also valid on t

u k 1 -1 , t s k 1 and implies that q(t) ∈ S s f (ω),h ′ k1(ω)
. This is a contradiction. This concludes the proof of claim 1.

Claim 2. The era transition map Φω,h 0 is well defined at every point of

S s 0 ∩ B II and Φω,h 0 (x) = Φ * k 1 -1 (x) for every x ∈ S s 0 ∩ B II . Proof of claim 2. Let x ∈ S s 0 ∩ B II . In particular, x u = 0. Iteration of formula (4.8) gives Φ * k 1 -1 (x) = (0, h, 0, f k 1 (x c )) if k 1 ≥ 2 (0, 0, h, f k 1 (x c )) if k 1 = 1 Moreover, if we denote by [1; k 1 (x c ), k 2 (x c ), .
. . ] the continued fraction associated with x c , formula (4.6) can be rewritten as follows:

Φω,h (x) = (0, h, 0, f k 1 (x c ) (x c )) if k 1 (x c ) ≥ 2 (0, 0, h, f k 1 (x c ) (x c )) if k 1 (x c ) = 1
We are left to prove that k 1 (x c ) = k 1 . This is a consequence of Proposition A.6 together with the fact that |x c -ω| ≤ h 0,// < (10k

1 (ω) 2 k 2 (ω)k 3 (ω)) -1
. This concludes the proof of claim 2.

Claim 3. The era transition map Φω,h 0 is well defined at every point of

S s 0 ∩ B VII 0 and Φω,h 0 (x) = Φ * k 1 -1 (x) for every x ∈ S s 0 ∩ B VII 0 .
Proof of claim 3. This is a mix of claim 1 (before the orbit starting at x converges to a point of the Kasner circle) and claim 2 (after the orbit starting at x converges to a point of the Kasner circle). This concludes the proof of Lemma 8.8, since

S s 0 = (S s 0 ∩ B II ) ⊔ (S s 0 ∩ B VII 0 ) ⊔ (S s 0 ∩ B IX ).
Proof of Proposition 8.4. As in the preceding proof, we will denote k 1 = k 1 (ω) and we will assume that k 1 ≥ 2. The proof of Proposition 8.4 relies on the decomposition (8.23) of the era transition map Φω,h 0 as a product of k 1 epoch transition maps, together with the estimates on these epoch transition maps stated in Proposition 7.1. In other words, estimates (8.17), . . . , (8.21) will be obtained by applying k 1 times the corresponding estimates of Proposition 7.1. More precisely, we are going to prove Proposition 8.4 for a restriction of the map Φω,h 0 . Let C ≥ C5 and define h0,

⊥ def = hC(k 1 +1) 3 m(ω) 2 , h 0,// def = h C5 (k 1 +1) m(ω) 2 k 2 (ω)k 3 (ω) Let 0 < h ⊥ ≤ h0,⊥ , h = h, h ⊥ , h 0,// , x, x ∈ S s ω,h , Φ = Φω,h and ΦA = ΦA ω,h .
We are left to prove the following statement: provided that C is large enough, estimates (8.17), . . . , (8.21) hold true. From now on, we will use the notation Φ 0 ∶= Φ ω,h,h ′ 1 . Beware of the fact that this is the restriction of the former epoch transition map Φ 0 to the smaller section S s ω,h . Proof of estimate (8.17). Define a 0 = h ⊥ and

a j = sup x∈S s ω,h Φ * j-1 (x) -Φ * A j-1 (x) ⊥ , 1 ≤ j ≤ k 1 Claim 1. For all 1 ≤ j ≤ k 1 , a j ≤ h 5 4 k1+4 k1-j+5 ⊥ (8.26)
Proof of claim 1. Recall that k 0 (ω) = 1. Applying (7.5) to the epoch transition map Φ 0 , we get that

a 1 ≤ h 5 4 ⊥ . Assume that a j ≤ h 5 4 k1 +4 k1 -j+5 ⊥
for some 1 ≤ j ≤ k 1 -1. We are now going to apply Proposition 7.1 with h ⊥ = a j . More precisely, we apply (7.5) to the epoch transition map Φ j restricted to the section S s f j (ω),(h j ,a j ,h j,/ / ) . We get that

a j+1 ≤ a k0 (f j (ω))+4 k0 (f j (ω))+3 j = a k1-j+5 k1-j+4 j ≤ h 5 4 k1+4 k1 -j+4
⊥ By induction on j, claim 1 holds true.

Since Φ = Φ * k 1 -1 , estimate (8.17) is a direct consequence of estimate (8.26) with j = k 1 . Proof of estimate (8.18). Define

b j = sup x∈S s ω,h Φ * j-1 (x) -Φ * A j-1 (x) / / , 1 ≤ j ≤ k 1
Using (7.6) and the fact that Lip f (ω) = 16k 2 1 (see the explicit formula (7.3)), we get

b 1 ≤ 2h ⊥ Lip f (ω) ≤ 32k 2 1 h ⊥ (8.27)
We are now going to find a relation between b j+1 and b j . Let 1

≤ j ≤ k 1 -1. Remark that Φ * j (x) -Φ * A j (x) / / ≤ Φ j Φ * j-1 (x) -Φ A j Φ * j-1 (x) / / + Φ A j Φ * j-1 (x) -Φ A j Φ * A j-1 (x) / / (8.28)
As a direct consequence of estimate (7.6) applied to the epoch transition map Φ j restricted to the section S s f j (ω),( h,a j ,h j,/ / ) , we get that

Φ j Φ * j-1 (x) -Φ A j Φ * j-1 (x) / / ≤ 2a j Lip f (f j (ω)) ≤ 2a j (8.29)
since f j (ω) > 2 implies that Lip f (f j (ω)) = 1 by the explicit formula (7.3). Moreover, recall that the x c -coordinate of Φ A j is essentially the Kasner map (see (4.9)). Hence, Proposition 7.3 implies that

Φ A j Φ * j-1 (x) -Φ A j Φ * A j-1 (x) / / ≤ Lip f (f j (ω)) Φ * j-1 (x) -Φ * A j-1 (x) / / ≤ b j (8.30)
Plugging (8.29) and (8.30) in (8.28), we get that b j+1 ≤ 2a j + b j . Hence, using (8.26) and (8.27), it follows that for all 1

≤ j ≤ k 1 , b j ≤ 2 j-1 r=1 a r + b 1 ≤ 2k 1 h ⊥ + 32k 2 1 h ⊥ ≤ 34k 2 1 h ⊥ (8.31) Since Φ = Φ * k 1 -1 , estimate (8.18) is a direct consequence of estimate (8.31) with j = k 1 . Proof of estimate (8.22). Let 0 ≤ l ≤ k 1 -1. Recall that Φ *
l is defined as the product of l + 1 epoch transition maps. Using l + 1 times inequality (7.10), we obtain the following Lipschitz estimate for Φ * l :

∥Φ * l (x) -Φ * l (x)∥ ∞ ≤ l j=0 4 Lip f (f j (ω)) ∥x -x∥ ∞
Recall that Lip f (.) is defined by the explicit formula (7.3) which yields

l j=0 4 Lip f (f j (ω)) = 4 l+1 16k 2 1 since f j (ω) > 2 for 1 ≤ j ≤ k 1 -1. Hence, we obtain ∥Φ * l (x) -Φ * l (x)∥ ∞ ≤ 4 l+3 k 2 1 ∥x -x∥ ∞ (8.32)
Taking l = k 1 -1, we obtain the desired estimate (8.22) for the era transition map Φ = Φ * k 1 -1 . Proof of estimate (8.19). Let us turn to the contraction estimate (8.19). The idea is to decompose the era transition map Φ

= Φ * k 1 -1 as Φ k 1 -1 • Φ * k 1 -2
and to use the contraction estimate (7.7) for the epoch transition map Φ k 1 -1 restricted to the section S s f k1-1 (ω),( h,a k1 -1 ,h k1-1,/ / ) as well as the Lipschitz estimate (8.32) for Φ *

k 1 -2 . For 1 ≤ l ≤ k 1 -1, ∥Φ * l (x) -Φ * l (x)∥ ⊥ = ∥Φ l (Φ * l-1 (x)) -Φ l (Φ * l-1 (x))∥ ⊥ ≤a 1 k0 (f l (ω))+3 l ∥Φ * l-1 (x) -Φ * l-1 (x)∥ ∞ using (7.7) for Φ l restricted to S s f l (ω),( h,a l ,h l,/ / ) ≤a 1 k1 -l+4 l 4 l+2 k 2 1 ∥x -x∥ ∞ using (8.32) and k 0 (f l (ω)) = k 1 -l + 1 (8.33) Taking l = k 1 -1, we get ∥Φ * k 1 -1 (x) -Φ * k 1 -1 (x)∥ ⊥ ≤ a 1 5 k 1 -1 4 k 1 +1 k 2 1 ∥x -x∥ ∞ ≤ h k1 24 ⊥ 4 k 1 +1 k 2 1 ∥x -x∥ ∞ using (8.26) ≤ h k1 25 ⊥ ∥x -x∥ ∞ provided that C is large enough Since Φ = Φ * k 1 -1 , estimate (8.19
) is a direct consequence of the above inequality.

Proof of estimate (8.21). The idea is to decompose the era transition map Φ as the product of the k 1 epoch transition maps Φ j and then to apply recursively the expansion estimate (7.9) to these epoch transition maps. Define,

Λ 1 = 4 3 k 2 1 a 1 k1+3 1 + C5 k ñ5 1 hh 1 4 ⊥ Λ j = 4 j+1 k 2 1 4a 1 k1-j+4 j + C5 (k 1 -j + 1) ñ5 ha 1 k1-j+5 j-1 2 ≤ j ≤ k 1 -1
According to (7.9), we have

∥Φ 0 (x) -Φ 0 (x)∥ / / ≥ K f (ω) ∥x -x∥ / / -h 1 4 ⊥ ∥x -x∥ ∞ -C5 h ∥x -x∥ ⊥ Let 1 ≤ j ≤ k 1 -1. Remark once again that ∥Φ * j (x) -Φ * j (x)∥ / / = ∥Φ j Φ * j-1 (x) -Φ j Φ * j-1 (x) ∥ / /
Hence, the expansion estimate (7.9) applied to the epoch transition map Φ j restricted to the section S s f j (ω),( h,a j ,h j,/ / ) gives

∥Φ * j (x) -Φ * j (x)∥ / / ≥ K f (f j (ω)) ∥Φ * j-1 (x) -Φ * j-1 (x)∥ / / -a 1 k1-j+4 j ∥Φ * j-1 (x) -Φ * j-1 (x)∥ ∞ -C5 (k 1 -j + 1) ñ5 h ∥Φ * j-1 (x) -Φ * j-1 (x)∥ ⊥ Remark that f j (ω) > 5 3 , hence K f (f j (ω)) = 1 by the explicit formula (7.2). If j = 1, use (8.32) to estimate the term ∥Φ * j-1 (x) -Φ * j-1 (x)∥ ∞ = ∥Φ * 0 (x) -Φ * 0 (x)∥ ∞ and (7.7) to estimate the term ∥Φ * j-1 (x) -Φ * j-1 (x)∥ ⊥ = ∥Φ 0 (x) -Φ 0 (x)∥ ⊥ . This gives ∥Φ * 1 (x) -Φ * 1 (x)∥ / / ≥ ∥Φ * 0 (x) -Φ * 0 (x)∥ / / -Λ 1 ∥x -x∥ ∞ If 2 ≤ j ≤ k 1 -1, use (8.32) to estimate the term ∥Φ * j-1 (x) -Φ * j-1 (x)∥ ∞ and (8.33) to estimate the term ∥Φ * j-1 (x) -Φ * j-1 (x)∥ ⊥ . This gives ∥Φ * j (x) -Φ * j (x)∥ / / ≥ ∥Φ * j-1 (x) -Φ * j-1 (x)∥ / / -Λ j ∥x -x∥ ∞
By induction on j, it follows that

∥Φ * k 1 -1 (x) -Φ * k 1 -1 (x)∥ / / ≥ K f (ω) ∥x -x∥ / / -h 1 4 ⊥ + k 1 -1 j=1 Λ j ∥x -x∥ ∞ -C5 h ∥x -x∥ ⊥ Using (8.26), one can see that if C is large enough, then h 1 4 ⊥ + k 1 -1 j=1 Λ j ≤ h 1 k1 +4 ⊥ Hence, if C is large enough, then ∥Φ * k 1 -1 (x) -Φ * k 1 -1 (x)∥ / / ≥ K f (ω) ∥x -x∥ / / -h 1 k1+4 ⊥ ∥x -x∥ ∞ -C5 h ∥x -x∥ ⊥
which is precisely the desired estimate (8.21).

Proof of estimate (8.20). Define, for 0

≤ j ≤ k 1 -1, N j,// = Φ * j (x) -Φ * j (x) -Φ * A j (x) -Φ * A j (x) / / Let 1 ≤ j ≤ k 1 -1. Decompose Φ * j as Φ j • Φ * j-1 and Φ * A j as Φ A j • Φ * A j-1 .
Using the standard triangle inequality, we get

N j,// ≤ Φ j (Φ * j-1 (x)) -Φ j (Φ * j-1 (x)) -Φ A j (Φ * j-1 (x)) -Φ A j (Φ * j-1 (x)) / / + Φ A j (Φ * j-1 (x)) -Φ A j (Φ * j-1 (x)) -Φ A j (Φ * A j-1 (x)) -Φ A j (Φ * A j-1 (x)) / / (8.34)
Let us begin with the second term. Recall that the x c -coordinate of Φ A j is essentially the Kasner map f (see (4.9)). Moreover, f (u) = u -1 for all u ≥ 2 and the x c -coordinates of the four points Φ * j-1 (x), Φ * j-1 (x), Φ * A j-1 (x) and Φ * A j-1 (x) are all larger than 2. Hence,

Φ A j (Φ * j-1 (x)) -Φ A j (Φ * j-1 (x)) -Φ A j (Φ * A j-1 (x)) -Φ A j (Φ * A j-1 (x)) / / = Φ * j-1 (x) -Φ * j-1 (x) -Φ * A j-1 (x) -Φ * A j-1 (x)
/ / = N j-1,// (8.35)

Let us now turn to the first term. Using the Lipschitz estimate (7.8) with the epoch transition map Φ j restricted to the section S s f j (ω),( h,a j ,h j,/ / ) , we get

Φ j (Φ * j-1 (x)) -Φ j (Φ * j-1 (x)) -Φ A j (Φ * j-1 (x)) -Φ A j (Φ * j-1 (x)) / / ≤ a 1 k1-j+4 j ∥Φ * j-1 (x) -Φ * j-1 (x)∥ ∞ + Lip f (f j (ω)) ∥Φ * j-1 (x) -Φ * j-1 (x)∥ ⊥ Since f j (ω) > 2, the explicit formula (7.3) implies that Lip f (f j (ω)) = 1. Now, use (8.26) to estimate a 1 k1 -j+4 j and (8.32) to estimate ∥Φ * j-1 (x) -Φ * j-1 (x)∥ ∞ . We obtain Φ j (Φ * j-1 (x)) -Φ j (Φ * j-1 (x)) -Φ A j (Φ * j-1 (x)) -Φ A j (Φ * j-1 (x)) / / ≤ h 5 4 1 k1 -j+4 ⊥ 4 j+2 k 2 1 ∥x -x∥ ∞ + ∥Φ * j-1 (x) -Φ * j-1 (x)∥ ⊥ If j = 1, use (7.7) to estimate the term ∥Φ * j-1 (x) -Φ * j-1 (x)∥ ⊥ = ∥Φ 0 (x) -Φ 0 (x)∥ ⊥ . This gives Φ 1 (Φ * 0 (x)) -Φ 1 (Φ * 0 (x)) -Φ A 1 (Φ * 0 (x)) -Φ A 1 (Φ * 0 (x)) / / ≤ h 5 4 1 k1+3 ⊥ 4 3 k 2 1 + h 1 4 ⊥ ∥x -x∥ ∞ ≤ h 5 4 1 k1 +3 ⊥ 4 4 k 2 1 ∥x -x∥ ∞ (8.36) If 2 ≤ j ≤ k 1 -1, use (8.33) to estimate the term ∥Φ * j-1 (x) -Φ * j-1 (x)∥ ⊥ . This gives Φ j (Φ * j-1 (x)) -Φ j (Φ * j-1 (x)) -Φ A j (Φ * j-1 (x)) -Φ A j (Φ * j-1 (x)) / / ≤ h 5 4 1 k1-j+4 ⊥ 4 j+2 k 2 1 + a 1 k1-(j-1)+4 j-1 4 j+1 k 2 1 ∥x -x∥ ∞ ≤h 5 4 1 k1+3 ⊥ 4 j+3 k 2 1 ∥x -x∥ ∞ (8.37)
Plugging (8.35), (8.36) and (8.37) into (8.34), we get

N j,// ≤ h 5 4 1 k1 +3 ⊥ 4 j+3 k 2 1 ∥x -x∥ ∞ + N j-1,//
By induction on j, it follows that

N k 1 -1,// ≤ h 5 4 1 k1 +3 ⊥ 4 k 1 +3 k 3 1 ∥x -x∥ ∞ + N 0,//
As a direct consequence of the Lipschitz estimate (7.8) applied to the epoch transition map Φ 0 , we get

N 0,// ≤ h 1 4 ⊥ ∥x -x∥ ∞ + 16k 2 1 ∥x -x∥ ⊥ Hence, provided that C is large enough, N k 1 -1,// ≤ h 1 k1+4 ⊥ ∥x -x∥ ∞ + 16k 2 1 ∥x -x∥ ⊥ which is the desired estimate (8.20).
This concludes the proof of Proposition 8.4.

Control of the double era transition map Φω,h

Recall that

s 2 (ω) = k 1 (ω) 2 + k 2 (ω) 2 + k 3 (ω) 2 + k 4 (ω) 2 s 4 (ω) = k 1 (ω) 4 + k 2 (ω) 4 + k 3 (ω) 4 + k 4 (ω) 4
Proof of Proposition 8.2. The general idea of this proof is to see the double era transition map as the composition of two era transition maps and to apply twice Proposition 8.4. Let

ĥ def = 1 4 K f -K c 1 + 1 σ 1 C6 (8.38) Observe that, since K f = 36 25 and K c = 1+ K f 2
(see (8.2) and (8.4)), we have ĥ < C-1 6 . This will allow us to use Proposition 8.4 with h = ĥ.

Let ω ∈ ]1, 2[ \ Q. To avoid clutter, denote k j ∶= k j (ω) for all j ≥ 1. Define

h ω = ĥ, h C6 (k 1 +1) 3 ω m(ω) 2 , h C6 (k 1 +1) ω m(ω) 2 k 2 k 3 where hω = ĥk -ñ 5 1
, and

h f (ω) = ĥ, h C6 (k 2 +1) 3 f (ω) m( f (ω)) 2 , h C6 (k 2 +1) f (ω) m( f (ω)) 2 k 3 k 4 where h f (ω) = ĥk -ñ 5 2
. According to Proposition 8.4, the era transition maps Φω,h ω and Φ f (ω),h f (ω) are well defined and satisfy the estimates (8.17), . . . , (8.22).

Let C ≥ C6 be some large constant. Define

h ′ ω = ĥ, h ⊥ , e -Cs 2 (ω)
where 0 < h ⊥ ≤ e -Cs 4 (ω)

and

h ′ f (ω) = ĥ, h ′ ⊥ , h C6 (k 2 +1) f (ω) m( f (ω)) 2 k 3 k 4 where h ′ ⊥ = h 1+ k1 4 ⊥
Using (8.15) together with the fact that ĥ is fixed, we get that, for all C large enough,

e -Cs 4 (ω) ≤ h C6 (k 1 +1) 3 ω m(ω) 2 and e -Cs 2 (ω) ≤ h C6 (k 1 +1) ω m(ω) 2 k 2 k 3 (we use that ln(k 1 ) = o k 1 →+∞ (k 1 )).
Hence, for all C large enough, the era transition map Φω,h ′ ω is a restriction of Φω,h ω . Analogously, for all C large enough,

e -Cs 4 (ω) 1+ k1 4 ≤ h C6 (k 2 +1) 3 f (ω) m( f (ω)) 2 so the era transition map Φ f (ω),h ′ f (ω) is a restriction of Φ f (ω),h f (ω)
. Applying (8.17) and (8.18) to Φ f (ω),h ′ f (ω) , we also get that, for all C large enough,

Φω,h ′ ω S s ω,h ′ ω ⊂ S s f (ω),h ′ f (ω)
and the decomposition Φω,h . We also get that for every

′ ω = Φ f (ω),h ′ f (ω)
x ∈ S s ω,h ′ ω , k 1 (x c ) = k 1 (ω) = k 1 and k 1 ( Φω,h ′ ω (x) c ) = k 1 ( f (ω))
= k 2 (it also can be seen as a consequence of Proposition A.7). We are left to prove estimates (8.8), . . . , (8.12) using the preceding decomposition. From now on, we will be using the simplified notation

Φ = Φω,h ′ ω Φω = Φω,h ′ ω Φ f (ω) = Φ f (ω),h ′ f (ω) ΦA = ΦA ω,h ′ ω ΦA ω = ΦA ω,h ′ ω ΦA f (ω) = ΦA f (ω),h ′ f (ω) Let x, x ∈ S s ω,h ′ ω .
Proof of estimate (8.8). Using (8.17) twice, we get

dist ∞ Φω (x), A = Φω (x) - ΦA ω (x) ⊥ ≤ h ′ ⊥ = h 1+ k1 4 ⊥ and then dist ∞ Φ(x), A = Φ(x) - ΦA (x) ⊥ ≤ h ′ ⊥ 1+ k2 4 = h 1+ k1 4 ⊥ 1+ k2 4 ≤ h 1+ k1 4 + k2 4 ⊥
Hence, estimate (8.8) holds true.

Proof of estimate (8.9). Using the triangle inequality, we get

Φ(x) - ΦA (x) / / = Φ f (ω) • Φω (x) - ΦA f (ω) • ΦA ω (x) / / ≤ Φ f (ω) ( Φω (x)) - ΦA f (ω) ( Φω (x)) / / + ΦA f (ω) ( Φω (x)) - ΦA f (ω) ( ΦA ω (x)) / /
(8.39) Applying (8.18) to Φ f (ω) , we get

Φ f (ω) ( Φω (x)) - ΦA f (ω) ( Φω (x)) / / ≤ 34h ′ ⊥ k 2 2 = 34h 1+ k1 4 ⊥ k 2 2
(8.40) Using Proposition 7.3 (we will prove below that the assumptions of this proposition are indeed satisfied) and the fact that k 1 ((

Φω (x)) c ) = k 2 = k 1 (( ΦA ω (x)) c ), we get ΦA f (ω) ( Φω (x)) - ΦA f (ω) ( ΦA ω (x)) / / ≤ 16k 2 2 Φω (x) - ΦA ω (x) / /
and then, applying (8.18) to Φω , we get

ΦA f (ω) ( Φω (x)) - ΦA f (ω) ( ΦA ω (x)) / / ≤ 16 × 34h ⊥ k 2 1 k 2 2
(8.41)

Plugging (8.40) and (8.41) into (8.39), we get

Φ(x) - ΦA (x) / / ≤ 34h 1+ k1 4 ⊥ k 2 2 + 16 × 34h ⊥ k 2 1 k 2 2 ≤ 34h ⊥ k 2 1 k 2 2 + 16 × 34h ⊥ k 2 1 k 2 2 ≤ 578h ⊥ k 2 1 k 2 2
Hence, estimate (8.9) holds true under the assumption that we can use Proposition 7.3. In order to use Proposition 7.3, we need to make sure that

Φω (x) c ∈] f (ω) -η, f (ω) + η[ and ΦA ω (x) c ∈] f (ω) -η, f (ω) + η[ where η = min f (ω)-1 2 , | f (ω)-2| 2 . We know that Φω (x) ∈ S s f (ω),h ′ f (ω) and ΦA ω (x) ∈ S s f (ω),h ′ f (ω)
. Moreover, the size of this section in the tangential direction is

h f (ω),// def = h C6 (k 2 +1) f (ω) m( f (ω)) 2 k 3 k 4
Hence, we have

| Φω (x) c -f (ω)| ≤ h f (ω),// and ΦA ω (x) c -f (ω) ≤ h f (ω),//
and we are left to prove that h f (ω),// ≤ η. On the one hand, we have h C6 (k 2 +1)

f (ω)
≤ 1 2 and m( f (ω))

2 k 3 k 4 ≤ m( f (ω)) 2 ≤ m( f (ω)) 2 = min 1, f (ω) -2 2 2 ≤ | f (ω) -2| so h f (ω),// ≤ | f (ω)-2| 2
. On the other hand, f (ω

) -1 ≥ 1 2k 2 and h C6 (k 2 +1) f (ω) ≤ 1 4k 2 so h f (ω),// ≤ f (ω)-1 2
. Hence, we can apply Proposition 7.3 in this context.

Proof of estimate (8.10). We have

Φ(x) -Φ(x) ⊥ = ∥ Φ f (ω) ( Φω (x)) -Φ f (ω) ( Φω (x))∥ ⊥ ≤ h ′ ⊥ k2 25 ∥ Φω (x) -Φω (x)∥ ∞ applying (8.19) to Φ f (ω) ≤ h 1+ k1 4 ⊥ k2 25 ∥ Φω (x) -Φω (x)∥ ∞ ≤ h k2 25 + k1k2 100 ⊥ 4 k 1 +2 k 2 1 ∥x -x∥ ∞ using (8.22) ≤ h k1 100 + k2 100 ⊥ 4 k 1 +2 k 2 1 h 1 100 ⊥ ∥x -x∥ ∞ using k 2 ≥ 1
Recall that h ⊥ ≤ e -Cs 4 (ω) . Hence, for all C large enough, we have

4 k 1 +2 k 2 1 h 1 100
⊥ ≤ 1 and it follows that, for all C large enough, estimate (8.10) holds true.

Proof of estimate (8.11). Let us introduce the point x = (x u , xs 1 , xs 2 , x c ). Observe that x is the unique point in S Using the triangle inequality, we get

Φ(x) -Φ(x) - ΦA (x) - ΦA (x) / / ≤ A 1 (x, x) + A 2 (x, x) + A 2 (x, x) (8.42)
where

A 1 (x, x) = Φ f (ω) ( Φω (x)) -Φ f (ω) ( Φω (x)) - ΦA f (ω) ( Φω (x)) - ΦA f (ω) ( Φω (x)) / / A 2 (x, x) = ΦA f (ω) ( Φω (x)) - ΦA f (ω) ( Φω (x)) - ΦA f (ω) ( ΦA ω (x)) - ΦA f (ω) ( ΦA ω (x)) / / A 2 (x, x) = ΦA f (ω) ( Φω (x)) - ΦA f (ω) ( Φω (x)) - ΦA f (ω) ( ΦA ω (x)) - ΦA f (ω) ( ΦA ω (x)) / /
Applying (8.20) to Φ f (ω) , we get

A 1 (x, x) ≤ h ′ ⊥ 1 k2+4 ∥ Φω (x) -Φω (x)∥ ∞ + 16k 2 2 ∥ Φω (x) -Φω (x)∥ ⊥ ≤ h 1+ k1 4 ⊥ 1 k2+4 ∥ Φω (x) -Φω (x)∥ ∞ + 16k 2 2 ∥ Φω (x) -Φω (x)∥ ⊥
and then, applying (8.19) and (8.22) to Φω , we get

A 1 (x, x) ≤ h 1+ k1 4 ⊥ 1 k2 +4 4 k 1 +2 k 2 1 ∥x -x∥ ∞ + 16k 2 2 h k1 25 ⊥ ∥x -x∥ ∞ (8.43)
Claim 1. We have

A 2 (x, x) ≤ 16k 2 2 h 1 k1 +4 ⊥ ∥x -x∥ ∞ + 16 2 k 2 1 k 2 2 ∥x -x∥ ⊥ (8.44) Proof of claim 1. Since ∥x -x∥ / / = 0, we have ΦA ω (x) = ΦA ω (x) (we use here that k 1 (x c ) = k 1 (x c
) and formula (4.6)). Hence,

A 2 (x, x) = ΦA f (ω) ( Φω (x)) - ΦA f (ω) ( Φω (x)) / / ≤ Lip f ( f (ω)) ∥ Φω (x) -Φω (x)∥ / /
using Proposition 7.3

≤ Lip f ( f (ω)) Φω (x) -Φω (x) - ΦA ω (x) - ΦA ω (x) / / ≤ Lip f ( f (ω)) h 1 k1 +4 ⊥ ∥x -x∥ ∞ + 16k 2 1 ∥x -x∥ ⊥ using (8.20) ≤ 16k 2 2 h 1 k1+4 ⊥ ∥x -x∥ ∞ + 16 2 k 2 1 k 2 2 ∥x -x∥ ⊥ using (7.3) ≤ 16k 2 2 h 1 k1+4 ⊥ ∥x -x∥ ∞ + 16 2 k 2 1 k 2 2 ∥x -x∥ ⊥ Claim 2. We have A 2 (x, x) ≤ 16k 2 2 h 1 k1 +4 ⊥ ∥x -x∥ ∞ + 16 × 34 × 128h ⊥ k 4 1 k 3 2 ∥x -x∥ ∞ (8.45)
Proof of claim 2. Using formula (4.6), we get that Remark that ΦA c (t) = f (t). From now on, assume that xu ≠ 0, xs 1 ≠ 0 and xs 2 ≠ 0 (these cases can be obtained by continuity once inequality (8.44) is obtained in the general case). Using this assumption and the fact that for all t ∈ [x c , x c ], k 1 (t) = k 1 , k 2 (t) = k 2 and k 1 ( Φc (t)) = k 2 , we get that all the derivatives exist in the following formula :

ΦA f (ω) ( Φω (x)) c = f Φω (x) c ΦA f (ω) ( Φω (x)) c = f Φω (x) c ΦA f (ω) ( ΦA ω (x)) c = f ΦA ω (x) c ΦA f (ω) ( ΦA ω (x)) c = f ΦA ω (x) c Hence, A 2 (x, x) = f Φω (x) c -f Φω (x) c -f ΦA ω (x) c -f ΦA ω (x) c Recall that x = (x u ,
A 2 (x, x) = x c xc f • Φc ′ (t)dt - x c xc f • ΦA c ′ (t)dt = x c xc Φ′ c (t) f ′ ( Φc (t)) - ΦA c ′ (t) f ′ ΦA c (t) dt ≤ I 1 + I 2 (8.46)
where

I 1 = x c xc Φ′ c (t) - ΦA c ′ (t) f ′ ( Φc (t)) dt I 2 = x c xc ΦA c ′ (t) f ′ ( Φc (t)) - f ′ ( ΦA c (t)) dt
On the one hand, estimate (8.20) with ∥x -

x∥ ⊥ = 0 gives Φ′ c (t) - ΦA c ′ (t) ≤ h 1 k1+4 ⊥ Moreover, using Proposition 7.3, we get f ′ ( Φc (t)) ≤ Lip f ( f (ω)) ≤ 16k 2 2
Hence,

I 1 ≤ 16k 2 2 h 1 k1 +4 ⊥ |x c -xc | = 16k 2 2 h 1 k1 +4 ⊥ ∥x -x∥ ∞ (8.47)
On the other hand, using Proposition 7.3, we get

ΦA c ′ (t) = f ′ (t) ≤ 16k 2 1 and f ′ ( Φc (t)) - f ′ ( ΦA c (t)) ≤ Lip f ′ ( f (ω)) Φc (t) - ΦA c (t) Using (8.18), we get f ′ ( Φc (t)) - f ′ ( ΦA c (t)) ≤ 128k 3 2 × 34h ⊥ k 2 1
Hence,

I 2 ≤ 16 × 34 × 128h ⊥ k 4 1 k 3 2 |x c -xc | = 16 × 34 × 128h ⊥ k 4 1 k 3 2 ∥x -x∥ ∞ (8.
Φ(x) -Φ(x) - ΦA (x) - ΦA (x) / / ≤ h 1+ k1 4 ⊥ 1 k2 +4 4 k 1 +2 k 2 1 + 16k 2 2 h k1 25 ⊥ + 32k 2 2 h 1 k1 +4 ⊥ + 16 × 34 × 128h ⊥ k 4 1 k 3 2 ∥x -x∥ ∞ + 16 2 k 2 1 k 2 2 ∥x -x∥ ⊥
We are left to prove that, for all C large enough,

h 1+ k1 4 ⊥ 1 k2 +4 4 k 1 +2 k 2 1 + 16k 2 2 h k1 25 ⊥ + 32k 2 2 h 1 k1 +4 ⊥ + 16 × 34 × 128h ⊥ k 4 1 k 3 2 ≤ h 1 26k1 ⊥ + h 1 26k2

⊥

Let us give some details for the first term (computations are similar to the ones detailed in claim 3 below) :

h 1+ k1 4 ⊥ 1 k2+4 4 k 1 +2 k 2 1 ≤ h 1 5k2 ⊥ 4 k 1 +2 k 2 1 ≤ h 4 25k2 ⊥ 4 k 1 +2 k 2 1 h 1 25k2 ⊥ ≤ e -4 25 C k 2 1 +k 2 2 k2 4 3k 1 e 2k 1 h 1 25k2 ⊥ ≤ e -4 25 C 2k1k2 k2 4 3k 1 e 2k 1 h 1 25k2 ⊥ ≤ e 2+3 ln(4)-8 25 C k 1 h 1 25k2 ⊥ ≤ e 2+3 ln(4)-8 25 C h 1 25k2

⊥

Hence, for all C large enough,

h 1+ k1 4 ⊥ 1 k2+4 4 k 1 +2 k 2 1 ≤ 1 10 h 1 25k2 ⊥ ≤ 1 10 h 1 26k2

⊥

Conducting similar computations for the remaining terms, we get that estimate (8.11) holds true for all C large enough.

Proof of estimate (8.12). We have

Φ(x) -Φ(x) / / = ∥ Φ f (ω) ( Φω (x)) -Φ f (ω) ( Φω (x))∥ / /
so, applying (8.21) to Φ f (ω) , we get

Φ(x) -Φ(x) / / ≥K f ( f (ω)) ∥ Φω (x) -Φω (x)∥ / / -h ′ ⊥ 1 k2+4 ∥ Φω (x) -Φω (x)∥ ∞ -ĥ C6 ∥ Φω (x) -Φω (x)∥ ⊥ ≥K f ( f (ω)) ∥ Φω (x) -Φω (x)∥ / / -h 1+ k1 4 ⊥ 1 k2+4 ∥ Φω (x) -Φω (x)∥ ∞ -ĥ C6 ∥ Φω (x) -Φω (x)∥ ⊥ (8.
Φ(x) -Φ(x) / / ≥ K f ( f (ω))K f (ω) ∥x -x∥ / / -K f ( f (ω))h 1 k1+4 ⊥ ∥x -x∥ ∞ -K f ( f (ω)) ĥ C6 ∥x -x∥ ⊥ -h 1+ k1 4 ⊥ 1 k2 +4 4 k 1 +2 k 2 1 ∥x -x∥ ∞ -ĥ C6 h k1 25 ⊥ ∥x -x∥ ∞ so Φ(x) -Φ(x) / / ≥ K f ( f (ω))K f (ω) ∥x -x∥ / / -K f ( f (ω))h 1 k1 +4 ⊥ + K f ( f (ω)) ĥ C6 + h 1+ k1 4 ⊥ 1 k2 +4 4 k 1 +2 k 2 1 + ĥ C6 h k1 25 ⊥ ∥x -x∥ ∞ (8.50)
Plugging (8.3) into (8.50), we get

Φ(x) -Φ(x) / / ≥ K f ∥x -x∥ / / -K f ( f (ω)) ĥ C6 + ε(ω, h ⊥ ) ∥x -x∥ ∞ (8.51) where ε(ω, h ⊥ ) = K f ( f (ω))h 1 k1+4 ⊥ + h 1+ k1 4 ⊥ 1 k2+4 4 k 1 +2 k 2 1 + ĥ C6 h k1 25
⊥ . By definition of ĥ, ĥ C6 = 1 4

K f -K c 1 + 1 σ so K f ( f (ω)) ĥ C6 ≤ 36 25 1 4 K f -K c 1 + 1 σ using (7.2) < 1 2 K f -K c 1 + 1 σ (8.
K f ( f (ω))h 1 k1 +4 ⊥ ≤ 36 25 e -C k 4 1 k1+4 ≤ 36 25 e -C k 4 1 5k1 ≤ 36 25 e -C 5 Moreover, h 1+ k1 4 ⊥ 1 k2+4 ≤ h 1 k2+4 ⊥ ≤ e -C k 2 1 +k 2 2 5k2 ≤ e -C 2k1 k2 5k2 ≤ e -2 5 Ck 1 so h 1+ k1 4 ⊥ 1 k2+4 4 k 1 +2 k 2 1 ≤ e -2 5 Ck 1 4 3k 1 e 2k 1 = e 2+3 ln(4)-2 5 C k 1 ≤ e 2+3 ln(4)-2 5 C
and finally, ĥ C6 h k1 25

⊥ ≤ ĥ C6 e

-C 25
Hence, claim 3 holds true.

It follows from (8.52) and (8.53) that, for all C large enough, we have

K f ( f (ω)) ĥ C6 + ε(ω, h ⊥ ) ≤ K f -K c 1 + 1 σ
Plugging this inequality into (8.51), we get that estimate (8.12) holds true for all C large enough. This concludes the proof of Proposition 8.2.

Shadowing of a heteroclinic chain along an era

Let ω ∈ ]1, 2[\Q. Recall that Φ * j = Φ j •⋅ ⋅ ⋅•Φ 0 . The heteroclinic chain starting at P s i ω, ĥ hits successively the local sections S s 2 1 , S s 1 2 , . . . , S s 1 k 1 (ω)-1 at the points Φ * 0 P s i ω, ĥ , Φ * 1 P s i ω, ĥ , . . . , Φ * k 1 (ω)-2 P s i ω, ĥ . If x ∈ Sĥ is close enough to P s i ω, ĥ, then the orbit starting at x will hit successively the local sections S s 2 1 , S s 1 2 , . . . , S s 1 k 1 (ω)-1 at the points Φ * 0 (x), Φ * 1 (x), . . . , Φ * k 1 (ω)-2 (x). The following proposition provides an upper bound for the distance between the intersection points Φ * j-1 P s i ω, ĥ and Φ * j-1 (x). Proposition 8.9 (Shadowing along an era). There exists a constant C9 ≥ C8 such that the property below holds for ω ∈ ]1, 2[\Q, i ∈ {1, 2} and 0 < ≤ 1. Let x ∈ S s i ω,h where h = ĥ, e -C8 s 4 (ω) , e -C8 s 2 (ω) .

If

x -P s i ω, ĥ ∞ ≤ e -C9 k 1 (ω) 4 +k 2 (ω) 4 +k 3 (ω) 4 then, for any 1 ≤ j ≤ k 1 (ω) -1, Φ * j-1 (x) -Φ * j-1 P s i ω, ĥ ∞ ≤ ĥk 1 (ω) -ñ 5 C5 f j (ω) m(f j (ω)) (8.54)
Proof. Recall that m(ω) = min 0≤j≤k 1 (ω) m(f j (ω)) (see (8.13)). Using the inequality (8.15), we get that there exists a constant C9 ≥ C8 such that for every

ω = [1; k 1 , k 2 , . . . ] ∈ ]1, 2[ \ Q and every 1 ≤ j ≤ k 1 -1, we have 50k 2 1 e -C9 k 4 1 +k 4 2 +k 4 3 ≤ ĥk -ñ 5 1 C5 f j (ω) m(ω) ≤ ĥk -ñ 5 1 C5 f j (ω) m(f j (ω)) (8.55) Let 0 < ≤ 1, ω = [1; k 1 , k 2 , . . . ] ∈ ]1, 2[ \ Q and x ∈ S s i
ω,h where h = ĥ, e -C8 s 4 (ω) , e -C8 s 2 (ω) . Assume that x -P

s i ω, ĥ ∞ e -C9 k 4 1 +k 4 2 +k 4 3 ≤ Let 1 ≤ j ≤ k 1 -1. Using estimate (8.26) with h ⊥ = x -P s i ω, ĥ ⊥ , we get Φ * j-1 (x) -Φ * j-1 P s i ω, ĥ ⊥ ≤ x -P s i ω, ĥ 5 4 ⊥ ≤ e -C9 k 4 1 +k 4 2 +k 4 3 ≤ ĥk -ñ 5 1 C5 f j (ω)
m(f j (ω)) using (8.55)

Remark that

Φ * j-1 (x) -Φ * j-1 P s i ω, ĥ / / ≤ Φ * j-1 (x) -Φ * A j-1 (x) / / + Φ * A j-1 (x) -Φ * j-1 P s i ω, ĥ / /
Using estimate (8.31) with h ⊥ = x -P s i ω, ĥ ⊥

, we get

Φ * j-1 (x) -Φ * A j-1 (x) / / ≤ 34k 2 1 x -P s i ω, ĥ ⊥ ≤ 34k 2 1 e -C9 k 4 1 +k 4 2 +k 4 3
Using Proposition 7.3 on the local Lipschitz constant for the Kasner map, we get

Φ * A j-1 (x) -Φ * j-1 P s i ω, ĥ / / ≤ j-1 l=0 Lip f (f l (ω)) x -P s i ω, ĥ / / ≤ 16k 2 1 e -C9 k 4 1 +k 4 2 +k 4 3
Hence, using (8.55), we get

Φ * j-1 (x) -Φ * j-1 P s i ω, ĥ / / ≤ 50k 2 1 e -C9 k 4 1 +k 4 2 +k 4 3 ≤ ĥk -ñ 5 1 C5 f j (ω) m(f j (ω))
This concludes the proof of Proposition 8.9.

The next proposition allows one to use Proposition 8.9 twice during a double era.

Proposition 8.10. There exists a constant C10 ≥ C9 such that the property below holds for ω ∈ ]1, 2[ \ Q, i ∈ {1, 2} and 0 < ≤ 1. Let x ∈ S s i ω,h where h = ĥ, e -C8 s 4 (ω) , e -C8 s 2 (ω) . Let Φ = Φω,h . If

x -P s i ω, ĥ ∞ ≤ e -C10 s 4 (ω) then, Φ(x) -Φ P s i ω, ĥ ∞ ≤ e -C9 k 1 ( f (ω)) 4 +k 2 ( f (ω)) 4 +k 3 ( f (ω)) 4 (8.

56)

Proof. There exists a constant C10 ≥ C9 such that, for all

k 1 ∈ N * , 4 k 1 +2 k 2 1 e -C10 k 4 1 ≤ 1 (8.57) Fix ω ∈ ]1, 2[ \ Q, i ∈ {1, 2} and 0 < ≤ 1. Let x ∈ S s i
ω,h where h = ĥ, e -C8 s 4 (ω) , e -C8 s 2 (ω) . Let Φ = Φω,h . Assume that

x -P s i ω, ĥ ∞ ≤ e -C10 s 4 (ω)
Using (8.22), we get Φ(x) -Φ P

s i ω, ĥ ∞ ≤ 4 k 1 (ω)+2 k 1 (ω) 2 x -P s i ω, ĥ ∞ ≤ 4 k 1 (ω)+2 k 1 (ω) 2 e -C10 s 4 (ω) ≤ 4 k 1 (ω)+2 k 1 (ω) 2 e -C10 k 1 (ω) 4 e -C10 k 2 (ω) 4 +k 3 (ω) 4 +k 4 (ω) 4 ≤ e -C10 k 2 (ω) 4 +k 3 (ω) 4 +k 4 (ω) 4 using (8.57) ≤ e -C9 k 1 ( f (ω)) 4 +k 2 ( f (ω)) 4 +k 3 ( f (ω))
The following corollary states that an orbit starting close to a type II orbit stays close during two eras.

Corollary 8.11 (Shadowing along two eras). Let

ω ∈ ]1, 2[ \ Q, i ∈ {1, 2} and 0 < ≤ 1. Let x ∈ S s i ω,h
where h = ĥ, e -C8 s 4 (ω) , e -C8 s 2 (ω) . If

x -P s i ω, ĥ ∞ ≤ e -C10 s 4 (ω)
then,

Φ * j-1 (x) -Φ * j-1 P s i ω, ĥ ∞ ≤ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ĥk 1 (ω) -ñ 5 C5 f j (ω) m(f j (ω)) if 1 ≤ j ≤ k 1 (ω) -1 ĥk 2 (ω) -ñ 5 C5 f j (ω) m(f j (ω)) if k 1 (ω) ≤ j ≤ k 1 (ω) + k 2 (ω) -1 (8.58)
Proof. For 1 ≤ j ≤ k 1 (ω) -1, the estimate (8.58) is a direct consequence of Proposition 8.9. For k 1 (ω) ≤ j ≤ k 1 (ω) + k 2 (ω) -1, it is a consequence of Proposition 8.9 applied to Φ(x) (together with Proposition 8.10 which proves that we can indeed apply Proposition 8.9 to Φ(x)).

Local stable manifolds of the double era return map

The purpose of this section is to construct some stable manifolds for the double era return map Φ ∶= Φĥ. These local stable manifolds play a central role in the proof of our main theorem. In Section 10, we will prove that any type IX orbit whose starting point lies in the local stable manifold of a point p will shadow the heteroclinic chain starting at p. In Section 11, we will prove that the union of these local stable manifolds over a positive 1-dimensional Lebesgue measure subset of the Kasner circle has positive 3-dimensional Lebesgue measure. Their construction rely on the estimates proven in Section 8. Recall that for any

ω ∈ ]1, 2[ \ Q, s 2 (ω) = k 1 (ω) 2 + k 2 (ω) 2 + k 3 (ω) 2 + k 4 (ω) 2 s 4 (ω) = k 1 (ω) 4 + k 2 (ω) 4 + k 3 (ω) 4 + k 4 (ω)
4

Recall that we have fixed a constant ĥ in the preceding section (see (8.38) and Proposition 8.2).

According to Proposition 8.2, the double era return map Φ is well defined on the set

⋃ ω∈]1,2[\Q S s ω,h ω ⊂ Sĥ
where h ω = ĥ, e -C8 s 4 (ω) , e -C8 s 2 (ω) .

Recall that P ω = (0, 0, 0, ω) denotes the point (in local coordinates) of Kasner parameter ω on the Kasner interval K 0 . Moreover, P s 1 ω, ĥ = (0, ĥ, 0, ω) and P s 2 ω, ĥ = (0, 0, ĥ, ω) denote the intersection points of the two type II orbits arriving at P ω with the global section Sĥ (see definition 4.10). Definition 9.1 (Local stable set). Let ω ∈ ]1, 2[ \ Q, i ∈ {1, 2} and η > 0. We call the local stable set of P s i ω, ĥ of size η and we denote by W s η P s i ω, ĥ, Φ the set of all x ∈ Sĥ such that for every n ≥ 0, Φn

is well defined and satisfies Φn (x) -Φn P

s i ω, ĥ ∞ ≤ η
We want to prove that for Lebesgue almost all ω ∈ ]1, 2[ and for η small enough, the local stable set W s η P s i ω, ĥ, Φ contains a Lipschitz graph. Definition 9.2 (Rooted graph). Let ω ∈ ]1, 2[ and 0 < a ≤ ĥ. A graph of size a rooted at P s 1 ω, ĥ is a set γ ⊂ S s 1 ĥ of the form

γ = Graph s 1 (ζ) def = x u , ĥ, x s 2 , ζ(x u , x s 2 ) | (x u , x s 2 ) ∈ [0, a] 2 where ζ ∶ [0, a] 2 → R is a map such that ζ(0, x s 2 ) = ω for all x s 2 ∈ [0, a].
We define analogously the graphs rooted at P s 2 ω, ĥ.

Remark 9.3. We say that γ = Graph s 1 (ζ) is a graph rooted at P s 1 ω, ĥ because P s 1 ω, ĥ = 0, ĥ, 0, ζ(0, 0) . Remark 9.4. Recall from Proposition 3.2 that the local coordinates x u , x s 1 and x s 2 are positive. This is why the map ζ is defined on [0, a] 2 .

We endow [0, a] 2 with the sup-norm ∥(x u , x s )∥ ∶= max(|x u |, |x s |). The map ζ is entirely determined by γ = Graph s 1 (ζ). Recall that σ is a constant that has been fixed in the preceding section (see (8.5)). We say that a graph γ is σ-Lipschitz if it is associated to a σ-Lipschitz map ζ.

We can now state the main theorem of this section, which describes the local stable manifolds of the double era return map Φ. We refer to Theorem 9.20 for a version that characterizes the size of the local stable manifolds. Recall that C8 is the constant fixed in Proposition 8.2 on the double era transition map.

Theorem 9.5 (Local stable manifolds of the double era return map). There exists a full Lebesgue measure set Ω graph ⊂ ]1, 2[ with the following properties. For all ω ∈ Ω graph and all i ∈ {1, 2}, for η small enough, the local stable set W s η P s i ω, ĥ, Φ of P s i ω, ĥ of size η contains a σ-Lipschitz graph of size η rooted at P s i ω, ĥ. Moreover, for all x belonging to this graph and all n ≥ 0, Φn (x) -Φn P 5 ≪ e -2 C8 n . Hence, we have a super exponential convergence to the Mixmaster attractor for the orbits starting in those graphs.

s i ω, ĥ ∞ ≤ x -P s i ω, ĥ ⊥ e -C8 ∑
We are going to prove Theorem 9.5 using the so-called Hadamard graph transformation method. Let us describe informally our strategy, which may not be the most standard one.

The first step is to show that the double era return map Φ satisfies some hyperbolic properties. We prove the existence of two invariant cone fields, namely: the unstable cone field containing the direction tangent to the Mixmaster attractor and the stable cone field containing the direction transverse to the Mixmaster attractor. The unstable cone field is forward invariant while the stable cone field is backward invariant. Moreover, the map Φ expands the length of the vectors in the unstable cone field and contracts the length of the vectors in the stable cone field. See Proposition 9.8.

Once we know that the double era return map Φ satisfies some hyperbolic properties, we can show that the preimage of a σ-Lipschitz graph rooted at P sî (ω) f (ω), ĥ by the double era transition map Φ S s i ω,h is a σ-Lipschitz graph rooted at P s i ω, ĥ. To make this statement correct, one must carefully choose the size of the graph rooted at P sî (ω) f (ω), ĥ and the size of the section S s i ω,h . See Lemma 9.11. The next step consists in constructing a space Γ of families of σ-Lipschitz graphs invariant by the preimage procedure described in the above paragraph. For Ω

graph ⊂ ]1, 2[ \ Q and ĥ⊥ ∶ Ω graph → ]0, +∞[ fixed, define Γ def = γ = γ ω,i ω∈Ω graph ,i∈{1,2} | γ ω,i is a σ-Lipschitz graph of size ĥ⊥ (ω) rooted at P s i ω, ĥ
Roughly speaking, the graph transformation Φ * ∶ Γ → Γ is defined as follows. For γ ∈ Γ, Φ * γ ω,i is the preimage of γ f (ω), î(ω) by a suitable restriction of the double era return map Φ S s i ω,h (see definition 9.17). Our goal is to find a full Lebesgue measure set Ω graph ⊂ ]1, 2[ invariant by the double era Kasner map f and a function ĥ⊥ ∶ Ω graph → ]0, +∞[ such that the graph transformation Φ * defined above is well defined. See Definition 9.14 and Proposition 9.15.

Using the hyperbolic properties of Φ, the graph transformation Φ * will be proved to be a contraction mapping. The standard contraction mapping theorem will provide a fixed point γ = γω,i ∈ Γ.

The final step consists in checking that γω,i is contained in the local stable set of the point P s i ω, ĥ

for the double era return map Φ for all ω and i.

Cone field invariant by the double era return map

Recall that σ has been fixed in the preceding section (see (8.5)). This parameter will serve as the angle of the invariant cone field.

S s i ω,h • x V / / ω,h,i (x) • x Φ S sî (ω) f (ω),h ′ • Φ(x) Φ V / / ω,h,i (x) • Φ(x)
Figure 18: Forward invariance of the tangential cone field. Definition 9.7 (Cones). Let ω ∈ ]1, 2[ \ Q, h = ( ĥ, h ⊥ , h / / ) with min(h ⊥ , h / / ) > 0, i ∈ {1, 2} and x ∈ S s i ω,h . We define the tangential cone at x as

V / / ω,h,i (x) def = x ∈ S s i ω,h | ∥x -x∥ ⊥ ≤ σ ∥x -x∥ / /
As usual in hyperbolic dynamical system theory, we define the interior of

V / / ω,h,i (x) as Int V / / ω,h,i (x) def = {x} ∪ x ∈ S s i ω,h | ∥x -x∥ ⊥ < σ ∥x -x∥ / /
In other words, Int V / / ω,h,i (x) is the union of the topological interior of V / / ω,h,i (x) and its vertex {x}. Analogously, we define the transverse cone at x as

V ⊥ ω,h,i (x) def = x ∈ S s i ω,h | ∥x -x∥ / / ≤ σ ∥x -x∥ ⊥ and the interior of V ⊥ ω,h,i (x) as Int V ⊥ ω,h,i (x) def = {x} ∪ x ∈ S s i ω,h | ∥x -x∥ / / < σ ∥x -x∥ ⊥
Recall that K c > 1 is an explicit constant fixed in the preceding section (see (8.4)).

Proposition 9.8 (Hyperbolic properties of the double era return map). There exists a constant C11 ≥ C10 such that the properties below hold for ω ∈ ]1, 2[\Q, i ∈ {1, 2}, h = ĥ, e -C11 s 4 (ω) , e

-C8 s 2 (ω)
and h ′ = ĥ, ĥ, ĥ .

(Forward invariance of the tangential cone field) For all x ∈ S

s i ω,h , Φ V / / ω,h,i (x) ⊂ Int V / / f (ω),h ′ , î(ω) Φ(x) (9.2) (Backward invariance of the transverse cone field) For all x ∈ S s i ω,h , Φ S s i ω,h -1 V ⊥ f (ω),h ′ , î(ω) Φ(x) ⊂ Int V ⊥ ω,h,i (x) (9.3)
(Expansion in the tangential cone field) For every x, x ∈ S

s i ω,h , if x ∈ V / / ω,h,i (x), then Φ(x) -Φ(x) / / ≥ K c ∥x -x∥ / / (9.4)
(Contraction in the transverse cone field) For every x, x ∈ S

s i ω,h , if Φ(x) ∈ V ⊥ f (ω),h ′ , î(ω) Φ(x) , then Φ(x) -Φ(x) ⊥ ≤ e -C8 k 1 (ω) 5 +k 2 (ω) 5 ∥x -x∥ ⊥ (9.5)
See figures 18 and 19.

S s i ω,h • x Φ S s i ω,h -1 V ⊥ f (ω),h ′ , î(ω) Φ(x) • x Φ S s i ω,h -1 S sî (ω) f (ω),h ′ • Φ(x) V ⊥ f (ω),h ′ , î(ω) Φ(x) • Φ(x)
Figure 19: Backward invariance of the transverse cone field.

Proof. Let C ≥ 100 C8 such that e -C 100 1 + 1 σ K -1 c < σ (9.6) Fix ω ∈ ]1, 2[ \ Q and x, x ∈ S s i ω,h (i ∈ {1, 2}
). Let h = ĥ, e -Cs 4 (ω) , e -C8 s 2 (ω) and h ′ = ĥ, ĥ, ĥ .

Recall that σ < 1. Hence,

x ∈ V / / ω,h,i (x) ⟹ ∥x -x∥ ∞ = ∥x -x∥ / / (9.7a) x ∈ V ⊥ ω,h,i (x) ⟹ ∥x -x∥ ∞ = ∥x -x∥ ⊥ (9.7b) Expansion estimate (9.4). According to (9.7a), if x ∈ V / / ω,h,i (x), then ∥x -x∥ ∞ = ∥x -x∥ / / ≤ 1 + 1 σ ∥x -x∥ / /
Hence, expansion estimate (8.12) implies that the expansion estimate (9.4) holds true. Backward invariance of the transverse cone field (9.3). Assume that ∥x -

x∥ ⊥ ≤ 1 σ ∥x -x∥ / / . Hence, ∥x -x∥ ∞ ≤ 1 + 1 σ ∥x -x∥ / / (9.8)
As a consequence, the expansion estimate (9.4) remains true in that case: which is the desired estimate (9.5).

Φ(x) -Φ(x) / / ≥ K c ∥x -x∥ / / (9
Remark 9.9. Let us describe precisely how we will use Proposition 9.8. The forward invariance of the tangential cone field (9.2) together with the expansion estimate (9.4) are used two times:

1. To show that the preimage of a σ-Lipschitz graph is a graph.

2. To show that the graph transformation is a contraction mapping.

Knowing that the preimage of a σ-Lipschitz graph rooted at P sî (ω) f (ω), ĥ by the double era transition map Φ S s i ω,h is a graph rooted at P s i ω, ĥ, the backward invariance of the transverse cone field (9.3) implies that this graph is also σ-Lipschitz. This property is essential to show that the set of σ-Lipschitz graphs families is invariant by the graph transformation.

Finally, the contraction estimate in the transverse cone field (9.5) is used to prove the exponential convergence (9.1) for a point in a graph constructed as the fixed point of the graph transformation. It also proves that this graph is contained in the local stable set of some point for the double era return map.

Local graph transformation

Our next task is to understand the preimage of a σ-Lipschitz graph by the double era return map Φ. This is the purpose of Lemma 9.11 below. To make the computations in coordinates easier to follow, let us identify the section S s 1 ĥ with a subset of R 3 , forgetting the coordinate x s 1 which is constant equal to ĥ on S s 1 ĥ . More precisely, we identify the point (x u , ĥ, x s 2 , x c ) ∈ S s 1 ĥ with (x ⊥ , x / / ) ∈ R 2 × R where x ⊥ = (x u , x s 2 ) and x / / = x c . We will use the same notation in S s 2 ĥ , letting x ⊥ = (x u , x s 1 ). We will not work in both sections at the same time, hence this notation will not be ambiguous. Remark that with these coordinates, a graph of size a rooted at P

s i ω, ĥ is a subset γ of R 2 × R of the form Graph(ζ) = (x ⊥ , ζ(x ⊥ )) | x ⊥ ∈ [0, a]
ω ∈ ]1, 2[ \ Q, i ∈ {1, 2}, 0 < a ≤ e -C12 s 4 (ω) , a ′ ≥ ae -C8 k 1 (ω) 5 +k 2 (ω)
5 and h = ĥ, a, e ω) , then Φ(x) is on the right side of γ. If

S s i ω,h ∂ ∂x / / ∂ ∂x ⊥ V ⊥ ω,h,i P s i ω, ĥ Φ S s i ω,h -1 (Graph (γ)) • P s i ω, ĥ Φ S s i ω,h * S sî (ω) f (ω),h ′ ∂ ∂x / / ∂ ∂x ⊥ V ⊥ f (ω),h ′ , î(ω) P sî (ω) f (ω), ĥ Graph (γ) • P sî (ω) f (ω), ĥ
S s i ω,h • x ω x / / ω + e -C8 s 2 (ω) e -C8 s 2 (ω) x ⊥ Φ S sî (ω) f (ω),h ′ • Φ(x) • ζ Φ(x) ⊥ Φ(x) / / λ(x / / ) Φ(x) ⊥ γ Figure 21: Interpretation of the map λ. If x / / = ω + e -C8 s 2 (
x / / = ωe -C8 s 2 (ω) , then Φ(x) is on the left side of γ.

Proof of Lemma 9.13. This is a straightforward consequence of the definition of the cone V ⊥ ω,h,i (x).

Proof of Lemma 9.11. Take C12 ≥ 4 C11 large enough so that for every

ω ∈ ]1, 2[ \ Q, σ e -C12 s 4 (ω) 1+ k1 4 + k2 4 + 578e -C12 s 4 (ω) k 1 (ω) 2 k 2 (ω) 2 ≤ 4e -C8 s 2 (ω) (9.11) Fix ω = [1; k 1 , k 2 , .
. . ], i, a, a ′ and γ as in the statement. There exists a unique map ζ ∶ 0, a (ζ). Let h = ĥ, a, e -C8 s 2 (ω) , h ′ = ĥ, ĥ, ĥ ,

′ 2 → R such that γ = Graph sî (ω)
Φs i = Φ S s i ω,h and ΦA,s i = ΦA S s i ω,h .
To prove that ( Φs i ) -1 (γ) is a graph of size a rooted at P s i ω, ĥ, we first need a technical claim. Define

I ω ∶= ω -e -C8 s 2 (ω) , ω + e -C8 s 2 (ω) . Claim 1. Fix x ⊥ ∈ [0, a] 2 . Define a map λ ∶ I ω → R by the formula λ(x / / ) def = Φs i (x ⊥ , x / / ) / / -ζ Φs i (x ⊥ , x / / ) ⊥
The map λ is well defined. Moreover, λ ω + e -C8 s 2 (ω) ≥ 0 and λ ωe -C8 s 2 (ω) ≤ 0. See figure 21.

Proof of claim 1. Recall that C12 ≥ 4 C11 ≥ 4 C8 . According to (8.8), for every x / / ∈ I ω , we have

Φs i (x ⊥ , x / / ) - ΦA,s i (x ⊥ , x / / ) ⊥ ≤ a 1+ k1 +k2 4 ≤ ae -C8 k 5 1 +k 5 2 ≤ a ′ This means that Φs i (x ⊥ , x / / ) ⊥ ∈ 0, a ′ 2 .
Since γ is a graph of size a ′ , the map λ is well defined.

Roughly speaking, since x ⊥ is "small", Φs i (x ⊥ , x / / ) / / is close to f (x / / ) and Φs i (x ⊥ , x / / ) ⊥ is close to (0, 0). Using this approximation and the fact that γ is a graph rooted at P sî (ω) f (ω), ĥ, we get

λ ω + e -C8 s 2 (ω) ≃ f ω + e -C8 s 2 (ω) -f (ω)
Using the fact that f is increasing on I ω (see Proposition A.7), we get λ ω + e -C8 s 2 (ω) ≥ 0

We are now going to make rigorous this computation. Remark that

λ ω + e -C8 s 2 (ω) = Φs i x ⊥ , ω + e -C8 s 2 (ω) / / -f ω + e -C8 s 2 (ω) + f ω + e -C8 s 2 (ω) -f (ω) + f (ω) -ζ Φs i x ⊥ , ω + e -C8 s 2 (ω)

⊥

We will compare the three terms in the right-hand side of the above equation. Recall from (4.7) that ΦA,s i (x) = (0, ĥ, 0, f

(x c )) if k 2 (x c ) ≥ 2 (0, 0, ĥ, f (x c )) if k 2 (x c ) = 1 where x c = [1; k 1 (x c ), k 2 (x c ), . . . ]
Hence, estimate (8.9) gives

Φs i x ⊥ , ω + e -C8 s 2 (ω) / / -f ω + e -C8 s 2 (ω) ≤ 578ak 1 (ω) 2 k 2 (ω) 2 (9.12)
According to Proposition A.7 about the Gauss transformation, we have the expansion estimate

f ω + e -C8 s 2 (ω) -f (ω) ≥ 4e -C8 s 2 (ω) (9.13)
Moreover, using the fact that γ is a graph rooted at P sî (ω) f (ω), ĥ, we get It follows from (9.12), (9.13), (9.17) and (9.11) that

f (ω) -ζ Φs i x ⊥ , ω + e -C8 s 2 (ω) ⊥ = ζ(0, 0) -ζ Φs i x ⊥ , ω + e -C8
Φs i x ⊥ , ω + e -C8 s 2 (ω) / / -f ω + e -C8 s 2 (ω) + f (ω) -ζ Φs i x ⊥ , ω + e -C8 s 2 (ω) ⊥ ≤ f ω + e -C8 s 2 (ω) -f (ω)
Hence, λ ω + e -C8 s 2 (ω) and f ω + e -C8 s 2 (ω) -f (ω) have the same sign. Since f is increasing on I ω , we get that λ ω + e -C8 s 2 (ω) ≥ 0. The arguments are analogous for λ ωe

-C8 s 2 (ω) Claim 2. ( Φs i ) -1 (γ) is a graph of size a rooted at P s i ω, ĥ.
Proof of claim 2. Fix x ⊥ ∈ [0, a] 2 . First, let us prove that there exists a unique x / / ∈ I ω , denoted by

( Φs i ) * ζ(x ⊥ ), such that Φs i (x ⊥ , x / / ) ∈ γ. This will show that Φs i -1 (γ) = x ⊥ , ( Φs i ) * ζ(x ⊥ ) | x ⊥ ∈ [0, a] 2 = Graph s i ( Φs i ) * ζ Remark that for x = (x ⊥ , x / / ) with x / / ∈ I ω , Φs i (x) ∈ γ ⟺ Φs i (x) / / = ζ Φs i (x) ⊥ ⟺ λ(x / / ) = 0
Since ζ is σ-Lipschitz and Φs i is continuous, λ is continuous on I ω . According to claim 1 and the intermediate value theorem, there exists x / / ∈ I ω such that λ(x / / ) = 0.

Let

x / / , x// ∈ I ω such that λ(x / / ) = λ(x / / ) = 0. Set x = (x ⊥ , x / / ) ∈ S s i ω,h and x = (x ⊥ , x// ) ∈ S s i ω,h
. By definition of the tangential cone, we have

x ∈ V / / ω,h,j (x)
and by forward invariance of the tangential cone field (see (9.2)), we get

Φs i (x) ∈ V / / f (ω),h ′ , î(ω) Φs i (x)
Moreover, Φs i (x) and Φs i (x) both belong to γ which is a σ-Lipschitz graph so Lemma 9.13 implies that Φs

i (x) ∈ V ⊥ f (ω),h ′ , î(ω) Φs i (x) It follows that Φs i (x) = Φs i (x)
Using the expansion estimate (9.4) in the direction tangent to the Mixmaster attractor, we get x / / = x// . Let z ∈ [0, a]. We have

ζ Φs i ((0, z), ω) ⊥ = ζ(0, 0) = f (ω) = Φs i ((0, z), ω) / /
Hence, Φs i ((0, z), ω) ∈ γ. By uniqueness, we get that ( Φs i ) * ζ(0, z) = ω. This concludes the proof of claim 2.

Claim 3. (

i ) -1 (γ) is a σ-Lipschitz graph. Proof of claim 3. Let x ⊥ , x⊥ ∈ [0, a] 2 . Set x = x ⊥ , ( Φs i ) * ζ (x ⊥ ) ∈ ( Φs i ) -1 (γ) and x = x⊥ , ( Φs i ) * ζ (x ⊥ ) ∈ ( Φs i ) -1 (γ) Φs
The graph γ is σ-Lipschitz so Lemma 9.13 implies that

Φs i (x) ∈ V ⊥ f (ω),h ′ , î(ω) Φs i (x)
and by backward invariance of the transverse cone field (see (9.3)), we get

x ∈ V ⊥ ω,h,i (x)
Using Lemma 9.13 once again, we get that (

Φs i ) -1 (γ) = Graph s i ( Φs i ) * ζ is σ-Lipschitz.
This concludes the proof of Lemma 9.11.

The set of admissible points for the graph transformation

According to Lemma 9.11, we have a graph transformation over one point ω ∈ ]1, 2[ \ Q, pulling-back a graph rooted at P sî (ω) f (ω), ĥ. Recall that our goal is to define a graph transformation over a full-measure set Ω graph ⊂ ]1, 2[ \ Q, called the set of admissible points for the graph transformation. To do this, we need to iterate the procedure described in Lemma 9.11. This means that the set Ω graph must be invariant under the Kasner double era map f . Hence, it is a union of orbits (ω, f (ω), f 2 (ω), . . . ). Roughly speaking, the idea is to attach a graph to each point ω ∈ Ω graph and to replace the graph rooted at P s i ω, ĥ by the preimage of the graph rooted at P sî (ω) f (ω), ĥ by the double era return map Φ. If we go into the technical details, there are two sections S s 1 ω and S s 2 ω above each point ω ∈ Ω graph so we need to consider two graphs rooted at each point. Let us temporarily simplify the discussion by acting as if there were only one section, say S s ω . The graph transformation acts above the orbit (ω, f (ω), f 2 (ω), . . . ) as follows: for all n ≥ 0, the graph rooted at P s f n (ω), ĥ is replaced by the preimage of the graph rooted at P s f n+1 (ω), ĥ by the double era return map. Informally, the graph transformation is well defined above the orbit (ω, f (ω), f 2 (ω), . . . ) if there exists a sequence (a n ) n≥0 of positive real numbers such that for any family (γ n ) n≥0 where γ n is a σ-Lipschitz graph of size a n rooted at P s f n (ω), ĥ, the following property holds: for all n ≥ 0, the preimage of the graph γ n+1 by the double era return map defines a σ-Lipschitz graph of size a n rooted at P s f n (ω), ĥ. Remark that for all n ≥ 0, k 1 (

f n (ω)) = k 2n+1 (ω), k 2 ( f n (ω)) = k 2n+2 (
ω), etc. Hence, Lemma 9.11 gives a sufficient condition: if there exists a sequence (a n ) n≥0 of positive real numbers such that for every n ≥ 0,

a n ≤ e -C12 s 4 ( f n (ω)) = e -C12 k 2n+1 (ω) 4 +k 2n+2 (ω) 4 +k 2n+3 (ω) 4 +k 2n+4 (ω) 4 and a n e -C8 k 2n+1 (ω) 5 +k 2n+2 (ω) 5 = a n e -C8 k 1 ( f n (ω)) 5 +k 2 ( f n (ω)) 5 ≤ a n+1
then the graph transformation is well defined above the orbit (ω, f (ω), f 2 (ω), . . . ). This leads to the following definition. Definition 9.14 (Admissible points for the graph transformation). Let ω ∈ ]1, 2[ \ Q and h ⊥ > 0. We associate with ω and h ⊥ a sequence (a n (ω, h ⊥ )) n≥0 defined by

a 0 (ω, h ⊥ ) = h ⊥ a n+1 (ω, h ⊥ ) = a n (ω, h ⊥ )e -C8 k 2n+1 (ω) 5 +k 2n+2 (ω) 5
We say that ω is admissible (for the graph transformation) if there exists h ⊥ > 0 such that for every

n ≥ 0, a n (ω, h ⊥ ) ≤ e -C12 s 4 ( f n (ω)) . If ω is admissible, we define ĥ⊥ (ω) def = sup h ⊥ > 0 | ∀n ≥ 0, a n (ω, h ⊥ ) ≤ e -C12 s 4 ( f n (ω)) (9.18)
We denote by Ω graph the set of all admissible points in ]1, 2[ \ Q.

Proposition 9.15. The set of the admissible points is invariant in the future and the past by the Kasner double era map, i.e. f -1 (Ω graph ) = Ω graph .

Proof. One can remark that for every ω ∈ ]1, 2[ \ Q, for every h ⊥ > 0 and for every n ≥ 0,

a n f (ω), a 1 (ω, h ⊥ ) = a n+1 (ω, h ⊥ ) (
]1, 2[ \ Q. Proof. Let ω ∈ ]1, 2[ \ Q. Observe that a n (ω, h ⊥ ) = h ⊥ e -C8 ∑ 2n i=1 k i (ω) 5 ,
As a consequence, ω ∈ Ω graph as soon as

s 4 ( f n (ω)) = o n→+∞ 2n i=1 k i (ω) 5 (9.20)
On the other hand, (MG) clearly implies (9.20).

Global graph transformation Φ *

Now that the set Ω graph and the function ĥ⊥ ∶ Ω graph → ]0, +∞[ are defined, recall that

Γ def = γ = γ ω,i ω∈Ω graph ,i∈{1,2} | γ ω,i is a σ-Lipschitz graph of size ĥ⊥ (ω) rooted at P s i ω, ĥ
Beware of the fact that in this definition, γ is not a graph but a family of graphs.

For ω ∈ Ω graph , define a "canonical" triplet of parameters ĥω def = ĥ, ĥ⊥ (ω), e

-C8 s 2 (ω)
The double era return map defines a natural transformation Φ * ∶ Γ → Γ.

Definition 9.17 (Graph transformation). The graph transformation

Φ * ∶ Γ → Γ is defined by the formula Φ * γ ω,i def = Φ S s i ω, ĥω -1 γ f (ω), î(ω)
for all γ ∈ Γ, all ω ∈ Ω graph and all i ∈ {1, 2}.

Proposition 9.18. The graph transformation Φ * ∶ Γ → Γ is well defined.

Proof. This is a straightforward consequence of Lemma 9.11 and Proposition 9.15.

Local stable manifolds of the double era return map

For γ ∈ Γ, there exists a unique family of maps ζ = ζ ω,i where ζ ω,i is a map from 0, ĥ⊥ (ω) 2 to R and γ ω,i = Graph s i ζ ω,i . We will denote γ = Graph (ζ). We endow Γ with the distance

d graph Graph (ζ) , Graph ζ def = sup ω∈Ω graph ,i∈{1,2} ∥ζ ω,i -ζω,i ∥ ∞, 0, ĥ⊥ (ω) 2 where ∥ζ ω,i -ζω,i ∥ ∞, 0, ĥ⊥ (ω) 2 def = sup x ⊥ ∈ 0, ĥ⊥ (ω) 2 |ζ ω,i (x ⊥ ) -ζω,i (x ⊥ )| .
Remark that Γ, d graph is a complete space.

i ∈ {1, 2}, Φ * ζ ω,i - Φ * ζ ω,i ∞, 0, ĥ⊥ (ω) 2 ≤ 1 K c (1 -σ2 ) ζ f (ω), î(ω) -ζ f (ω), î(ω) ∞, 0, ĥ⊥ ( f (ω)) 2 (9.21)
Let ω ∈ Ω graph , i ∈ {1, 2} and y ∈ [0, ĥ⊥ (ω)] 2 . Consider two points with the same first coordinate:

x = y, Φ * ζ ω,i (y) ∈ Φ * γ ω,i z = Φ(x) ∈ γ f (ω), î(ω) x = y, Φ * ζ ω,i (y) ∈ Φ * γ ω,i z = Φ(x) ∈ γ f (ω), î(ω) Since x ⊥ = x⊥ , we have ∥x -x∥ ⊥ = 0. It follows that x ∈ V / /
ω, ĥω ,i (x). By forward invariance of the tangential cone field (see (9.2)), we have z ∈

V / / f (ω), ĥ f (ω) , î(ω) (z) which means ∥z -z∥ ⊥ ≤ σ ∥z -z∥ / / (9.22) Hence, ∥z -z∥ / / = ζ f (ω), î(ω) (z ⊥ ) -ζ f (ω), î(ω) (z ⊥ ) ≤ ζ f (ω), î(ω) (z ⊥ ) -ζ f (ω), î(ω) (z ⊥ ) + ζ f (ω), î(ω) -ζ f (ω), î(ω) ∞, 0, ĥ⊥ ( f (ω)) 2 ≤ σ ∥z -z∥ ⊥ + ζ f (ω), î(ω) -ζ f (ω), î(ω) ∞, 0, ĥ⊥ ( f (ω)) 2 ≤ σ2 ∥z -z∥ / / + ζ f (ω), î(ω) -ζ f (ω), î(ω) ∞, 0, ĥ⊥ ( f (ω)) 2 using (9.22) or equivalently ∥z -z∥ / / ≤ 1 1 -σ2 ζ f (ω), î(ω) -ζ f (ω), î(ω) ∞, 0, ĥ⊥ ( f (ω)) 2 (9.23) Recall that x ∈ V / /
ω, ĥω ,i (x). By expansion in the tangential cone field (see (9.4)), we have

∥z -z∥ / / ≥ K c ∥x -x∥ / / = K c Φ * ζ ω,i (x ⊥ ) - Φ * ζ ω,i (x ⊥ ) (9.24)
Using (9.23) and (9.24), we get

Φ * ζ ω,i (x ⊥ ) - Φ * ζ ω,i (x ⊥ ) ≤ 1 K c (1 -σ2 ) ζ f (ω), î(ω) -ζ f (ω), î(ω) ∞, 0, ĥ⊥ ( f (ω)) 2
Hence, (9.21) holds true. According to (8.5), K c 1 -σ2 > 1 so Φ * is a contraction mapping. Using the standard contraction mapping theorem, we get that Φ * admits a unique fixed point in Γ. This concludes the proof.

j ∈ {1, 2}, Φ γ f n (ω),j ⊂ γ f n+1 (ω), î( f n (ω))
Using the fact that ζ f n+1 (ω), î( f n (ω)) is σ-Lipschitz with σ ≤ 1, we get that for every n ≥ 0,

γ f n+1 (ω), î( f n (ω)) ⊂ S s f n+1 (ω), ĥ f n+1 (ω)
By induction, we get that for every n ≥ 0, Φn (x) is well defined and belongs to S s f n (ω), ĥ f n (ω)

.

Let n ≥ 0. Since Φn+1 (x) and Φn+1 P s i ω, ĥ both belong to γ f n+1 (ω), î( f n (ω)) , it follows from Lemma 9.13 that Φn+1

(x) ∈ V ⊥ f n+1 (ω),h ′ , î( f n (ω))
Φn+1 P s i ω, ĥ

where h ′ = ĥ, ĥ, ĥ . Hence, the contraction estimate (9.5) in the transverse cone gives

Φn+1 (x) - Φn+1 P s i ω, ĥ ⊥ ≤ Φn (x) - Φn P s i ω, ĥ ⊥ e -C8 k 5 2n+1 +k 5 2n+2 (9.26) Moreover, Φn+1 (x) - Φn+1 P s i ω, ĥ / / = ζ f n+1 (ω), î( f n (ω)) Φn+1 (x) ⊥ -ζ f n+1 (ω), î( f n (ω)) Φn+1 P s i ω, ĥ ⊥ ≤ σ Φn+1 (x) - Φn+1 P s i ω, ĥ ⊥ (9.27)
Using (9.26), (9.27) and the fact that σ ≤ 1, we get that for every n ≥ 0,

Φn+1 (x) - Φn+1 P s i ω, ĥ ∞ ≤ e -C8 k 5 2n+1 +k 5 2n+2
Φn (x) -Φn P s i ω, ĥ ⊥ By induction, we get that for every n ≥ 0, Φn (x) -Φn P

s i ω, ĥ ∞ ≤ x -P s i ω, ĥ ⊥ e -C8 ∑ 2n i=1 k i (ω) 5 (9.28) Hence, x ∈ W s ĥ⊥ (ω) P s i ω,
ĥ, Φ and the convergence is exponential in the graph. This concludes the proof of Theorem 9.20.

Continuity of the local stable manifolds

We want to show that the graphs γω,i depend continuously on ω ∈ Ω graph . Equivalently, we can show that the maps ζω,i depend continuously on ω ∈ Ω graph . Now remark that if P s i ω, ĥ and P s j ω, ĥ are close to each other, then i = j. Hence, we can fix i = 1 and discuss the regularity of the map ζ1 ∶ ω ∈ Ω graph ↦ ζω,1 .

Recall that for all ω ∈ Ω graph , ζω,1 ∶ 0, ĥ⊥ (ω) 2 → R is a σ-Lipschitz map such that ζω,1 (0, z) = ω for all z ∈ 0, ĥ⊥ (ω) . We want to compare two different maps ζω,1 and ζω,1 when ω and ω are close together. The most natural way to compare ζω,1 and ζω,1 is to restrict them to 0, min ĥ⊥ (ω), ĥ⊥ (ω) 2 and then to use the sup-norm. We do not want the function min ĥ⊥ (ω), ĥ⊥ (ω) to collapse to 0 while ω tends to ω so we will restrict ourselves to points ω that satisfy ĥ⊥ (ω) ≥ h ⊥ where h ⊥ > 0 is an arbitrary fixed number. This leads us to define the following subset of Ω graph :

Ω graph (h ⊥ ) = ω ∈ Ω graph | ĥ⊥ (ω) ≥ h ⊥ One should note that Ω graph = ⋃ n≥1 Ω graph 1 n .
γ = Graph Φ * n ζ . We then have ζω,1 -ζω,1 ∞ ≤ ζω,1 - Φ * n ζ ω,1 ∞ + Φ * n ζ ω,1 - Φ * n ζ ω,1 ∞ + Φ * n ζ ω,1 -ζω,1 ∞ ≤ Φ * n ζ ω,1 - Φ * n ζ ω,1 ∞ + 2
One can remark that for every Graph (Λ) ∈ Γ and every z, z ∈ Ω graph (h ⊥ ) close enough, we have

Φ * Λ z,1 - Φ * Λ z,1 ∞,[0,h ⊥ ] 2 ≤ λ Λ f (z), î(z) -Λ f (z), î(z) ∞, 0,h ⊥ e -C8 k1 (z) 5 +k2 (z) 5 2 (9.29) where λ = 1 K c (1-σ 2 )
. This inequality follows from the very same argument as in Lemma 9.19. One just needs to check that if z is close enough to z, we can indeed use the invariant cone field from Proposition 9.8. Now remark that if ω is close enough to ω, then for every 0 ≤ j ≤ n, f j (ω) is close enough to f j (ω) so that estimate (9.29) holds true with z = f j (ω) and z = f j (ω). By induction, we get

Φ * n ζ ω,1 - Φ * n ζ ω,1 ∞,[0,h ⊥ ] 2 ≤ λ n ζ f n (ω), î( f n-1 (ω)) -ζ f n (ω), î( f n-1 (ω)) ∞, 0,h ⊥ e -C8 ∑ 2n i=1 k 5 i 2 ≤ λ n f n (ω) - f n (ω) Moreover, if ω is close enough to ω, then f n (ω) = f k 1 +⋅⋅⋅+k 2n (y c ) and f n (ω) = f k 1 +⋅⋅⋅+k 2n (ω)
Hence, using Proposition 7.3 on the Kasner map, we get that

Φ * n ζ ω,1 - Φ * n ζ ω,1 ∞,[0,h ⊥ ] 2 ≤ λ n 2n i=1 16k 2 i |ω -ω| Take η > 0 such that η λ n ∏ 2n i=1 16k 2 i ≤ . If |ω -ω| ≤ η, then ζω,1 -ζω,1 ∞ ≤ 3
which concludes the proof.

Shadowing of heteroclinic chains

If p is a point of the Kasner circle K , let us denote by Shad(p) the union of all the type IX orbits in B + shadowing the heteroclinic chain H (p) starting at p.

Recall that ω ∈ ]1, +∞[ \ Q satisfies the moderate growth condition if k n+4 (ω) 4 = o n→+∞ n i=1 k i (ω) 5 (MG)
We are now ready to prove the first part of Theorem 1.9 stated in the introduction. Let us recall the statement.

Theorem 10.1 (Theorem 1.9, first part). Let p be a point of the Kasner circle. If ω(p) verifies the moderate growth condition (MG), then Shad(p) contains a 3-dimensional ball Lipschitz embedded in the phase space B + .

We will reduce Theorem 10.1 to a more technical statement, see Theorem 10.4 below. Let us recall some notations. For any ω ∈ ]1, 2[ \ Q, we denote by P ω the unique point belonging to the Kasner interval K 0 whose Kasner parameter is ω and by H (ω) the heteroclinic chain starting at P ω (see definition 2.7). Recall that γ = (γ ω,i ) denotes the fixed point of the graph transformation, that is, the graph family invariant by the double era return map Φ constructed in Section 9. Roughly speaking, we will prove that the orbits starting in γω,i will shadow the heteroclinic chain H (ω) (see definition 1.4). In practice, we need to impose a stronger condition on ω : the moderate growth condition (MG). Definition 10.2. We denote by Ω shad the set of all the points ω ∈ ]1, 2[ \ Q satisfying the moderate growth condition (MG).

Proposition 10.3. Ω shad ⊂ Ω graph (see definition 9.14) and Ω shad is a Lebesgue full measure subset of ]1, 2[ \ Q. Moreover, if ω ∈ Ω shad , then k 2n+1 (ω) 4 + k 2n+2 (ω) 4 + k 2n+3 (ω) 4 + k 2n+4 (ω) 4 = o n→+∞ 2n i=1 k i (ω) 5 (10.1)
Proof. The first part of Proposition 10.3 is a direct consequence of Proposition 9.16 and Lemma A.1. The fact that (MG) implies (10.1) is straightforward.

Shadowing theorem

Recall that γ = (γ ω,i ) denotes the fixed point of the graph transformation. Recall that the type IX points are those satisfying, in local coordinates, the condition

x u > 0, x s 1 > 0, x s 2 > 0
In particular, any point in the interior of γω,i is of type IX. Recall that Shad(ω) is the union of all the type IX orbits in B + shadowing H (ω).

Theorem 10.4 (Partial description of the shadowing sets). For every ω ∈ Ω shad , every i ∈ {1, 2} and every point q 0 ∈ ξ -1 γω,i of type IX, the orbit of the Wainwright-Hsu vector field starting at q 0 shadows the heteroclinic chain H (ω). In particular, the shadowing set Shad(ω) contains a 3dimensional injectively immersed Lipschitz manifold, namely the set

X t (q) with q ∈ ξ -1 γω,i ∩ B IX and t ∈]t -(q), t + (q)[
where X t is the flow of the induced Wainwright-Hsu vector field X , and ]t -(q), t + (q)[ is the maximal domain of definition of the orbit of q for this flow.

Proof of Theorem 10.4. The proof relies on the following ingredients: Theorem 9.20, Proposition 8.10, Proposition 8.9, Proposition 6.4 and Proposition 5.9. To make the proof easier to read, we will sometimes identify a point in U ξ ⊂ B + with its image by the local coordinate system ξ.

Let ω = [1; k 1 , k 2 , . . . ] ∈ Ω shad , i ∈ {1, 2}, q 0 ∈ ξ - 1
γω,i be a type IX point and q ∶ t ↦ q(t) be the forward X -orbit of q 0 . Our goal is to prove that q ∶ t ↦ q(t) shadows the heteroclinic chain H (ω). Recall that H (ω) is the concatenation of the type II orbit O P ω →P f (ω) with O P f (ω) →P f 2 (ω) and so on. Hence, the orbit

(ω n ) n≥0 = ω, f (ω), f 2 (
ω), . . . will play a fundamental role. It will be convenient to gather the terms of this sequence by eras, that is, to look at it as the double sequence

(ω j,l ) (j,l)∈E ω = (ω 0,0 = ω, ω 0,1 = f (ω), . . . , ω 0,k 1 -1 = f k 1 -1 (ω), ω 1,0 = f (ω), ω 1,1 = f ( f (ω)), . . . , ω 1,k 2 -1 = f k 2 -1 ( f (ω)), ω 2,0 = f 2 (ω), ω 2,1 = f ( f 2 (ω)), . . . )
where

E ω = (j, l) ∈ N 2 | 0 ≤ l ≤ k j+1 -1
is endowed with the lexicographical order. We will alternate between those two points of view, using the increasing bijection ϕ ∶ E ω → N defined by

ϕ(j, l) = l + j m=1 k m
In other words, we associate with any formal sequence (a n ) n∈N a sequence (a j,l ) (j,l)∈E ω where a j,l ∶= a ϕ(j,l) and conversely.

According to Theorem 9.20, q 0 belongs to the local stable manifold of P s i ω, ĥ of size ĥ⊥ (ω). In particular, Φj (q 0 ) is well defined for all j ≥ 0 and, a fortiori, Φj ĥ(q 0 ) is also well defined for all j ≥ 0. Let T 0,0 = 0, T 1,0 , T 2,0 , . . . be the successive times when the orbit q intersects the section Sĥ. For j ≥ 0,

define h ⊥,2j,0 = e -C10 s 4 ( f j (ω)) , h 2j,0 = ĥ, h ⊥,2j,0 , e -C8 s 2 ( f j (ω)) and S 2j,0 = S s f j (ω),h 2j,0 .
Claim 1. For all j ≥ 0, X T 2j,0 (q 0 ) ∈ S 2j,0 .

Proof of claim 1. Recall that ĥ⊥ (ω) ≤ h ⊥,0,0 and ζω,i is σ-Lipschitz, hence the claim is trivial for j = 0. For j ≥ 1, remark that

X T 2j,0 (q 0 ) = Φj (q 0 ) ∈ Graph sî ( f j-1 (ω)) ζ f j (ω), î( f j-1 (ω)) ⊂ S s f j (ω), ĥ f j (ω) ⊂ S 2j,0 For j ≥ 0, define h ⊥,2j+1,0 ∶= e -C9 s 4 ( f 2j+1 (ω)) , h 2j+1,0 ∶= ĥ, h ⊥,2j+1,0 , e -C8 s 2 ( f 2j+1 (ω)) and S 2j+1,0 ∶= S s f 2j+1 (ω),h 2j+1,0 .
Claim 2. For all j ≥ 0, X T 2j+1,0 (q 0 ) ∈ S 2j+1,0 .

Proof of claim 2. This is an immediate consequence of claim 1 and Proposition 8.10.

For j ≥ 0 and 1 ≤ l ≤ k j+1 -1, define h ⊥,j,l = ĥk

-ñ 5 j+1 C5 f l ( f j (ω)) m(f l ( f j (ω))), h j,l = ĥk -ñ 5
j+1 , h ⊥,j,l , h ⊥,j,l and S j,l = S s f l ( f j (ω)),h j,l . Let j ≥ 0. According to Lemma 8.8 and Proposition 8.9, the orbit segment X T j,0 (q 0 ), X T j+1,0 (q 0 ) passes through all the sections S j,1 , S j,2 , . . . , S j,k j+1 -1 in that order. We denote by

T j,0 < T j,1 < ⋅ ⋅ ⋅ < T j,k j+1 -1 < T j+1,0
the successive first times T j,l such that X T j,l (q 0 ) ∈ S j,l for all 1 ≤ l ≤ k j+1 -1. More precisely, T j,1 < ⋅ ⋅ ⋅ < T j,k j+1 -1 are defined recursively as follows

T j,1 = min t > T j,0 | X t (q 0 ) ∈ S j,1 T j,2 = min t > T j,1 | X t (q 0 ) ∈ S j,2 . . . T j,k j+1 -1 = min t > T j,k j+1 -2 | X t (q 0 ) ∈ S j,k j+1 -1
Let (P n ) n≥0 be the sequence of the successive intersection points of the heteroclinic chain H P s i ω, ĥ with the sections S 0 , S 1 , etc. According to Proposition 8.10, if X T 2j,0 (q 0 ) is close to P 2j,0 (relatively to the size h ⊥,2j,0 of the section S 2j,0 in the direction transverse to the Mixmaster attractor), then X T 2j+1,0 (q 0 ) is close to P 2j+1,0 (relatively to the size h ⊥,2j+1,0 ). More precisely, if

X T 2j,0 (q 0 ) -P 2j,0 ∞ h ⊥,2j,0 ≤ with 0 < ≤ 1, then X T 2j+1,0 (q 0 ) -P 2j+1,0 ∞ h ⊥,2j+1,0 ≤
According to Proposition 8.9, if X T j,0 (q 0 ) is close to P j,0 (relatively to the size h ⊥,j,0 ), then, for every 1 ≤ l ≤ k j+1 -1, X T j,l (q 0 ) is close to P j,l (relatively to the size h ⊥,j,l ). For (j, l) ∈ E ω , define

h j,l = ĥ if l = 0 ĥk -ñ 5 j+1 if l ≥ 1 and t n = T n + τ * f n (ω),h n X T n (q 0 )
where τ * is defined in Proposition 5.9. According to Proposition 5.9 together with Proposition 5.1 and Proposition 6.4, if X

T n (q 0 ) is close to P n (relatively to the size h ⊥,n ), then 1. X t n (q 0 ) is close to P f n (ω) .

2. The orbit segment X T n (q 0 ), X T n+1 (q 0 ) is close to the heteroclinic chain segment [P n , P n+1 ] for the Hausdorff distance.

Hence, we are left to prove that the ratio between X T 2j,0 (q 0 ) -P 2j,0 ∞ and h ⊥,2j,0 tends to 0 as j tends to +∞. One can rewrite (9.25) as follows:

X T 2j,0 (q 0 ) -P 2j,0 ∞ ≤ ∥q 0 -P 0 ∥ ⊥ e -C8 ∑ 2j m=1 k 5 m Hence, X T 2j,0 (q 0 ) -P 2j,0 ∞ h ⊥,2j,0 ≤ ∥q 0 -P 0 ∥ ⊥ e C10 s 4 ( f j (ω))-C8 ∑ 2j m=1 k 5 m
To conclude, recall that ω ∈ Ω shad . Equation (10.1) implies that

lim j→+∞ e C10 s 4 ( f j (ω))-C8 ∑ 2j m=1 k 5 m = 0
Hence, the orbit q ∶ t ↦ q(t) shadows the heteroclinic chain H (ω). This concludes the proof since Shad(ω) is clearly invariant by the flow of the Wainwright-Hsu vector field X .

Proof of Theorem 10.1. Let p be a point of the Kasner circle such that ω(p) verifies the moderate growth condition (MG). One can find an iterate F

j (p) such that ω(F j (p)) ∈ ]1, 2[ \ Q. Moreover, Shad(p) = Shad(F j (p))
and ω(F j (p)) verifies the moderate growth condition (MG). Hence, one can assume that ω = ω(p) ∈ ]1, 2[ \ Q without loss of generality. According to Proposition 10.3, ω ∈ Ω shad . Remark that γω,i ∩ B IX is a 2-dimensional Lipschitz manifold. Since the local coordinate system ξ is a diffeomorphism, it follows that ξ -1 γω,i ∩ B IX is a 2-dimensional Lipschitz manifold as well. According to Theorem 10.4, for all point q 0 ∈ ξ -1 γω,i of type IX, the orbit of the Wainwright-Hsu vector field starting at q 0 shadows the heteroclinic chain H (ω). In other words, denoting by ]t -(q), t + (q)[ the domain of definition of the orbit of a point q for the Wainwright-Hsu flow, we have the inclusion

X t (q) with q ∈ ξ -1 γω,i ∩ B IX and t ∈]t -(q), t + (q)[ ⊂ Shad(ω)
Moreover, for > 0 small enough, 1. the interval ] -, [ is contained in ]t -(q), t + (q)[ for every q ∈ ξ -1 γω,i ∩ B IX (because the domain of definition of the orbits varies semi-continuously, and ξ -1 γω,i ∩ B IX is contained in a compact subset of the phase space), 2. the set

X t (q) with q ∈ ξ -1 γω,i ∩ B IX and t ∈] -, [ = ⋃ t∈]-, [ X t ξ -1 γω,i ∩ B IX is a 3-dimensional ball Lipschitz embedded in the phase space B + .
Hence, the shadowing set Shad(ω) contains a 3-dimensional ball Lipschitz embedded in the phase space B + . Recall that B + is a quotient of B + (see section 2.6). As a consequence, Shad(p) contains a 3dimensional ball Lipschitz embedded in the phase space B + . This concludes the proof of Theorem 10.1.

Absolute continuity of the stable manifolds foliation

If p is a point of the Kasner circle K , recall that we denote by Shad(p) the reunion of all the type IX orbits in B + shadowing the heteroclinic chain starting at p. The purpose of this last section is to prove the second part of Theorem 1.9 stated in the introduction. Let us recall the statement.

Theorem 11.1 (Theorem 1.9, second part). If E ⊂ K has positive 1-dimensional Lebesgue measure, then ⋃ p∈E Shad(p) has positive 4-dimensional Lebesgue measure in the phase space B + .

We can reduce Theorem 11.1 to the following proposition. Recall that Ω shad ⊂ Ω graph ⊂ ]1, 2[. In Section 9, we constructed a graph included in the local stable set of P s i ω, ĥ for all ω ∈ Ω graph , denoted by γω,i = Graph ζω,i (see Theorem 9.20). For F ⊂ Ω graph and i ∈ {1, 2}, let

W s i F, Φ def = ⨆ ω∈F γω,i Proposition
-1 W s 1 E, Φ ⊂ S
s 1 ĥ has positive 3-dimensional Lebesgue measure as well. Hence, the set

X t (q) | q ∈ ξ -1 W s 1 E, Φ and t ∈ t ∈]t -(q), t + (q)[
(where ]t -(q), t + (q)[ is the domain of definition of the orbit of q) has positive 4-dimensional Lebesgue measure in B + . Moreover, according to Theorem 10.4, ⋃ ω∈E Shad(ω) contains the above set. Recall that B

+ is a finite quotient of B + (see Section 2.6). As a consequence, ⋃ p∈E Shad(p) contains a 4-dimensional Lebesgue measure set. Hence, Theorem 11.1 holds true.

To prove Proposition 11.2, we will use a strategy due to Pesin, which consists in considering the holonomy along the "foliation" in local stable manifolds, and proving that this holonomy is made of absolutely continuous maps. This strategy is well-known in the context of non-uniformly hyperbolic maps. We call "foliation in local stable manifolds" the set

γω,i | ω ∈ E, i ∈ {1, 2} S s 1 ω,h H y R L y W s 1 E, Φ γω,1 γω ′ ,1 • ω • H y (ω) • H y (ω ′ ) • ω ′ Figure 22:
The map H y .

According to Proposition 9.21, the map ζω depends continuously on ω. Now let us explain why Proposition 11.2 is not a direct consequence of this continuity. Consider a set E ⊂ Ω shad of positive 1-dimensional Lebesgue measure. For y ∈ 0, ĥ 2 , introduce the horizontal line

L y def = (x u , ĥ, x s 2 , x c ) ∈ S s 1 ĥ | (x u , x s 2 ) = y
Recall that ĥ⊥ (ω) is the "size" of the graph γω,i (see definition (9.18) and Theorem 9.20). To simplify the discussion, suppose that there exists h ⊥ > 0 such that for all ω ∈ E, the graph γω,1 has a size larger thanh ⊥ , i.e. ĥ⊥ (ω) ≥ h ⊥ . According to Fubini's theorem, the set W s 1 E, Φ has positive 3-dimensional Lebesgue measure if and only if there exists a set Y ⊂ 0, ĥ 2 of positive 2-dimensional Lebesgue measure such that for all y ∈ Y , the intersection

L y ∩ W s 1 E, Φ has positive 1-dimensional Lebesgue measure. For y = (y 1 , y 2 ) ∈ [0, h ⊥ ] 2 , define the map H y ∶ E ⟶ L y ∩ W s 1 E, Φ ω ⟼ L y ∩ γω,1 = (y 1 , ĥ, y 2 , ζω,1 (y))
See figure 22. Since γω,1 is a graph of size bigger than h ⊥ , the map H y is well-defined, one-to-one and onto. Remark that

H y (E) = L y ∩ W s 1 E, Φ
If one wants to deduce the fact that W s 1 E, Φ has positive 3-dimensional Lebesgue measure from the fact that E has positive 1-dimensional Lebesgue measure using the maps H y , one needs to show that these maps send positive Lebesgue measure sets onto positive Lebesgue measure sets for all

y ∈ Y ⊂ [0, h ⊥ ]
2 where Y has positive 2-dimensional Lebesgue measure. Proposition 9.21 shows that H y is continuous, which implies that the restriction of H y to any compact set is a homeomorphism on its image. Unfortunately, it is well known that there exist homeomorphisms that send non-zero Lebesgue measure sets onto zero Lebesgue measure sets. Hence, Proposition 11.2 is not a straightforward consequence of Proposition 9.21. We must show that the map H y sends positive Lebesgue measure sets onto positive Lebesgue measure sets, using a specific method. Let us now describe this specific method, which is due to Y. Pesin. From now on and until the end of this section, we fix a set E ⊂ Ω shad of positive 1-dimensional Lebesgue measure. We are going to replace E by a subset Ẽ such that we have some uniform estimates on the continued fraction expansion of points of Ẽ and such that Ẽ still has positive 1-dimensional Lebesgue measure. Define

F s 1 def = γω,1 | ω ∈ Ẽ Remark that Ω shad is totally disconnected, hence F s 1 is the family of connected components of W s 1 Ẽ, Φ . Even if F s 1
is not a foliation of the section S s 1 ĥ , we will call F s 1 the local stable manifolds "foliation" of the double era return map Φ. Remark that F s 1 is leaf-invariant by Φ. The uniform estimates on points of Ẽ will be crucial to prove that the local stable manifolds "foliation" of the double era return map Φ is absolutely continuous. We now proceed to define Ẽ. According to Lemma A.1, there exists n 0 and l 0 such that the set

π S s 1 ĥ R • z • y W s 1 Ẽ, Φ • π(z) • π(y) •x • π(x)
Ẽ def = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ω ∈ E ∀n ≥ n 0 , 2n i=1 k i (ω) 5 ≥ n 5-1 10 ∀n ≥ n 0 , k 2n+1 (ω) 4 + k 2n+2 (ω) 4 + k 2n+3 (ω) 4 + k 2n+4 (ω) 4 ≤ n 4+ 1 10 ∀1 ≤ n ≤ 2n 0 , k n (ω) ≤ l 0 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (11.1)
has positive 1-dimensional Lebesgue measure. Remark that the quantity In other words, the size of the graph γω,1 is uniformly bounded from below by δ ⊥,0 for ω ∈ Ẽ. Now, let us introduce a projection map π which is somehow the inverse of H y . Roughly speaking, we will project points of W s 1 Ẽ, Φ onto the Mixmaster attractor along the foliation F s 1 and then project to the last coordinate. See figure 23.

e C12 s 4 ( f n (ω))-C8 ∑ 2n i=0 k i (ω)

Definition 11.3 (Projection map). The projection map

π ∶ W s 1 Ẽ, Φ = ⨆ ω∈ Ẽ γω,1 → R is defined by π(x) = ω for all x ∈ γω,1 .
Remark 11.4. The restriction π |L y of the projection map is the inverse of H y .

Remark 11.5. To make the reading easier, we will write π(G) instead of π G ∩ W s 1 Ẽ, Φ for any set G ⊂ Sĥ.

We denote by Leb n the n-dimensional Lebesgue measure. Lemma 11.6 states precisely that the projection map π is absolutely continuous in restriction to horizontal lines. For y ∈ 0, ĥ 2 , let π y be the restriction of π to the horizontal line L y . We are left to prove Lemma 11.6. Let us explain the general strategy of the proof. Consider a set G ⊂ L y ∩ W s 1 Ẽ, Φ such that Leb 1 (G) = 0. We will cover G by a countable union of little horizontal segments. We need two definitions to make this idea precise. ĥ . We say that D is a horizontal segment if there exists y ∈ 0, ĥ 2 such that D is a compact and connected subset of L y . If this is the case, there exists a unique pair (x, x) ∈ L y such that |D| = xcx c . In other words,

Lemma 11.6. There exists

0 < h ⊥ ≤ δ ⊥,0 such that for all y ∈ [0, h ⊥ ] 2 and all G ⊂ L y ∩ W s 1 Ẽ, Φ , Leb 1 (G) = 0 ⟹ Leb 1 (π(G)) = 0 (11.3) Proof of
D = z = (z u , ĥ, z s 2 , z c ) ∈ S s 1 ĥ | (z u , z s 2 ) = y, x c ≤ z c ≤ xc
We call x and x the end points of D. Moreover, we say that D is centered above ω if y ∈ 0, ĥ⊥ (ω) Since G has zero 1-dimensional Lebesgue measure, one can find a countable family (D i ) i∈N of horizontal segments centered above points of Ẽ covering G and satisfying

+∞ i=0 Leb 1 (D i ) ≤
where is an arbitrary fixed positive number. Assume that there exists a constant M (independent of the choice of the segments D i ) such that for all i ∈ N,

Leb 1 (π (D i )) ≤ M Leb 1 (D i )
We get that π(G) is covered by the countable union of sets π (D i ) whose total measure is arbitrary small. As a consequence, π(G) has zero 1-dimensional Lebesgue measure. Hence, we are left to control the projection of an horizontal segment D by the map π. Informally, we will prove the following statement. As stated earlier, the strategy used to prove the above statement is borrowed from Pesin's work on non-uniformly hyperbolic dynamical systems. First, remark that if D has a diameter δ / / and is positioned at distance δ ⊥ from the Mixmaster attractor with δ ⊥ ≤ δ / / , then the above result is easy to prove. Indeed, recall that the graphs γω,1 are all 1 2 -Lipschitz. Hence, the projection π (D) has a diameter less than δ ⊥ + δ / / ≤ 2δ / / . For the general case, one can try to "push by Φ" the horizontal segment D so that Φn (D) is in the configuration of the previous situation. Indeed, recall that Φ contracts the direction transverse to the Mixmaster attractor and expands the direction tangent to the Mixmaster attractor. Hence, for n large enough, Φn (D) will have a "large" diameter and will be "close" to the Mixmaster attractor. As a consequence, the result should hold true if we replace D by Φn (D). To conclude, one needs to tackle two difficulties.

The first one is the ability to "come back to D". In other words, we need to prove that if the result holds true for Φn (D), then it holds true for D as well. This amounts to prove that Φn expands D in the tangent direction to the Mixmaster attractor and f n expands π (D) with almost the same factor, independantly of n. This is the distorsion estimate proved in Proposition 11.9.

The second one is the fact that n must be "well chosen": large enough so that Φn (D) has a "large" diameter and is "close" to the Mixmaster attractor but not too large because we need to ensure that for all 0 ≤ j ≤ n, Φj (D) is contained in a small section S s f j (ω),h j (ω) where all the objects of interest are well defined and well controlled.

Distorsion estimate

Recall that we introduced a constant C12 in Lemma 9.11, when we described the preimage of a σ-Lipschitz graph by the double era return map Φ. Now, let us fix C13 ≥ C12 large enough so that for all a ≥ 1, The property (H ω,D,n ) implies that all the objects playing a role in the distorsion estimate are well defined and well controled for n iterates. The next proposition gives a precise statement about the distorsion estimate we need.

Proposition 11.9 (Distorsion estimate). There exists a constant ∆ ≥ 1 and a constant δ ⊥,2 > 0 such that the following property holds true for ω ∈ Ẽ, y ∈ 0, δ ⊥,2 2 and n ≥ 0. Let D ⊂ L y be an horizontal segment centered above ω. If (H ω,D,n ) holds true, then Φn (D)

f n (π (D)) ≤ ∆ |D| |π (D)|
Roughly speaking, the distorsion estimate means that D and π (D) are "similarly" expanded by Φn and f n respectively, uniformly with respect to n. The first step to prove this distorsion estimate is to show that under the hypotheses of Proposition 11.9, the j-th iterate of the horizontal segment D by the double era return map is almost horizontal.

Recall that C8 is the constant defined in Proposition 8.2 on the double era transition map. Remark that δ ⊥,0 ≤ inf Since the coefficients k 1 (ω), . . . , k 2n 1 (ω) of any point ω ∈ Ẽ admit a uniform upper bound depending only on n 0 , n 1 and l 0 (see (11.1)), one can find a constant 0 < δ ⊥,1 ≤ δ ⊥,0 such that for every ω = [1; k 1 , k 2 , . . . ] ∈ Ẽ and every j ≥ 0, we have δ ⊥,1 h⊥,j

α j = Φj (x) - Φj (x) ⊥ Φj (x) -Φj (x) / /
According to (8.8) and the fact that both Φj (x) and Φj (x) belong to the section S s f j (ω),h f j (ω) for 0 ≤ j ≤ n -1, we get by induction on j that for every 0 ≤ j ≤ n, h ⊥,j ≤ δ ⊥,1 e Proof of claim 1. By hypothesis, x and x belong to a same horizontal line, hence ∥x -x∥ ⊥ = 0. In other words, α 0 = 0 so α 0 ≤ α max 0

(ω) holds true. Fix 0 ≤ j ≤ n -1 and assume that α j ≤ α max j

(ω). We apply (8.11) to the map Φ restricted to the section S s f j (ω), ĥ,h ⊥,j ,h / /,j : Φ Φj (x) -Φ Φj (x) -ΦA Φj (x) -ΦA Φj (x)

α j+1 ≤ α max j+1 (ω)
This concludes the proof of claim 1.

In particular, α n ≤ α max n (ω), which is the desired result. This concludes the proof of Proposition 11.10.

Proof of Proposition 11.9. We begin with the definition of the size δ ⊥,2 . For ω = [1; k 1 , k 2 , . . . ] ∈ Ẽ and j ≥ 0, recall that h⊥,j = e Proof of claim 1. Let 0 ≤ j ≤ n -1. We apply (8.11) to the map Φ restricted to the section S s f j (ω), ĥ,h ⊥,j ,h / /,j : Φ(x j ) -Φ(x j ) -ΦA (x j ) -ΦA (x j ) This concludes the proof of claim 1.

Recall that K c > 1 is the expansivity constant in the tangential cone field (see (9.4)). Let

C 2 def = 1 4 +∞ j=0 K -j c + +∞ j=0 e -j Claim 2. R 2 ≤ e C 2 .
Proof of claim 2. Recall that the last coordinate of the double era return map restricted to the Mixmaster attractor is exactly the double era Kasner map, hence

R 2 = n-1 j=0 f (x j ) c -f (x j ) c |(x j ) c -(x j ) c | x ′ j - x′ j f (x ′ j ) -f (x ′ j )
Applying the mean value theorem to the function f , we get that

R 2 = n-1 j=0 f ′ (z j ) f ′ (z ′ j )
where z j ∈ (x j ) c , (x j ) c and z (note the hypothesis of Proposition 7.3 are satisfied, since |z j -ω| ≤ h / /,j , |z j -ω| ≤ h / /,j and we know that ω-1 2 ≥ 1 4k 1 (ω) >> h / /,j and and analogously if G is a subset of S s 2 ĥ . The next lemma gives a sufficient condition so that we can control all the iterates Φj (D), 0 ≤ j ≤ n -1, for a time n sufficiently large so that the distance between the Mixmaster attractor and Φn (D) is smaller than its diameter in the direction tangential to the Mixmaster attractor. Proof of claim 1. We need to prove that for all 0 ≤ j ≤ n(D) -1, Φj (D) ⊂ S s f j (ω),h j (ω)

For 0 ≤ j ≤ n(D) -1, define the property

(P j ) ∶ Φj (D) ⊂ S s f j (ω),h j (ω)
Recall that D is centered above ω. Hence, for all x ∈ D,

|x c -ω| ≤ |x c -v c | + |v c -ω| ≤ δ / / + δ ⊥,3 ≤ 2δ / /
where v denotes the intersection point of D with γω,1 . Using (11.31a) and the definition of δ / / , we get that D ⊂ S s ω,h 0 (ω) . Hence, (P 0 ) holds true. Fix 0 ≤ l ≤ n(D) -2 and assume that for all 0 ≤ j ≤ l, (P j ) holds true. It follows that l + 1 ≤ N (D). Let x ∈ D. Plugging (11.32) Hence, Φl+1 (x) ∈ S s f l+1 (ω),h l+1 (ω) and (P l+1 ) holds true. By induction, we get that for all 0 ≤ j ≤ n(D) -1, (P j ) holds true. This concludes the proof of claim 1.

Claim 2. N (π (D)) ≥ n(D).

Proof of claim 2. Assume that N (π (D)) < n(D). Recall that the local stable manifolds "foliation" F s 1 of the double era return map Φ is made of σ-Lipschitz graphs with σ ≤ 1 2 (see (8.5)). Using (11.32) with j = N (π (D)) ≤ N (D) (see claim 1), we get that Recall that the local stable manifolds "foliation" F s 1 is leaf-invariant by Φ. Hence, π semi-conjugate Φ and f : The inequality (A.2) is a consequence of a standard fact: for Lebesgue almost every point x ∈ Ω, the sequence (k i (x)) i≥0 of the partial quotients of x does not grow "too fast" (see corollary A.3). The inequality (A.1) is a consequence of a less standard result: for Lebesgue almost every point x ∈ Ω and for every n ∈ N large enough, there is at least one partial quotient among k 1 (x), . . . , k n (x) which is "large" (see Proposition A.4). More precisely, the standard result can be rigorously stated as follows.

π • Φ = f • π
E x def = n ∈ N * | k n (x) ≥ ϕ(n)
is finite for Lebesgue almost all x ∈ Ω, or it is infinite for Lebesgue almost all x ∈ Ω. More precisely, this dichotomy depends on ϕ as follows:

1. If ∑ 1 ϕ(n) is divergent, then for Lebesgue almost all x ∈ Ω, there exists infinitely many n ∈ N * such that k n (x) ≥ ϕ(n).

2. If ∑ 1 ϕ(n) is convergent, then for Lebesgue almost all x ∈ Ω, there exists n 0 (x) ∈ N * such that for every n ≥ n 0 (x), k n (x) < ϕ(n).

Proof. See [START_REF] Iakovlevich | Continued Fractions[END_REF].

Corollary A.3. Let > 0. For Lebesgue almost every point x ∈ Ω, there exists n 0 ∈ N such that for every n ≥ n 0 , k n (x) ≤ n 1+ .

Proof. For any > 0, the serie ∑ n -1-is convergent.

We now give a precise formulation of the second result needed to prove Lemma A.1.

Proposition A.4. For Lebesgue almost all x ∈ Ω, for every > 0, there exists n 0 (x, ) ≥ 1 such that for every n ≥ n 0 (x, ), there exists an integer 1 ≤ j ≤ n such that k j (x) ≥ n 1-.

Let us introduce some tools that will be needed to prove Proposition A.4. We denote by τ ∶ Ω → Ω the Gauss transformation defined by τ (x) = 1

x where {x} = x -⌊x⌋ denotes the fractional part of x. The very definition of τ implies that, for every continued fraction where the supremum is taken on all the finite sequences 0 ≤ t 1 < ⋅ ⋅ ⋅ < t n ≤ 1, n ≥ 2. If var f < +∞, then we say that f is of bounded variation. For any map f ∈ L ∞ λ ([0, 1]), we call essential variation of f and we denote by v(f ) the number inf var f where the infimum is taken on all the maps f equal to f mod 0. If v(f ) < +∞, then we say that f is of bounded essential variation. Let us denote by BEV([0, 1]) the set of all maps f ∈ L ∞ λ ([0, 1]) such that v(f ) < +∞. Let us equip BEV([0, 1]) with the norm ∥f ∥ BEV = v(f ) + ∥f ∥ 1

We define the Perron-Frobenius operator U as the "dual" of the composition operator induced by τ . More precisely, U is defined as the unique bounded linear operator L

U n f - 1 0 f dγ G 1 ≤ Cα n ∥f ∥ BEV Proof. See [IK13].
Proof of Proposition A.4. 13 Let us define, for n ≥ 2 and > 0 small:

X n, = x ∈ Ω | k 1 (x) < n 1- Y n, = x ∈ Ω | ∀1 ≤ j ≤ n, k j (x) < n 1-
According to the Borel-Cantelli lemma, it is enough to prove that n≥1 γ G Y n, < +∞ (A.4)

One can remark that

Y n, = ⋂ 0≤j≤n-1 τ -j X n, so γ G Y n, = 1 0 n-1 j=0 1 X n, • τ j (x) dγ G (x)
Let c = ⌊n 2 ⌋ and K = ⌊ n-1 n 2 ⌋. We can estimate the above integral by keeping only the terms whose indices are multiples of c:

γ G Y n, ≤ 1 0 K j=0 1 X n, • τ jc (x) dγ G (x) = 1 0 1 X n, (x). K-1 j=0 1 X n, • τ jc • τ c (x) dγ G (x) = 1 0 U c 1 X n, (x) . K-1 j=0 1 X n, • τ jc (x) dγ G (x)
However, the family 1 X n, n is uniformly bounded by 2 in BEV([0, 1]) and ∏ By induction, we get

γ G Y n, ≤ γ G X n, K+1 + O n→∞ Kα c
However, X n, = Ω ∩ 1 ⌊n 1-⌋+1 , 1 and using (A.3), we get that γ G X n, = 1 ln 2

1 1 ⌊n 1-⌋+1 1 x + 1 dx = 1 - 1 n 1-ln 2 + O n→+∞ 1 n 2-2
Moreover, ⌊ n -1

n 2 ⌋ + 1 ln 1 - 1 n 1-ln 2 + O n→+∞ 1 n 2-2 = - 1 ln 2 n 2 + o n→+∞ (1)
Hence,

γ G X n, K+1 ∼ n→+∞ e -1 ln 2 n 2
and γ G X n, K+1 is the general term of a convergent series. Analogously, Kα c is the general term of a convergent series. Hence, (A.4) holds true. This concludes the proof of Proposition A.4. The following result provides some explicit conditions ensuring that the continued fraction expansion of two nearby real numbers start by the same integer. It is used to prove Lemma 8.8. In particular, it is useful to find a sufficiently small size for the section S s ω,h ω so that all the points (in fact, their coordinate x c ) in S s ω,h ω have the same first partial quotient.

Proof of Lemma

Proposition A.6. For x, x ′ ∈ Ω, if

x -x ′ < 1 10 1 k 1 (x) 2 k 2 (x)k 3 (x)
then k 1 (x ′ ) = k 1 (x).

Proof. Fix x = [k 1 , k 2 , . . . ] ∈ Ω. Let x ′ = [k ′ 1 , k ′ 2 , . . . ] ∈ Ω such that x -x ′ < 1 3 1 k 2 1 k 2 k 3 One can remark that [k 1 + 1] < [k 1 , k 2 , k 3 + 1] < x < [k 1 , k 2 , k 3 ] < [k 1 ]
By a straightforward computation, one gets

[k 1 ] -[k 1 , k 2 , k 3 ] ≥ 1 3k 2 1 k 2 k 3 and [k 1 , k 2 , k 3 + 1] -[k 1 + 1] ≥ 1 10k 2 1 k 2 k 3 It follows that [k 1 + 1] < x ′ < [k 1 ] Hence, k ′ 1 = k 1 .
The following results provide some explicit conditions ensuring that the continued fraction expansion of two nearby real numbers start by the same first two integers. Moreover, it shows that the double Gauss transformation τ 2 is expansive. It is particularly useful to prove Lemma 9.11.

Proposition A.7 (Expansivity of τ 2 ). For x, x ′ ∈ Ω, if

xx ′ < 1 24 1 k 1 (x) 2 k 2 (x) 2 k 3 (x)k 4 (x)

then k 1 (x ′ ) = k 1 (x), k 2 (x ′ ) = k 2 (x) and τ 2 (x) -τ 2 (x ′ ) ≥ 4 x -x ′ Proof. Fix x = [k 1 , k 2 , . . . ] ∈ Ω. Let x ′ = [k ′ 1 , k ′ 2 , . . . ] ∈ Ω such that x -x ′ < 1 24 1 k 2 1 k 2 2 k 3 k 4 One can remark that [k 1 , k 2 ] < [k 1 , k 2 , k 3 , k 4 ] < x < [k 1 , k 2 , k 3 , k 4 + 1] < [k 1 , k 2 + 1]
By a straightforward computation, one gets

[k 1 , k 2 + 1] -[k 1 , k 2 , k 3 , k 4 + 1] = 1 + (k 3 -1)(k 4 + 1) (k 1 (k 2 + 1) + 1) ((k 1 k 2 k 3 + k 3 + k 1 )(k 4 + 1) + k 1 k 2 + 1) ≥ 1 24 1 k 2 1 k 2 2 k 3 k 4 and [k 1 , k 2 , k 3 , k 4 ] -[k 1 , k 2 ] = k 4 (k 1 k 2 + 1)((k 1 k 2 + 1)(k 3 k 4 + 1) + k 1 k 4 ) ≥ 1 10 1 k 2 1 k 2 2 k 3 It follows that [k 1 , k 2 ] < x ′ < [k 1 , k 2 + 1] Hence, k ′ 1 = k 1 and k ′ 2 = k 2 . Writing x = 1 k 1 + 1 k 2 +τ 2 (x) , x ′ = 1 k 1 + 1 k 2 +τ 2 (x ′ ) leads to x -x ′ = τ 2 (x) -τ 2 (x ′ ) (k 1 k 2 + k 1 τ 2 (x) + 1)(k 1 k 2 + k 1 τ 2 (x ′ ) + 1) Since k 1 k 2 + 1 ≥ 2, we get τ 2 (x) -τ 2 (x ′ ) ≥ 4 x -x ′

B Statement of the main theorem in the entire phase space

In this appendix, we explain how to extend Theorem 1.9 to type VIII orbits. To this end, we show how some objects defined in the introduction (especially type II orbits, the Kasner map and heteroclinic chains) can be generalized to the entire phase space. A technical complication arises since most abstract heteroclinic chains cannot be shadowed by any type VIII or IX orbit for elementary reasons. This will lead us to introduce a notion of coherent heteroclinic chain.

Type II orbits. Recall that in B + , for every point p of the Kasner circle that is not a Taub point, there is exactly one type II orbit starting at p. When looking at the full phase space B, we have the following result. For every point p of the Kasner circle that is not a Taub point, there are exactly two type II orbits starting at p. These two orbits are exchanged by the symmetry

(N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 ) ↦ (-N 1 , -N 2 , -N 3 , Σ 1 , Σ 2 , Σ 3 )
fixing the points of the plane (N 1 = N 2 = N 3 = 0) containing the Kasner circle. As an immediate consequence, these two type II orbits converge to the same point of K in the future.

For every point p of the Kasner circle (such that, for every k ≥ 0, F k (p) is not a Taub point), there are exactly eight coherent heteroclinic chains starting at p corresponding to the eight different choices of three hemiellipsoids (or, analogously, corresponding to the eight different choices of three signs for the variables N i ). One should remark that a type VIII orbit cannot shadow the same coherent heteroclinic chain as a type IX orbit. Among the eight coherent heteroclinic chains starting at p, six can be shadowed by type VIII orbits and two by type IX orbits.

Having this definition in mind, it is clear that Theorem 1.9 must be generalized by replacing the unique heteroclinic chain in B + starting at p by one of the eight coherent heteroclinic chains in B starting at p. Recall that K (M G) denotes the set of all the points p ∈ K such that ω(p) verifies the moderate growth condition (MG).

Main

Figure 1 :

 1 Figure 1: A type II orbit connecting two points of the Kasner circle K .

  2 (p), etc. Formally, this is the sequence O p→F (p) , O F (p)→F 2 (p) , O F 2 (p)→F 3 (p) , . . . (1.4)

Definition 1. 7 (

 7 Moderate growth condition). Let ω = [k 0 ; k 1 , k 2 , . . . ] ∈ ]1, +∞[ \ Q be a continued fraction.We say that ω verifies the moderate growth condition if

Figure 3 :

 3 Figure 3: Bowen's eye attractor.

F

  (p) , • apply Grönwall's lemma to obtain some (very) rough control during this travel.Dynamics along an epoch. Given a point p of the Kasner circle, the epoch transition map Φ p is the transition map of the orbits of X from a section S s p at the entrance of a neighbourhood of the point p to a section S s F (p) at the entrance of a neighbourhood of the point F

2 II and B 3 II

 3 are defined analogously. The Mixmaster attractor. The reunion of the Kasner circle K and the stratum B II is called the Mixmaster attractor and is denoted by A . We denote by A + ∶= K ∪ B + II the positive part of the Mixmaster attractor. Generic orbits The strata B + IX corresponding to semi-simple Lie algebras is open and dense in B + . Generic orbits of the Wainwright-Hsu vector field X are contained in B + IX .

Definition 2. 3 (

 3 Kasner map). Let p ∈ K \ {T 1 , T 2 , T 3 }. The type II orbit O u p starting at p converges (in the future) to a point of the Kasner circle denoted by F (p). Set F (T i ) = T i for all i ∈ {1, 2, 3}. This defines a continuous map F ∶ K → K , called the Kasner map (sometimes also called the BKL map). The orbit O u p will also be denoted by O p→F (p) .

Figure 8 :

 8 Figure 8: Projections on the (Σ 1 , Σ 2 , Σ 3 )-plane of type II orbits contained in B 1 II , B 2 II and B 3 II (left to right, top to bottom).

s 1 ω and O s 2 ω

 2 , which converge to the point P ω as time goes to +∞. They are respectively asymptotically tangent to the directions ∂ N s1 and ∂ N s2 .

Definition 3. 1 .

 1 For any ω ∈ ]1, +∞[, C > 0 and n ∈ N, let us denote by B ω,C,n def = B P ω , 1 Cω n ⊂ B + 0 the ball of center P ω and radius 1 Cω n in the phase space B + 0 (for the distance d B , see (2.11)) and by

  2. Properties (3.1) and (3.2) follow immediately from (3.14) and (3.21). Properties (3.3) follow immediately from from (3.18) and (3.24). The fact that there exist two constants C > 0 and n ∈ N such that estimates (3.4) and (3.5) hold true for any ω ∈ ]1, +∞[ is an immediate consequence of (3.15), (3.22) and (3.23).

  coordinate system ξ. The pseudonorms ∥.∥ ⊥ , ∥.∥ / / and the projection Proj A

  and similarly for the two other type II orbits. By an abuse of notation, we will call the sets O u ω , O s 1 ω and O s 2 ω "type II orbits".

  oriented by the flow of the Wainwright-Hsu vector field X .

  Definition 4.10. Let ω ∈ ]1, +∞[ and h > 0. We denote respectively by P u ω,h , P s 1 ω,h , P s 2 ω,h the points on the type II orbits O u ω , O s 1 ω and O s 2

  s 1 ω,h to the Kasner interval K 0 . h ⊥ is the size of S s 1 ω,h in the direction transversal to the Mixmaster attractor. h / / is the size of S s 1 ω,h in the direction parallel to the Kasner interval K 0 . The section S s 1 ω,h cuts the type II orbit O s 1 ω at the point P s 1 ω,h . Its intersection with the Mixmaster attractor A is the segment {0} × {h} × {0} × ωh / / , ω + h / / . See figure 12. • Moreover, one can interpret the terms x -P s 1 ω,h ⊥ and x -P s 1 ω,h / / as follows: x -P s 1 ω,h ⊥ is the distance between x and the type II orbit O s 1 ω in the direction transverse to the Mixmaster attractor while x -P s 1 ω,h / / is the distance between x and the type II orbit O s 1 ω in the direction tangent to the Mixmaster attractor. Similar remarks hold for the sections S u ω,h and S s 2 ω,h as well.

Figure 13 :

 13 Figure 13: A type IX orbit O (in black) traveling close to the first era of the heteroclinic chain starting at P ω (in melon), where ω = [1; k 1 , k 2 , . . .]. The local sections S

Figure 14 :

 14 Figure 14: A type IX orbit O (in red) traveling between the section S s ω,h and the section S s f (ω),h ′ . During this travel, the orbit O stays close to a piece of heteroclinic chain (in melon) passing through the point P ω .If the parameters h, h′ and h ′′ are well-chosen (roughly speaking, h ′ must be small enough, h ′′ much smaller than h ′ and h much smaller than h ′ ; explicit bounds will be given in the next sections), the orbit O will first cross the section S

Figure 15 :

 15 Figure 15: Notations for local shadowing.

  s where h s = (h, min(h, d(ω)), min(h, d(ω))) and x in u ≠ 0 (hence x in u > 0). Let C 0 > 0 and n ∈ N be large enough such that we can apply Lemma 5.16 with these two constants and such that for any ω ∈ ]1, +∞[ the local vector field X ω is well defined on the open ball B ω,C 0 ,n in (R + ) 3 × ]1, +∞[ (see definition 3.1 and Proposition 3.8). Let C 1 = 100C 0 and

  Estimate of |A 2 |. According to (5.10), for every C ≥ C 1 , every n ≥ n 1 and every (ω, h, x, x) ∈ E C,n , we have e ∫ 0 w -μ ω,s i (x c (σ)) dσ ≤ e αµ s1 (ω)w

  .41) It follows from (5.40) and (5.41) that for every C ≥ C 3 , every n ≥ n 3 and every (ω, h, x, x) ∈ E C,n , we have xs i (0) -xin s i ≤ 4ωh 1 x in u x in u -xin u so the estimate (5.36) holds true with C = max(C 3 , 4) and n = n 3 .

  u -synchronized orbits, where h s ∶= (h, h ⊥ , min(h, d(ω))) and h u ∶= (h, h, 2h). According to Remark 5.25, we have x in u = xin u . By continuity of the map Υ, it is enough to prove the estimates (5.3) and (5.4) in the case x in u > 0. So, we assume x in u = xin u > 0 in the sequel.

  n . Let us denote by I ∶ M ↦ M -1 the inversion in GL(R 4 ). We have Dθ = I • Dϕ • θ and D 2 θ = DI(Dϕ • θ)D 2 ϕ(θ)Dθ. According to claims 2 and 4 and the inequality ∥DI(Dϕ • θ)∥ ∞ ≤ ∥Dθ∥ 2 ∞ , we get the desired result. Next claim is a double statement. First part is a standard consequence of the mean value theorem. Second part is a direct consequence of the definition of ϕ and S s 2 ω,(h,r ′ ,r ′ ) .

u 0 , t s 1 .

 1 Recall that the local coordinates (x u , x s 1 , x s 2 , x c ) are defined on the open ball B ω, C5 ,ñ 5 , i.e. B ω, C5 ,ñ 5 ⊂ U ξ (see definition 3.1). Let t out 0 be the first time t > t u 0 when the orbit ξ • q leaves the open ball B ω,2 C5 ,ñ 5 . On t u 0 , t out 0

  s ω,h ω which satisfies ∥x -x∥ / / = 0 and ∥x -x∥ ⊥ = 0

  Remark 9.6. Note that e-C8 ∑ 2n j=1 k j (ω)

Forward invariance of the

  tangential cone field (9.2). If x ∈ V / / ω,h,i (x), then Φ(x) -Φ(x) (x) ∈ Int V / / f (ω),h ′ , î(ω) Φ(x) .

2

  where ζ ∶ [0, a] 2 → R satisfies ζ(0, z) = ω for all z ∈ [0, a].Remark 9.10. Recall from Proposition 3.2 that the local coordinates x u , x s 1 and x s 2 are positive. This is why the map ζ is defined on [0, a] 2 .Lemma 9.11 (Graph transformation over one point).There exists a constant C12 > C11 such that the property below holds for

Figure 20 :

 20 Figure 20: The graph transformation.

Lemma 9. 19 (

 19 Fixed point of the graph transformation). The graph transformation Φ * is a contraction mapping of the complete metric space Γ, d graph with Lip Φ * ≤ 1 K c (1 -σ2 ) As a consequence, Φ * admits a unique fixed point in Γ, denoted by γ = Graph ζ . Proof. Let γ = Graph (ζ) , γ = Graph ζ ∈ Γ. We are going to prove that d graph Φ * γ, Φ * γ ≤ 1 K c (1 -σ2 ) d graph (γ, γ) If we denote Graph Φ * ζ = Φ * γ and Graph Φ * ζ = Φ * γ, then it is enough to prove that for all ω ∈ Ω graph and all

Figure 23 :

 23 Figure 23: The projection map π.

5

  is uniformly bounded from above for ω ∈ Ẽ. Hence, according to the very definition of ĥ⊥ (ω) (see (9.18)),

  Definition 11.7 (Diameter). Let G ⊂ Sĥ. We define the diameter (in the direction tangential to the Mixmaster attractor) of G by |G| def = sup x,x∈G ∥x -x∥ / / = sup x,x∈G |x c -xc | Definition 11.8 (Horizontal segment centered above ω). Let ω ∈ Ẽ and D ⊂ S s 1

2

  and if the middle of the segment [x c , xc ] coincides with v c , where v denotes the intersection point between L y and γω,1 . See figure24.

Figure 24 :

 24 Figure 24: Segment D centered above ω with its end points.

  ω ∈ ]1, 2[ \ Q and j ≥ 0, define m j (ω) def = s 2 ( f j (ω)) + max 1≤l≤2j k l (ω), h j (ω) def = ĥ, e -C13 s 4 ( f j (ω)) , e -C13 m j (ω)and the intervalI j (ω) def = f j (ω) -e -C13 m j (ω) , f j (ω) + e -C13 m j (ω)Note that m 0 (ω) = s 2 (ω). Given ω ∈ ]1, 2[ \ Q, a horizontal segment D and a positive integer n, we define the property(H ω,D,n ) ∶ ∀0 ≤ j ≤ n -1, Φj (D) ⊂ S s f j (ω),h j (ω)∀0 ≤ j ≤ n -1, f j (π (D)) ⊂ I j (ω)

- 4 ×

 4 C8 s 4 (ω)For ω ∈ Ẽ and n ≥ 0, let 162 k 2n+1 (ω) 2 k 2n+2 (ω) 2 For ω ∈ ]1, 2[ \ Q, let h ω def = ĥ, e -C8 s 4 (ω) , e -C8 s 2 (ω)Proposition 11.10 (Decreasing angle with the Mixmaster attractor). There exists 0 < δ ⊥,1 ≤ δ ⊥,0 such that the following property holds for ω ∈ Ẽ, y ∈ 0, δ ⊥,1 2 , x, x ∈ L y and n ≥ 0. Suppose that both Φj (x) and Φj (x) are well-defined and belong to the section Ss f j (ω),h f j (ω) for 0 ≤ j ≤ n -1, then Φn (x) -Φn (x)Remark 11.11. For n = 0, the above proposition is trivial since ∥x -x∥ ⊥ = 0.Remark 11.12. The ratioΦn (x)-Φn (x) ⊥ Φn (x)-Φn (x) / /measures the angle between the segment Φn (x), Φn (x) and the horizontal direction (the direction tangent to the Mixmaster attractor). Proposition 11.10 states that this angle decreases at a rate of a "stretched exponential".Proof of Proposition 11.10. The proof relies on the fact that the double era return map Φ contracts the direction transverse to the Mixmaster attractor and expands the direction tangent to the Mixmaster attractor. We begin with the definition of the size δ ⊥,1 . For any ω = [1; k 1 , k 2 , . . . ] ∈ Ẽ and any j ≥ 0Using the uniform estimates (11.1) for points of Ẽ, we get that there exists n 1 ≥ n 0 (depending only on n 0 ) such that for every ω = [1; k 1 , k 2 , . . . ] ∈ Ẽ and every j ≥ n 1 , we have h

Claim 1 .

 1 For all 0 ≤ j ≤ n, α j ≤ α max j (ω).

Figure 25 :

 25 Figure 25: Iteration of the two horizontal segments. To avoid clutter, we denote î = î( f j-1 (ω)).

  figure 25). Let 0 ≤ j ≤ n -1. According to Proposition 7.3, f ′ (z j ) -f ′ (z ′ j ) ≤ 128 2 k 3 2j+1 k 3 2j+2 z j -z ′ j

1 k 1 -For G a subset of S s 1 ĥ

 111 (ω)k 2 (ω) >> h / /,j ). Let us denote by v the intersection point of D with γω,1 . Letv j = Φj (v). Remark that (v j ) c ∈ (x j ) c , (x j ) c and f j (ω) ∈ x ′ j , x′ j . According to the above estimate, f ′ (z j ) -f ′ j -(v j ) c | + (v j ) c -f j (ω) + f j (ω) -z j -xj ∥ / / + (v j ) c -f j (ω) +x forward invariance of the tangential cone field and the expansion estimate (9.4),∥x j -xj ∥ / / ≤ K j+1-n c ∥x n-1 -xn-1 ∥ / / ≤ K j+1-n c e C13 m n-1 (ω) (11.27)One has an analogous estimate for x ′ j -x′ j . Putting together (11.27) and (11.5) (with a = max 1≤l≤2n k l )v belongs to the graph γω,1 , (9.25) implies that(v j ) c -f j (ω) ≤ δ ⊥,2 eUsing (11.18b), it follows from the above estimate that128 j ) c -f j (ω) ≤ 1 4 e -j(11.29) Plugging (11.28) and (11.29) into (11.26), we getf ′ (z j ) -f ′ Since f is expansive (see Proposition 7.2), it follows that f ′ (z ′ j ) ≥ 1 and f ′ (z j ) f ′ (z ′ j )As in the proof of claim 1, we get as a consequence of the above estimate that ln RThis concludes the proof of claim 2.Using claim 1 and claim 2 together with (11.19), we get that Φn (D)f n (π (D)) |π (D)| |D| ≤ e C 1 +C 2Hence, Proposition 11.9 holds true with ∆ ∶= e C 1 +C 2 . , define the "maximal gap" between G and the Mixmaster attractor by dist ⊥ (G, A) def = sup x∈G dist ∞ (x, A) = sup x∈G max x u , x s 2

Lemma 11. 13 .

 13 There exists a constant 0 < δ ⊥,3 ≤ δ ⊥,2 such that the following property holds for ω ∈ Ẽ and y ∈ 0, δ ⊥,32 . Let D ⊂ L y be an horizontal segment centered above ω.If |D| ≤ δ / /then there exists an integer n (D) ≥ 0 such that the property (H ω,D,n(D) ) holds true and such thatdist ⊥ Φn(D) (D) , A ≤ Φn(D) (D) (11.30)Proof. We begin with the definition of the size δ ⊥,3 . For ω = [1; k 1 , k 2 , . . . ] ∈ Ẽ and j ≥ 0, recall that h⊥,j = eUsing the uniform estimates (11.1) for points of Ẽ, we get that there exists n 1 ≥ n 0 (depending only on n 0 ) such that for every ω = [1; k 1 , k 2 , . . . ] ∈ Ẽ and every j ≥ n 1 , we have the following estimates h⊥,j ≤ e-C13 s 4 ( f j (ω)) h⊥,j ≤ e -C13 m j (ω)2Since the coefficients k 1 (ω), . . . , k 2n 1 (ω) of any point ω ∈ Ẽ admit a uniform upper bound depending only on n 0 , n 1 and l 0 (see (11.1)), one can find a constant 0 < δ ⊥,3 ≤ min δ / / , δ ⊥,2 such that for every ω = [1; k 1 , k 2 , . . . ] ∈ Ẽ and every j ≥ 0, we haveδ ⊥,3 h⊥,j ≤ e -C13 s 4 ( f j (ω)) (11.31a) δ ⊥,3 h⊥,j ≤ e -C13 m j (ω) 2 (11.31b) Fix ω = [1; k 1 , k 2 , . . . ] ∈ Ẽ, y ∈ 0, δ ⊥,32 and D ⊂ L y an horizontal segment centered above ω.Assume that |D| ≤ δ / /Recall thatI j (ω) = f j (ω) -e -C13 m j (ω) , f j (ω) + e -C13 m j (ω) h j (ω) = ĥ, e -C13 s 4 ( f j (ω)) , e -C13 m j (ω) Define n (D) def = max n ∈ N | ∀0 ≤ j ≤ n -1, Φj (D) ≤ e -C13 m j (ω) 2 N (D) def = max n ∈ N | ∀0 ≤ j ≤ n -1, Φj (D) ⊂ S s f j (ω),h j (ω) N (π (D)) def = max n ∈ N | ∀0 ≤ j ≤ n -1, f j (π (D)) ⊂ I j (ω)Sayingthat (H ω,D,n(D) ) holds true amounts to saying that N (D) ≥ n(D) and N (π (D)) ≥ n(D). According to (8.8), we get by induction on j that for every 0 ≤ j ≤ N (D), dist ⊥ Φj (D) , A ≤ δ ⊥,3 e Claim 1. N (D) ≥ n(D).

-

  C13 m l+1 (ω)

π

  Figure 26: π ΦN(π(D)) (D) ⊂ J with |J| ≤ ΦN(π(D)) (D) + δ ⊥,3 h⊥,N(π(D)) . To avoid clutter, we denote î = î( f N (π(D))-1 (ω)).

  As a consequence, f N (π(D)) (π (D)) ≤ e -C13 m N (π(D)) (ω) Moreover, f N (π(D)) (ω) ∈ f N (π(D)) (π (D)), hence f N (π(D)) (π (D)) ⊂ I N (π(D)) (ω) This contradicts the maximality of N (π (D)) and this concludes the proof of claim 2. It follows from claim 1 and claim 2 that (H ω,D,n(D) ) holds true. Using (11.32) with j = n (D) ≤ N (D) (see claim 1), we get that dist ⊥ Φn(D) (D) , A ≤ δ ⊥,3 h⊥,n(D) By definition of n(D), we have Φn(D) (D) > e -C13 m n(D) (ω) 2 (11.33)According to (11.31b) and (11.33), we haveδ ⊥,3 h⊥,n(D) ≤ Φn(D) (D) Hence, dist ⊥ Φn(D) (D) , A ≤ Φn(D) (D)This concludes the proof of Lemma 11.13.Proposition 11.14 (Absolute continuity of the projection map). Let ω ∈ Ẽ, y ∈ 0, δ ⊥,3 2 and D ⊂ L y be an horizontal segment centered above ω.If |D| ≤ δ / / then Leb 1 (π (D)) ≤ 2∆ Leb 1 (D) (11.34)Proof. According to Lemma 11.13, we can apply Proposition 11.9 to get Φn(D) (D)f n(D) (π (D)) f n(D) (π (D)) ≤ 2 Φn(D) (D) .Proof of claim 1. Recall that the local stable manifolds "foliation" F s 1 of the double era return map Φ is made of σ-Lipschitz graphs with σ ≤ 1 2 (see (8.5)). Hence, estimate (11.30) implies thatπ Φn(D) (D) ≤ Φn(D) (D) + dist ⊥ Φn(D) (D) , ANow recall that the integer n(D) was chosen so that dist ⊥ Φn(D) (D) , A ≤ Φn(D) (D) (see Lemma 11.13) so we getπ Φn(D) (D) ≤ 2 Φn(D) (D)The conjugacy relation π• Φ = f • π implies that π Φn(D) (D) = f n(D) (π (D)). This concludes the proof of claim 1. Claim 1 together with (11.35) gives |π (D)| ≤ 2∆ |D| Remark that Leb 1 (π (D)) ≤ |π (D)| and Leb 1 (D) = |D|. Hence Leb 1 (π (D)) ≤ 2∆ Leb 1 (D) which is the desired estimate. This concludes the proof of Proposition 11.14. Proof of Lemma 11.6. Let y ∈ 0, δ ⊥,3 2 and G ⊂ L y ∩ W s 1 Ẽ, Φ . Assume that Leb 1 (G) = 0. To show that Leb 1 (π(G)) = 0, cover G by a countable union of small horizontal segments and use the estimate (11.34).Lemma A.1. For Lebesgue almost every x ∈ Ω, there exists n 0 ∈ N such that for every n ≥ n 0 ,

Proposition A. 2 .

 2 Let ϕ ∶ N * → R * + . Either the set

  [k 1 , k 2 , . . . ], τ ([k 1 , k 2 , . . . ]) = [k 2 , k 3 , . . . ]In other words, τ is conjugated to the left shift on the space of sequences (k n ) n≥1 of integers larger than 1.Let us denote by γ G the Gauss measure, defined byγ G (A) = 1 ln 2 A 1 x + 1 dλ(x) for every Borel set A of [0, 1] (A.3)where λ denotes the Lebesgue measure. One can remark that the Gauss measure γ G is equivalent to the Lebesgue measure λ on[0, 1]. The fundamental fact is that γ G is τ -invariant, i.e. γ G τ -1 (A) = γ G (A) for every Borel set A of [0, 1]. For any map f ∶ [0, 1] → C, let var f def = sup n-1 i=1 |f (t i+1 ) -f (t i )|

  1]) satisfying, for every f ∈ L 1 λ ([0, 1]) and for every g ∈ L ∞ λ ([0, 1]), 1 0 (g • τ ) ⋅ f dγ G = 1 0 g ⋅ U f dγ GProposition A.5 (Spectral gap for the Perron-Frobenius operator). The Perron-Frobenius operator has a spectral gap: there exists 0 < α < 1 and C > 0 such that, for every f ∈ BEV([0, 1]),

1

  1]) so according to the Proposition A.5,γ G Y n, ≤ γ G X n, X n, • τ jc (x) dγ G (x) + O n→∞ α c 13We would like to thank Sébastien Gouëzel for explaining to us how to use the Perron-Frobenius operator here.

  A.1. Inequalities (A.1) and (A.2) are straightforward consequences of corollary A.3 and Proposition A.4 respectively, with = 10 -2 .

  theorem B.4. Let p be a point of the Kasner circle and let H be a coherent heteroclinic chain starting at p. If ω(p) verifies the moderate growth condition (MG), then the union of all the type VIII or IX orbits shadowing the heteroclinic chain H contains a 3-dimensional ball D(p, H ) Lipschitz embedded in the phase space B + . Moreover, for any E ⊂ K (M G) of positive 1-dimensional Lebesgue measure, the union of all the balls D(p, H ) for p ∈ E and H a coherent heteroclinic chain starting at p has positive 4-dimensional Lebesgue measure.

Table 1 :

 1 Table 1 summarizes the preceding description. Stratification of the phase space.

	Bianchi type	Name of the stratum	Dimension of the stratum	Signs of N 1 , N 2 , N 3 modulo permutation of the indices	Corresponding Lie algebra up to isomorphism
	I	K or B I	1	0, 0, 0	R 3
	II	B II	2	+, 0, 0 or -, 0, 0	Heisenberg's algebra
	VI 0 VII 0	B VI 0 B VII 0	3	+, -, 0 +, +, 0 or -, -, 0	isom(Min 2 ) isom(R 2 )
	VIII IX	B VIII B IX	4	+, +, -or -, -, + +, +, + or -, -, -	sl(2, R) su(2)

  the orbit O will travel side to side with the type II orbit O F (p)→F 2 (p) until entering a small neighbourhood B 2 of the point F

2 (p). It is impossible to control precisely the distance between O and O F (p)→F 2 (p) during this travel: we face the global behaviour of a non-linear vector field. But in any case, the travel from B 1 to B 2 will take a finite time T , and therefore the distance will grow at most linearly, the dilatation factor λ being the upper bound of the derivative of the time T map of the flow. As a consequence, the orbit O should enter the neighbourhood B 2 roughly at distance d 2 ∶= λd

µ s1 /µ u 1 of the orbit O F (p)→F 2 (p) ,

which is much smaller than d 1 (if d 1 is small enough). See figure 4. Iterating the argument, the orbit O should go through the small neighbourhood B 2 of F 2 (p), follow the type II orbit O F 2 (p)→F 3 (p) , and enter in a neighbourhood of F 3 (p) at a distance d 3 ≪ d 2 , go through the small neighbourhood of F 3 (p), follow the type II orbit O F 3 (p)→F 4 (p) , and enter in a neighbourhood of F 4 (p) at a distance d 4 ≪ d 3 , . . . So we can hope to keep some control of the behaviour of O forever and prove that it shadows the heteroclinic chain O p→F (p) , O F (p)→F 2 (p) , . . . .

Induced coordinate functions. The

  Q}be the (induced) Kasner interval. Any point p ∈ K 0 possesses a fiber containing six points, one in each sixth of the Kasner circleK (ijk) . Observe that B 0 is an open neighbourhood of K 0 . coordinates N 1 , N 2 , N 3 , Σ 1 , Σ 2 , Σ 3 on B induce a set of smooth coordinates functions N u , N s 1 , N s 2 , Σ u , Σ s 1 , Σ s 2on B 0 (here smooth stands for C ∞

	Distance on the quotient phase space. The Euclidean distance d E on R	6 induces a distance d B
	on B:	d B (π(p), π(q))	def = inf σ∈S 3	d E (p, σ.q)	(2.11)
	which we will always use to measure the radius of balls in B.

Hsu vector field. The

  Wainwright-Hsu vector field X on B is equivariant under the action of S 3 and therefore induces a vector field X on B. Let p ∈ K 0 . According to the discussion about the eigenvalues of the Wainwright-Hsu vector field X (see section 2.3), DX (p) is diagonalizable. More precisely, ∂ N u , ∂ N s1 , ∂ N s2 and the direction tangent to K at p are four eigendirections of DX (p) associated with the eigenvalues

	µ u (p)	def = -(2 + 2Σ u (p)), -µ s 1 (p)	def = -(2 + 2Σ s 1 (p)), -µ s 2 (p)	def = -(2 + 2Σ s 2 (p)) and 0
	Beware of the fact that µ s 1 and µ s 2 denote the modulus of the stable eigenvalues. As a consequence
	of Proposition 2.1, we have
		0 < µ

u < µ s 1 < µ s 2 in K 0 Induced Kasner map. The Kasner map is equivariant under the action of S 3 and therefore

	induces
	a map
	F ∶ K → K
	called the (induced) Kasner map. We have in particular

  small enough, the points P

	u ω,h , P ω,h and P s 1	s 2 ω,h are in the range U ξ of the local coordinate system
	ξ. In this case, we denote P ω,h ∶= ξ u	-1 (P ω,h ), P u ω,h ∶= ξ s 1	-1 (P ω,h ) and P s 1 ω,h ∶= ξ s 2	-1 (P ω,h ). s 2
	We now define three local sections that intersect respectively the type II orbits O ω , O u ω and O s 1	s 2

  We omit the x s 2 -direction since we cannot draw in four dimensions. The Mixmaster attractor A is represented in green. The Kasner interval ξ(K 0 ) is represented in red. The type II orbits arriving and starting at P ω are represented in blue. are included in the range U ξ of the local coordinate system ξ as soon as the parameters h, h ⊥ and h / / are chosen small enough. More precisely, there exists C > 0 and n ∈ N such that for every ω ∈ ]1, +∞[ and every h = h, h ⊥ , h / / , if max h, h ⊥ , h / / ≤ are included in U ξ . This is a direct consequence of Proposition 3.2 on the local coordinate system ξ. Remark 4.13. Let ω ∈ ]1, +∞[ and h = h, h ⊥ , h / / where h, h ⊥ and h / / are positive. Let x ∈ S

	Remark 4.12. The local sections S ω,h , S u ω,h and S s 1 ω,h (Cω s 2 n ) -1 , then the local sections S u ω,h , S s 1 s 2 ω,h and S ω,h s 1

12: Sections S s 1 ω,h and S u ω,h .

  and S h are included in U ξ and so that the projection Proj A is well defined on the section S

	Definition 4.15 (Maps	ΦA ω,h and	ΦA	s ω,h ω,h . We s
	define the map	ΦA ω,h ∶ S ω,h ∩ S h → S h by the formula s
					ΦA ω,h	def = Φω,h • Proj A
	Analogously, we define the map	ΦA ω,h ∶ S ω,h ∩ S h → S h by the formula s
					ω,h ΦA	def

ω,h ). Let ω ∈ ]1, 2[ and h = (h, h ⊥ , h / / ) so that the sections S

  etc. Hence, for any choice of the parameters h and h ′ , the point Φ ω,h,h ′ (x) is the first intersection point of the type II orbit O P xc →P f (xc ) with the section S

	s f (ω),h ′ . Using Remark 2.9 and the formulas (4.1b) and (4.1c), one gets the following explicit expression:

  h u is well defined. Moreover, for every x, x ∈ S s ω,h s , we have the following estimates where we denote Υ ∶= Υ ω,h s ,h u and Υ

A ∶= Υ A ω,h s ,h u :

(Distance to the Mixmaster attractor)

  and τ ↦ x(τ ) be an orbit of the local vector field X ω whose initial condition x in ∶= x(0) belongs to the section S According to corollary 5.18, x is well defined on

										in u ≠ 0 (hence x	in u > 0).
	Let h	u = (h, h, 2h). Using (2.14a), (2.14b), (5.6) and the fact that x	in u ≤ 1 3 h, it is straightforward to
	check that 0 ≤ τ 0, τ out (x, ω, h) . According to (5.27a), (5.27b) and (5.27c), we have
		∥x(τ	µ u (ω)τ	*	x	in u + e	-αµ s1 (ω)τ	*	h + 2h ⊥	(5.30)
	Using (2.14a), (2.14b), we get that
			e	µ u (ω)τ	*	= e	µu (ω)

s ω,h s (where h s = (h, h ⊥ , h ⊥ )) and satisfies x * ≤ τ out where τ out = τ out (x, ω, h). * ) -P ω ∥ ∞ ≤ e

  ). With the same notation as above, if x intersects S s ω,h s then we denote by τ

	Definition 5.22 (Time of intersection with S ω,h s s

h, 2h) and for any orbit τ ↦ x(τ ) of the local vector field X ω , if x intersects S s ω,h s at least once, then it intersects S s ω,h s and S u ω,h u exactly once. in (x, ω, h s ) the unique time τ ∈ R such that x(τ ) ∈ S s ω,h s . Definition 5.23 (Pair of synchronized orbits)

  64)Let C 10 = 2C 9 and n 10 = n 9 . For every C ≥ C 10 , every n ≥ n 10 and every (ω, α, h, x, x) ∈ E

					C,n , we
	have	C 9 ω	n 9 h ≤	1 6	(5.65)
	Let C ≥ C 10 , n ≥ n 10 and (ω, α, h, x, x) ∈ E C,n . Assume that T (x, x) < τ	out . Using (5.64) and (5.65),
	we have,

  For every C ≥ C 0 , every n ≥ n 0 and every(ω, h, h ⊥ , t ↦ x(t)) ∈ E C,n , we denote h u ∶= (h, h, 2h) and Υ ∶= Υ ω,h s ,h u . For every ω ∈ ]1, +∞[, let

	is an orbit of the local vector field X ω whose initial condition x	d(ω)) and t ↦ x(t) in ∶= x(0) belongs to the section S s ω,h s
	where h	s = (h, h ⊥ , min(h, d(ω))) and such that x u ≠ 0. α(ω) in

  h s is well defined and takes its values in S u we have the following estimates, where Ψ ∶= Ψ ω,h u ,h s and Ψ

	s i(ω)

f (ω),h s . Moreover, for every y, ỹ ∈ S u ω,h A ∶= Ψ A ω,h u ,h s :

(Control of the distance to the Mixmaster attractor)

  Lower and upper bounds for t 2 -t 1 .

	By an analogous reasoning on t 2 , τ ω,h s (P ω,h u ) , we get u
	τ ω,h s (P ω,h u ) -t 2 ≤ ω ln u	1 h s ≤ ω ln	1 h	(6.15)
	ω ln	1 h		(6.14)

  n . According to Proposition 6.1, Ψ ω,h u ,h f (ω) is well

	defined on S ω,h u and takes its values in S u	s i(ω)

  C5 ,ñ 5 ⊂ U ξ . It follows from the evolution equations (5.5) that t ↦ p u (t) is increasing on t

				24)
	According to (8.24) together with the estimate (3.5b), for every t ∈ t	u 0 , t out 0	, we have ξ • p(t) ∈
	B ω, u 0 , t out 0	.
	Since p is a type II orbit, p(t) = (p u (t), 0, 0, ω) on t	u 0 , t out 0

  • Φω,h ′ ω holds true. Applying twice Proposition 8.4, we get that Φω,h ′ ω takes its values in S

	sî (ω)
	ĥ

  xs 1 , xs 2 , x c ) and x = (x u , xs 1 , xs 2 , xc ). Define Φc ∶ t ↦ Φω (x u , xs 1 , xs 2 , t) c

	and define analogously
	ΦA c ∶ t ↦	ΦA ω (x u , xs 1 , xs 2 , t)

c

  52)Proof of claim 3. Recall that 0 < h ⊥ ≤ e -Cs 4 (ω) = e

	Claim 3.
	lim C→+∞	ω,h ⊥ sup	ε(ω, h ⊥ ) = 0	(8.53)
			-C k	4 1 +k 2 +k 4 3 +k 4 4 . Using (7.2), we get 4

  Genericity of the admissible points). Any point ω ∈ ]1, 2[ \ Q satisfying the moderate growth condition (MG) is admissible. In particular, Ω graph is a Lebesgue full measure subset of

	Proposition 9.16 (
		9.19)
	Proposition 9.15 is a straightforward consequence of formula (9.19).
	Recall that ω ∈ ]1, 2[ \ Q is said to satisfy the moderate growth condition if
	n
	k n+4 (ω) 4 = o n→+∞	k i (ω)
	i=1

5

(MG) Also, recall that the moderate growth condition is Lebesgue generic (see Lemma A.1).

  Theorem 9.20 (Local stable manifolds of the double era return map). For every ω ∈ Ω graph and every i ∈ {1, 2}, the local stable set ofP Let ω = [1; k 1 , k 2 , . . . ] ∈ Ω graph , i ∈ {1, 2} and x ∈ γω,i = Graph

	Proof. s i	ζω,i . By definition,
	γ =	Φ * γ. Hence, for every n ≥ 0 and every
		s i ω, ĥ of size ĥ⊥ (ω) contains a Lipschitz submanifold of dimension 2.
	More precisely,
		γω,i ⊂ W	s ĥ⊥ (ω) P ω, s i ĥ, Φ
		s i ω, ĥ	∞	≤ x -P ω, ĥ s i	⊥	e	-C8 ∑ 2n i=1 k i (ω) 5	(9.25)
				123

Moreover, the convergence is exponential in the graph: for every x ∈ γω,i and every n ≥ 0, Φn (x) -Φn P

  According to Proposition 9.16, for h ⊥ small enough, Ω graph (h ⊥ ) has positive Lebesgue measure. In the following proposition, Lip h ⊥ denotes the set of all real valued σ-Lipschitz maps defined on [0, h ⊥ ] 2 . For everyh ⊥ > 0, the map ζ1,h ⊥ ∶ ω ∈ Ω graph (h ⊥ ) ↦ ζω,1 |[0,h ⊥ ] 2 ∈ Lip h ⊥ is continous for the sup-norm topology on Lip h ⊥ . Proof. Let > 0, h ⊥ > 0 and ω = [1; k 1 , k 2 , . . . ] ∈ Ω graph (h ⊥ ).We are going to show that there exists η > 0 (depending only on and ω) such that for all ω ∈ Ω graph (h ⊥ ), if |ω -ω| ≤ η, then ζω,1 -ζω,1 ∞ ≤ (where the sup-norm is to be understood over [0, h ⊥ ] 2 ). Let γ = Graph (ζ) ∈ Γ be the "constant" graph family, defined by ζ z,i ≡ z for all z ∈ Ω graph and all i ∈ {1, 2}.

	Proposition 9.21. Since	Φ * is a
	contraction mapping (see Lemma 9.19), there exists an integer n such that
		d graph γ,	Φ * n	γ ≤
	From now on, we fix such a n. Denote	Φ * n

  Proposition 11.2 using Lemma 11.6. Assume that Lemma 11.6 holds true. Take 0 < h ⊥ ≤ δ ⊥,0 as in the statement of Lemma 11.6. Assume that Proof of claim 1. The inclusion π y W s 1 Ẽ, Φ ⊂ Ẽ is obvious by definition of π. Let ω ∈ Ẽ. Since h ⊥ ≤ δ ⊥,0 , the size ĥ⊥ (ω) of the graph γω,1 is larger than h ⊥ (see (11.2)). Hence, γω,1 intersects the horizontal line L y exactly one time, say at x. By definition, we have π y (x) = ω. This concludes the proof of claim 1.

	Using Fubini's theorem, we get that for Lebesgue almost all y ∈ [0, h ⊥ ] 2 ,
		Leb 1 L y ∩ W	s 1 Ẽ, Φ = 0	(11.4)
	Fix such a transversal L y .
	Claim 1. π y W	s 1 Ẽ, Φ = Ẽ.
	According to (11.3) and (11.4), we have
		Leb 1 π y W	s 1 Ẽ, Φ	= 0
	Using claim 1, we get that	Leb 1 Ẽ = 0
	This is the desired contradiction. Hence,
		Leb 3 W	s 1 E, Φ > 0
	and Proposition 11.2 holds true.
		Leb 3 W	s 1 E, Φ = 0
	This implies that
		Leb 3 W	s 1 Ẽ, Φ = 0

  Let ω = [1; k 1 , k 2 , . . . ] ∈ Ẽ, y, x, x as in the statement of Proposition 11.10. Assume that x ≠ x. Let n ≥ 0 such that for every 0 ≤ j ≤ n -1, we have

		1 26k 2j+1 + δ ⊥,1 h⊥,j	1 26k 2j+2 ≤	1 4	(11.7a)
	and	δ ⊥,1 h⊥,j	k 2j+1 100 +	k 2j+2 100	≤	1 2	α j+1 (ω) max	(11.7b)
	Φj (x) ∈ S	s f j (ω),h f j (ω)	and		Φj (x) ∈ S	s f j (ω),h f j (ω)

For 0 ≤ j ≤ n, let h ⊥,j = max Φj (x) -( ΦA ) j (x) ⊥ , Φj (x) -( ΦA ) j (x) ⊥ h / /,j = e -C8 s 2 ( f j (ω))

  Now we apply (8.10) to the map Φ restricted to the section S s f j (ω), ĥ,h ⊥,j ,h / /,j

	It follows from (11.12) and (11.13) that
	Φj+1 (x) -	Φj+1 (x)		/ / ≥	1 2	Φj (x) -	Φj (x)
											:
	Φ Φj (x) -Φ Φj (x)		⊥ ≤ h ⊥,j k 2j+1 100 +	k 2j+2 100	Φj (x) -	Φj (x)	∞	(11.15)
	Plugging (11.8) into (11.7b), we get
			h ⊥,j	k 2j+1 100 +	k 2j+2 100	≤	1 2	α j+1 (ω) max	(11.16)
	Plugging (11.16) into (11.15), we get
	Φj+1 (x) -	Φj+1 (x)	⊥ ≤	1 2	α	max j+1 (ω)	Φj (x) -	Φj (x)	∞	(11.17)
	According to (11.14) and (11.17), we have
											/ / ≤
				1							1
			h ⊥,j 26k 2j+1	+ h ⊥,j 26k 2j+2	+ 16	2 k	2 2j+1 k	2 2j+2 α j	Φj (x) -	Φj (x)	∞	(11.9)
	Plugging (11.8) into (11.7a), we get
				h ⊥,j 26k 2j+1 1	+ h ⊥,j 26k 2j+2 1	≤	1 4	(11.10)
	Using the hypothesis α j ≤ α j max	(ω), we get
				16	2 k	2 2j+1 k	2 2j+2 α j ≤	1 4	(11.11)
	Plugging (11.10) and (11.11) into (11.9), we get
											/ / ≤	1 2	Φj (x) -	Φj (x)

Φ Φj (x) -Φ Φj (x) -ΦA Φj (x) -ΦA Φj (x) ∞ (11.12) Moreover, the Kasner map being expansive (see Proposition 7.2), we have ΦA Φj (x) -ΦA Φj (x) / / ≥ Φj (x) -Φj (x) / / = Φj (x) -Φj (x) ∞ (11.13) ∞ (11.14)

  ∥x j -xj ∥ ∞ (11.23) where we used ∥x j -xj ∥ / / = ∥x j -xj ∥ ∞ . Plugging (11.22) and (11.23) into (11.21), we getΦ(x j ) -Φ(x j ) -ΦA (x j ) -ΦA (x j )Recall that the map ΦA is essentially the double era Kasner map (see (4.7)). Moreover, the Kasner map is expansive (see Proposition 7.2), henceΦA (x j ) -ΦA (x j ) / / ≥ ∥x j -xj ∥ / / = ∥x j -xj ∥ ∞ (11.25)It follows from (11.24) and (11.25) that Φ(x j ) -Φ(x j )

	Plugging (11.20) into (11.18a), we get
						h ⊥,j 26k 2j+1 1	+ h ⊥,j 26k 2j+2 1	≤	1 4	e	-j	(11.22)
	According to (11.6),
	16	2 k	2 2j+1 k 2j+2 ∥x j -xj ∥ 2 ⊥ ≤ 16	2 k	2 2j+1 k	2 2j+2 α j max	(ω) ∥x j -xj ∥ / / ≤ -j / / ≤ 1 4 e 1 2 e -j ∥x j -xj ∥ ∞	(11.24)
									-1 ≤	1 2	e	-j
									/ /
	As a consequence of the above estimate, we get
			ln R 1 =	n-1 j=0	ln		Φ(x / /	≤	n-1 j=0	1 2	e -j ≤ C 1
									/ / ≤
						1			1
				h ⊥,j 26k 2j+1	+ h ⊥,j 26k 2j+2	∥x j -xj ∥ ∞ + 16 2 k 2j+1 k 2	2 2j+2 ∥x j -xj ∥	⊥	(11.21)

/ / ΦA (x j ) -ΦA (x j ) j ) -Φ(x j ) / / ΦA (x j ) -ΦA (x j )

It is denoted by -τ in[START_REF] Heinzle | Mixmaster: Fact and Belief[END_REF] (they choose to respect the "physical" time-orientation.)

Here, Bianchi spacetimes are considered up to isomorphism, metric rescaling, time orientation reversal and time

This formulation is classic and is based on the work of Beliinski, Khalatnikov and Lifschitz on one hand and Misner on the other hand.

A heteroclinic connexion is an orbit "joining two different points". More precisely it is an orbit t ↦ O(t) such that there exists two distinct points p and q verifying lim t→+∞ O(t) = q and lim t→-∞ O(t) = p.

The word smooth will always stand for C ∞ in this work.

using the fact that k i+1 (ω) = k i ( f (ω)) for all i ≥ 1.

Gronwall's lemma implies that there exist C 2 ≥ C 1 , n 2 ≥ n 1 such that for every C ≥ C 2 , every n ≥ n 2 , every (ω, h u , h s , η) ∈ E C,n , every q ∈ S u ω,h u and every t ∈ [0, min 4τ ω,h s (P u ω,h u ), τ C (q) ], we have d B X t (q), X t (P u ω,h u ) ≤ e 4Aτ ω,h s (P u ω,h u ) d B q, P u ω,h u (6.33)

By definition of the size of the section S u ω,h u in the direction transverse to the Mixmaster attractor, the distance between q and the Mixmaster attractor in local coordinates is less than ηh Cω m(ω). Hence, according to the estimate (3.5b) on the derivative of the local coordinate system and the mean value theorem,

Recall from (6.6) that

It follows from (6.33), (6.34), (6.35) and (6.6) that for every

Using (6.36) with η = h (C-C 3 )ω , we get that there exists C4 ≥ C 3 such that for every C ≥ C4 , every

This inequality implies that the point X t (q) remains in the interior of the compact set C, and therefore min 4τ ω,h s (P

So we can set t = τ ω,h s (P u ω,h u ) in (6.37). By doing this, we obtain

Using point 6 of Lemma 6.8, it follows from the above inequality that for every C ≥ C4 , every n ≥ ñ4 , every (ω, h u , h s , 1) ∈ E C,n and every q ∈ S u ω,h u we have

A Continued fractions

In this appendix, we gather the results about continued fractions that are used in the memoir. The main result is Lemma 1.8. We also prove a result on the expansivity of the Gauss transformation.

We first need to introduce some notations. Set Ω = [0, 1] \ Q. For every x ∈ Ω, there exists a unique sequence

We use the notation

Lemma 1.8 is a straightforward consequence of the following lemma.

Kasner map

Let p be a point of the Kasner circle which is not a Taub point. When we restrict ourselves to B + , there is exactly one type II orbit starting at p and this orbit converges to a point denoted by F (p) (the image of p by the Kasner map). This is indeed how we defined the Kasner map (see section 2.5). As stated above, in B, there are two (symmetrical) type II orbits starting at p. Since they are symmetrical, they both converge to the same point of the Kasner circle, that is, the point F (p). We will denote these two type II orbits by O

Coherent heteroclinic chains

Definition B.1 (Heteroclinic chains). Let p be a point of the Kasner circle (such that, for every k ≥ 0, F k (p) is not a Taub point). A heteroclinic chain (starting at p) is a concatenation of one type II orbit starting at p and arriving at F (p), then one type II orbit starting at F (p) and arriving at F 2 (p), etc. Formally, this is a sequence of the form

where n ∈ {±} corresponds to a choice of one of the two symmetrical type II orbits starting at F n (p).

As we will see, some heteroclinic chains cannot be shadowed by type VIII or type IX orbits. First, let us recall the definition of shadowing, generalized to the full phase space in a straightforward manner. Definition B.2 (Shadowing). Let t ↦ O(t) be a type VIII or IX orbit in B, p be a point of the Kasner circle (such that, for every k ≥ 0, F k (p) is not a Taub point) and H be a heteroclinic chain (B.1) starting at p. We say that O shadows H (or H attracts O) if there exists a strictly increasing sequence

2. The Hausdorff distance between the orbit interval {O(t) | t n < t < t n+1 } and the type II orbit O n F n (p)→F n+1 (p) tends to 0 when n → +∞. Recall that any type II orbit is contained in a subset of the phase space of the form

where {i, j, k} = {1, 2, 3}. Consider for example a heteroclinic chain made of an infinite number of type II orbits traveling in {N 1 > 0, N 2 = 0, N 3 = 0} and an infinite number of type II orbits traveling in

) be a type VIII or IX orbit. Recall that the signs of the variables N i are constant. Hence, it is obvious that O cannot shadow this heteroclinic chain, as it would violate the fact that the sign of N 1 is constant along O. This means that any heteroclinic chain "alternating" between two signs as in the example above has zero chance to attract some type VIII or IX orbits. This leads us to the definition of coherent heteroclinic chains. Recall that the Mixmaster attractor is the union of three ellipsoids and each of these ellipsoids is the union of two symmetrical hemiellipsoids (they correspond to opposite signs for one of the three variables N i ). In other words,

and analogously for the other hemiellipsoids.

Definition B.3 (Coherent heteroclinic chain).

A heteroclinic chain of type II orbits is coherent if it is included in the union of three hemiellipsoids (in three different directions) bounded by the Kasner K , that is, if it is included in a set of the form