Sébastien Biebler

Almost blenders and parablenders

A blender for a surface endomorphism is a hyperbolic basic set for which the union of the local unstable manifolds contains robustly an open set. Introduced by Bonatti and Díaz in the 90s, blenders turned out to have many powerful applications to differentiable dynamics. In particular, a generalization in terms of jets, called parablenders, allowed Berger to prove the existence of generic families displaying robustly infinitely many sinks. In this paper, we introduce analogous notions in a measurable point of view. We define an almost blender as a hyperbolic basic set for which a prevalent perturbation has a local unstable set having positive Lebesgue measure. Almost parablenders are defined similarly in terms of jets. We study families of endomorphisms of R 2 leaving invariant the continuation of a hyperbolic basic set. When some inequality involving the entropy and the maximal contraction along stable manifolds is satisfied, we obtain an almost blender or parablender. This answers partially a conjecture of Berger. The proof is based on thermodynamic formalism: following works of Mihailescu, Simon, Solomyak and Urbański, we study families of fiberwise unipotent skew-products and we give conditions under which these maps have limit sets of positive measure inside their fibers.

Introduction

Blenders and almost blenders

Fractal sets have played a central role in the development of differentiable dynamics.

Among several examples, a central notion is that of blender, cast by Bonatti and Díaz in the 90s. It was first introduced in the invertible setting in [START_REF] Bonatti | Persistent nonhyperbolic transitive diffeomorphisms[END_REF] to construct robustly transitive nonhyperbolic diffeomorphisms. A blender is a hyperbolic basic set on which the dynamics has a special behavior: its unstable set forms an impenetrable wall in the sense that it intersects any perturbation of a submanifold of dimension lower than the stable dimension. In the case of surface endomorphisms, this notion takes the following simpler form: Definition. A C r -blender for a C r -endomorphism F of a surface S is a hyperbolic basic set K s.t. an union of its local unstable manifolds has C r -robustly a non-empty interior: there exists a non-empty open set U ⊂ S included in an union of local unstable manifolds of the continuation K of K for any map F C r -close to F. This property turned out to have many other powerful applications: for example C 1density of stable ergodicity [ACW], robust homoclinic tangencies [START_REF] Bonatti | Abundance of C 1 -robust homoclinic tangencies[END_REF][START_REF] Biebler | Newhouse phenomenon for automorphisms of low degree in C 3[END_REF] and thus Newhouse phenomenon, the existence of generic families displaying robustly infinitely many sinks [START_REF] Berger | Generic family with robustly infinitely many sinks[END_REF], robust bifurcations in complex dynamics [START_REF] Dujardin | Non density of stability for holomorphic mappings on C 2[END_REF][START_REF] Taflin | Blenders near polynomial product maps of C 2[END_REF][START_REF] Biebler | Lattès maps and the interior of the bifurcation locus[END_REF], fast growth of the number of periodic points [START_REF] Berger | Generic family displaying robustly a fast growth of the number of periodic points[END_REF][START_REF] Asaoka | Fast growth of the number of periodic points arising from heterodimensional connections[END_REF] ... Thus the following question is of fundamental interest: when do blenders appear ?

In this direction, Berger proposed the following conjecture:

Conjecture A (Berger [START_REF] Berger | ERC Project: Emergence of wild differentiable dynamical systems[END_REF]). Let F be a C r -local diffeomorphism of a manifold M, for r ≥ 2. Let K be a hyperbolic basic set for F. Suppose that the topological entropy hF of F|K satisfies: ||DzF(u)|| , and E s the stable bundle of K. Then there exists a C r -neighborhood U of F and an infinite codimensional subset N ⊂ U such that for every F ∈ U \ N , the continuation K of K is a C r -blender.

Note that we cannot hope that K is itself systematically a C r -blender under the assumptions of Conjecture A. Here is an easy counterexample.

Counterexample: Let us consider the doubling map f : x → 2x mod 1 on the circle S := R/Z. The whole circle is a hyperbolic basic set of repulsive type and f is a C ∞ -local diffeomorphism. The topological entropy h f is equal to log 2 > 0. Let us pick λ < 1 close to 1 s.t. h f > | log λ|. The map F : (x, y) ∈ S × R → (f (x), λy) ∈ S × R leaves invariant the hyperbolic basic set of saddle type K := S × {0}. Moreover F is a C ∞ -local diffeomorphism and hF = h f > | log λ|. However the unstable set of K is included in S × {0} and thus has empty interior.

Conjecture A is also linked to a program proposed by Díaz [Di] on the thermodynamical study of blenders.

In this article, we give an answer to both questions, in a measurable point of view.

We are going to define a measurable variant of the notion of blender, called almost blender . This will be a hyperbolic basic set having its unstable set of positive Lebesgue measure instead of being with non-empty interior. Also, this property will be asked to be robust in a measurable way instead of topological. Let us precise this measurable robustness . We recall that this notion is tricky since there is no canonical measure on the space C r (M, M) of C r -endomorphisms of a manifold M. But we need an analogous of the finite-dimensional notion of Lebesgue almost every in an infinite-dimensional setting. Nevertheless, there are several notions of prevalence or typicity which generalize this concept. A panorama has been drawn by Hunt and Kaloshin in [HK], by Ott and Yorke in [OY] or by Ilyashenko and Li in [IL]. Here is one of these notions of prevalence, particularly adapted to our case:

Definition ([HK] P.53). We say that a set E in a Banach space B is finitedimensionally prevalent if there exists a continuous family (vq)q∈Q of vectors vq ∈ B, for q varying in a neighborhood Q of 0 in R m with m > 0 and v0 = 0, s.t. for every fixed v ∈ B, we have that v + vq ∈ E for Lebm a.e. q ∈ Q.

In other terms, we require that for some finite-dimensional family of perturbations, if we start at any point in B, then by adding a perturbation randomly chosen with respect to the Lebesgue measure, we are in E with probability 1. A similar notion, simply called prevalence , has been designed by Sauer, Ott and Casdagli (see Definition 3.5 in [OY], or also [SYC] and [HSY]), for completely metrizable topological vector spaces and with the additive condition that vq is a linear function of q.

In our results, we will have this additional linearity but since we do not need it, we will take inspiration from the above definition. See Remark 1 P.53 in [HK] for details.

We restrict ourselves in this article to the case where the manifold M is equal to R 2 and endowed with its usual Euclidean metric. The vector space C r (R 2 , R 2 ) of C rendomorphisms of R 2 is endowed with the topology given by the uniform C r -norm:

||F||Cr := sup 0≤i≤r,z∈R 2 ||D i z F||
when 0 ≤ r < ∞. The space of C r -bounded C r -endomorphisms endowed with this norm is a Banach space. Since we are interested in properties depending only on perturbations of a map on a compact set, we can restrict ourselves to C r -bounded C r -endomorphisms if necessary and so the above definition of prevalence from [HK] could fit to our setting. Similarly, for r = ∞, we endow C ∞ (R 2 , R 2 ) with the union of uniform C s -topologies on Kj among integers s and j, for an exhausting sequence of compact sets Kj of R 2 , which gives to C ∞ (R 2 , R 2 ) a complete metrizable topology.

However we cannot hope that a blender-like property, even in a weak sense, holds true densely in C r (R 2 , R 2 ), even less in a prevalent way. This is why we introduce the following immediate adaptation for perturbations of a map:

Definition. A property (P ) holds true for a prevalent C r -perturbation of a C rendomorphism F of R 2 if there exists a C r -neighborhood U of F and a continuous family (Σq)q∈Q of C r -endomorphisms Σq of R 2 , for q varying in a neighborhood Q of 0 in R m with m > 0 and Σ0 = 0, s.t. for every fixed G ∈ U, the map G + Σq satisfies (P ) for Lebm a.e. q ∈ Q.

In particular, property (P ) holds true for an arbitrary small perturbation of F.

Here is now a new concept, which formalizes a measurable variant of blenders.

Definition. A hyperbolic basic set K for a C r -endomorphism F of R 2 , with r ≥ 1, is an almost C r -blender if an union of local unstable manifolds of the continuation K of K has positive measure for a prevalent C r -perturbation F of F:

Leb2(W u loc ( K)) > 0 .
A hyperbolic basic set for F is a compact, F-invariant, hyperbolic, transitive set K s.t. periodic points of F|K are dense in K (basic notions about hyperbolic sets for endomorphisms are in the Appendix).

Our first result gives an answer to Conjecture A, in a measurable point of view:

Theorem A. Let F be a C r -local diffeomorphism of R 2 , with 2 ≤ r ≤ ∞. Let K be a hyperbolic basic set for F. Suppose that the topological entropy hF of F|K satisfies hF > | log m(DF)|. Then K is an almost C r -blender.

Conjecture A seems a very difficult problem in its full generality. A related question is the following long-standing open problem:

Conjecture B ([Ho] Conj. 3.1). Let µ be the self similar measure associated to some IFS Ψ = (ψ a )a∈A formed by a finite number of contracting similarities ψ a on R. Suppose that there are no exact overlaps and that the similarity dimension of the IFS is strictly larger than 1. Then µ is absolutely continuous with respect to Leb1.

One can refer to the survey of Hochman [Ho] for more details. Let us also point out that the creation of blenders had also been investigated by Moreira and Silva [MS].

Parablenders and almost parablenders

Berger introduced in [START_REF] Berger | Generic family with robustly infinitely many sinks[END_REF] a variant of blenders, defined for families of maps this time, where not only the unstable set of a hyperbolic set, but also the set of jets of points inside unstable manifolds contains an open set. Such sets were named parablenders ( para standing for parameter ). Parablenders were introduced to prove the existence of generic families displaying robustly infinitely many sinks, which gave a counter-example to a conjecture of Pugh and Shub from the 90s [PS].

Definition ([Be1, BCP]

). A C r -parablender at p0 ∈ P for a C r -family (Fp)p∈P of endomorphisms of a surface S, r ≥ 1, parametrized by a parameter p in an open subset P ⊂ R d , is a continuation (Kp)p∈P of hyperbolic basic sets Kp for Fp s.t.:

• for every (γp)p∈P in a non-empty open set of C r -families of points γp ∈ S,

• for every C r -family ( Fp)p∈P of endomorphisms C r -close to (Fp)p∈P , there exists a C r -family (ζp)p∈P of points ζp ∈ S s.t.:

• there is a local unstable manifold of Kp 0 whose continuation for Fp contains ζp, for any p ∈ P,

• the r-jets of ζp and γp at p0 are equal:

(ζp, ∂pζp, . . . , ∂ r p ζp) |p=p 0 = (γp, ∂pγp, . . . , ∂ r p γp) |p=p 0 .
In particular, Kp 0 is a C r -blender for Fp 0 if (Kp)p∈P is a C r -parablender for (Fp)p∈P at p0. In a subsequent work [START_REF] Berger | Generic family displaying robustly a fast growth of the number of periodic points[END_REF], Berger used parablenders to prove the existence of generic families of maps displaying robustly a fast growth of the number of periodic points, solving a problem of Arnold [Ar] in the finitely differentiable case.

From now on, we work with C r -families (Fp)p of endomorphisms Fp of R 2 , with 2 ≤ r ≤ ∞, parametrized by a parameter p varying in P := (-1, 1) d with 1 ≤ d < ∞.

In fact, we will need to work with families which admit some extension on a larger parameter space. We therefore fix an open set P ⊂ R d s.t. P P . We then define a C r -family (Fp)p of endomorphisms Fp of R 2 as an element of C r (P × R 2 , R 2 ). We endow this space with the uniform C r -topology when 0 ≤ r < ∞, and with the union of uniform C s -topologies on Kj among integers s and j, for an exhausting sequence of compact sets Kj of P × R 2 when r = ∞. Note that for simplicity we denote often in the following (Fp)p∈P this family since we are interested mainly in the dynamics when p ∈ P but keep in mind that it admits such an extension. Let (Kp)p∈P be the (hyperbolic) continuation of a hyperbolic basic set (extending to P ). Let E s p and E u p be the one-dimensional stable and unstable bundles of Kp.

Our main result deals with jets of points inside local unstable manifolds of Kp. Let (Mp)p be a C r -curve of points Mp in the continuation of one local unstable manifold of Kp. For any integer s ≤ r, one can consider the s-jet of Mp at any p0 ∈ P:

J s p 0 Mp := (Mp, ∂pMp, . . . , ∂ s p Mp) |p=p 0 .
An interesting set is then the set J s p 0 W u loc (Kp) of all the s-jets among such curves (Mp)p. When this set has robustly a non-empty interior, (Kp)p∈P is a C s -parablender at p0. Let δ d,s be the dimension of the set of jets in d variables of order s in one dimension, which is that of the space Rs[X1, • • • , X d ] of polynomials in d variables of degree at most s. In particular, notice that the space of jets of order s of maps from P to R 2 is of dimension 2δ d,s .

Here is the counterpart for families of the definition of a prevalent C r -perturbation: Definition. A property (P ) holds true for a prevalent C r -perturbation of a C rfamily (Fp)p∈P of endomorphisms of R 2 if there exists a C r -neighborhood U of (Fp)p∈P and a continuous family (Σq)q∈Q of C r -families Σq of endomorphisms of R 2 , for q in a neighborhood Q of 0 in R m with m > 0 and Σ0 = (0)p∈P , s.t. for every fixed family (Gp)p ∈ U , the family (Gp)p + Σq satisfies (P ) for Lebm a.e. q ∈ Q.

In particular, property (P ) holds true for an arbitrary small perturbation of (Fp)p∈P .

The following is an analogous of C s -parablenders, in a measurable point of view:

Definition. The continuation (Kp)p∈P of a hyperbolic basic set for a C r -family (Fp)p∈P of endomorphisms of R 2 , with r ≥ 1, is an almost C r,s -parablender, with s an integer s.t. s ≤ r, if for a prevalent C r -perturbation ( Fp)p∈P of (Fp)p∈P , the continuation ( Kp)p∈P of (Kp)p∈P satisfies:

Leb 2δ d,s (J s p 0 W u loc ( Kp)) > 0 for Leb d a.e. p0 ∈ P .
Note that if (Kp)p∈P is an almost C r,s -parablender and p is a parameter in P, the set Kp is an almost C r -blender.

Here is our main result, which generalizes Theorem A in terms of jets:

Theorem B. Let (Fp)p∈P be a C r -family of local diffeomorphisms of R 2 , with 2 ≤ r ≤ ∞.
Let (Kp)p∈P be the continuation of a hyperbolic basic set for (Fp)p∈P . Take an integer s ≤ r -2 and suppose that the topological entropy hF p of Fp|Kp satisfies:

( ) hF p > δ d,s • | log m(DFp)| ∀p ∈ P . Then (Kp)p∈P is an almost C r,s -parablender.
This second result also goes in the direction of Conjecture A, both in a measurable point of view and in terms of jets this time. Let us mention that both Theorems A and B still hold true if we assume that the maps involved are local diffeomorphisms only in a neighborhood of the basic sets. Last but not least, we hope to use Theorem B to solve the conjecture of Pugh and Shub [PS] in the smooth C ∞ case which is not handled by [START_REF] Berger | Generic family with robustly infinitely many sinks[END_REF]. Finally, let us mention the following immediate question:

Acknowledgements:

The author would like to thank Pierre Berger for introducing him this topic, and also for many invaluable encouragements and suggestions which improved a lot this manuscript. The author is also grateful to François Ledrappier for helpful discussions.

Combinatorics and notations

Let A be a finite alphabet of cardinality at least 2. Let

-→ A = A N , ← - A = A Z * -and ← → A = A Z
be the sets of infinite forward, backward and bilateral words with letters in A. We consider the left full shift on -→ A or ← → A :

σ : α = (αi)i ∈ -→ A ← → A → σ(α) = (αi+1)i ∈ -→ A ← → A
and the right full shift on ← -A :

σ : α = (αi)i ∈ ← - A → σ(α) = (αi-1)i ∈ ← - A .
In particular these full shifts are of positive entropy and topologically mixing. We also define A * as the set of finite words with letters in A and denote by e the empty word. We endow ← → A with the distance given by d∞(α, β) = D q for every α = (αi)i ∈ ← → A and β = (βi)i ∈ ← → A , where D ∈ (0, 1) is a fixed number and q is the largest integer such that αi = βi for every |i| < q if α = β. We endow -→ A with a metric defined similarly.

For α ∈ A * ∪ -→ A ∪ ← - A ∪ ← →
A , let |α| ∈ N ∪ {+∞} be the number of letters in α. When |α| > n for some integer n > 0, we call αi the i th letter of α and denote

α |n := (α0, • • • , αn-1) when α ∈ A * ∪ -→ A and α |n := (α-n, • • • , α-1) when α ∈ ← - A ∪ ← → A . Finally, for α = (α-n, • • • , α-1) ∈ A * , let [α] be the corresponding cylinder in ← → A : [α] := {β ∈ ← → A : βi = αi ∀ -n ≤ i ≤ -1} .
We define similarly cylinders in ← -A and -→ A and use the same notation. Greek (resp. gothic) letters will be used for finite or backward infinite (resp. forward infinite) words. For a ∈ -→ A , α ∈ ← -A , β ∈ A * , we denote by αa, βa, αβ their concatenations. The topological closure of a set relatively to the Euclidean distance is denoted with an overline. The notation stands for the usual domination relation and f g means that f g and g f .

Example

We give here an application of our results. More precisely, we provide simple examples of almost blenders and parablenders.

Let us consider the segment X = [-1, 1]. We pick three integers n ≥ 2, d ≥ 1 and s ≥ 0. We choose n := (n + 1) δ d,s disjoint subsegments Xj X and n numbers 0 < rj < 1 -1/n, for 1 ≤ j ≤ n . Let gj be the affine preserving order map sending Xj onto X. We pick a C ∞ -map g : R → R which is equal to gj on a small neighborhood of each interval Xj and also a C ∞ -map h : R → R which is equal to rj on a small neighborhood of each interval Xj. The following C ∞ -endomorphism

F : (x, y) ∈ R 2 → (g(x), y n + h(x)) ∈ R 2
is a local diffeomorphism on a small neighborhood of U := 1≤j≤n Xj × X. It is easy to verify that the set

K := n∈Z F n (U ) (1)
is a compact, hyperbolic, invariant, locally maximal set, with stable and unstable dimensions equal to 1. This remains true for any C ∞ -endomorphism F which is C ∞ -close to F, with the same formula.

We set A := {1, • • • , n } and call Fj the restriction of F on Xj ×X. For a = (ai) i≥0 ∈ -→ A and α = (αi)i<0 ∈ ← -A , the following are local stable and unstable manifolds of K:

W a = j≥0 Dom(Fa j • • • • • Fa 0 ) and W α = j<0 Im(Fα j • • • • • Fα -1 ) , ( 2
)
where the domains Dom(Fa

j • • • • • Fa 0 ) and Dom(Fα j • • • • • Fα -1 ) of Fa j • • • • • Fa 0 and Fα j • • • • • Fα -1 are (ga j • • • • • ga 0 ) -1 (X) × X and (gα j • • • • • gα -1 ) -1 (X) × X.
It is immediate that W a is a vertical segment of second coordinate projection X. By hyperbolic continuation, for every C ∞ -endomorphism F which is C ∞ -close to F and every a ∈ -→ A , we can define a local stable manifold W a which is a vertical graph of class C ∞ over y ∈ X with small slope. We notice that these local stable manifolds are pairwise disjoint. We have analogous properties for local unstable manifolds (except their disjointness) and W a and W α intersect in exactly one point.

Let us now consider any d-unfolding (Fp)p∈P of F, i.e. a C ∞ -family of endomorphisms Fp s.t. F0 = F and P = (-1, 1) d . Up to restricting and then rescaling the parameter space, this family leaves invariant the continuation (Kp)p∈P of the hyperbolic set K and Fp is a local diffeomorphism on a neighborhood of Kp. We can define families of local stable and unstable manifolds W a p and W α p as above, and we denote by φp the map sending β = αa ∈ ← → A (with α ∈ ← -A and a ∈ -→ A ) to the unique intersection point φp(αa) ∈ Kp between W a p and W α p . We denote:

Φp : β ∈ ← → A → φp(σ i (β)) i ∈ ← → Kp .
We notice that Φp conjugates the full shift ( ← → A , σ) to the dynamics ( ← → Kp, ← → Fp) on the inverse limit and so periodic points are dense in Kp and Kp is transitive, and thus a hyperbolic basic set. The entropy hF p of Fp|Kp is equal to log(n ) and its stable contraction is close to 1/n. We recall that:

log(n ) |log(1/n)| = δ d,s • log(n + 1) log(n) > δ d,s ≥ 1 .
Thus hF > |log(1/n)| and assumption ( ) holds true for the family (Fp)p∈P . By Theorems A and B, we conclude the following:

Proposition. The set K is an almost C ∞ -blender and for any d-unfolding (Fp)p∈P of F, (Kp)p∈P is an almost C ∞,s -parablender, up to restricting and rescaling P.

Skew-product Formalism and Strategy

Our method is based on a method introduced by Mihailescu, Simon, Solomyak and Urbański. Let us give some details. For IFS without overlaps, the Hausdorff dimension of the limit set is given by Bowen's formula [START_REF] Bowen | Hausdorff dimension of quasi-circles[END_REF]. In [SSU], Simon, Solomyak and Urbański introduced a method to compute it even in the presence of overlaps. The key ingredient in their proofs (see Section 4) is a transversality property (see also [So] and [PeSo] for more on transversality). This also allows to get parameters for which the limit set has positive measure, which is our interest. Later these results were extended by Mihailescu and Urbański to the case of hyperbolic and fiberwise conformal skew-products in [MU]. Here we extend these to the setting of families of skew-products fiberwise unipotent.

Skew-products

We work with (N -dimensional) C r -skew-products acting on -→ A × [-1, 1] N , where N > 0. Here A is a fixed finite alphabet of cardinality at least 2. For simplicity, we denote X := [-1, 1] N . The regularity r of the maps is given by either an integer at least 2 or +∞. Definition 3.1.1. A pre-C r -skew-product is a map of the form:

F : (a, x) ∈ -→ A × X → (σ(a), fa(x)) ∈ -→ A × X
satisfying that there exists an open set X ⊂ R N independent of a s.t. X X and s.t. fa : X → X extends to a C r -diffeomorphism from X to fa(X ) X for every a.

The map F is a C r -skew-product if moreover the two following maps

a ∈ -→ A → fa ∈ C 0 (X , R N ) and a ∈ -→ A → Dfa ∈ C 0 (X , L(R N , R N ))
are Hölder with positive exponent, and the following third map is continuous:

a ∈ -→ A → D 2 fa ∈ C 0 (X , L 2 (R N , R N )) .
In the latter definition, -→ A is endowed with its distance and the spaces of C 0 -maps from X to R N , to the space L(R N , R N ) of linear maps from R N to R N and to the space L 2 (R N , R N ) of bilinear maps from R N × R N to R N endowed with the uniform C 0 -metric. In the following, we suppose that the extensions of the maps fa are fixed.

We are even more interested in families (Fp)p of (pre-)C r -skew-products, indexed by p varying in [-1, 1] d , for 1 ≤ d < ∞. We define such a family as a family of maps:

Fp : (a, x) ∈ -→ A × X → (σ(a), fp,a(x)) ∈ -→ A × X s.t. F : (a, (p, x)) → (σ(a), (p, fp,a(x))) is a (N + d-dimensional) (pre-)C r -skew- product.
In particular, there exist open neighborhoods X and P of X and P in R N and R d s.t. (p, x) → (p, fp,a(x)) extends to a diffeomorphism from P × X into P × X for each a and the map Fp is a (N -dimensional) (pre-)C r -skew-product for every p ∈ P . Again, we denote (Fp)p∈P this family since we are interested mainly on the dynamics when p ∈ P but still keep in mind that it admits such an extension.

We will say that such a family (Fp)p∈P of (pre-)C r -skew-products satisfies the Unipotent assumption (U) when the following is satisfied:

(U): For any p ∈ P , a ∈ -→ A and x ∈ X , the differential Dfp,a(x) is inferior unipotent, that is an inferior triangular matrix with all its diagonal coefficients equal the one other, and its unique eigenvalue is strictly bounded between 0 and 1 in modulus.

We will adopt the following formalism. For every p ∈ P , a ∈ -→ A , n > 0 and α = (α-n, . . . , α-1) ∈ A * , we set:

∀x ∈ X , ψ α p,a (x) := fp,α -1 a • . . . • fp,α -n •••α -1 a(x) .
We show below that a consequence of (U) is that ψ α p,a is a C r -contraction from X to ψ α p,a (X ) X when |α| is large enough. If we now take an infinite backward sequence α = (. . . , α-n, . . . , α-1) ∈ ← -A , we see that the points ψ α |n p,a (0) converge to a point πp,a(α) ∈ X. This defines a C 0 -map πp,a :

← -A → X.

Definition 3.1.2. The limit set Kp,a of the skew-product Fp inside the a-fiber is:

Kp,a := πp,a( ← - A ) .
We will give conditions under which this set has positive measure.

For a C 1 -map f : X → X, let m(Df ) and M (Df ) be the respective minimum and maximum of ||Df (x) • u|| among x ∈ X and u ∈ R N s.t. ||u|| = 1. We need the following thermodynamical formalism:

Definition 3.1.3. The pressure at the parameter p in the a-fiber is the map:

Πp,a : s ∈ R+ → lim n→+∞ 1 n log α∈A n M (Dψ α p,a ) s .
When Πp,a has a unique zero, we call it the similarity dimension inside the a-fiber.

In Proposition 5.1.4, we show that both the pressure and the similarity dimension are well-defined and independent of a. In particular, we denote them by Πp and ∆(p).

We adopt the following terminology to denote perturbations with special properties: Definition 3.1.4. Let (Fp)p be a family of (pre-)C r -skew-products and let us fix neighborhoods X and P of X and P s.t. (p, x) → (p, fp,a(x)) extends to a diffeomorphism from P × X into P × X for every a ∈ -→ A . For any ϑ > 0, a ϑ-perturbation of (Fp)p is a family of pre-C r -skew-products ( Fp)p s.t. the map (p, x) → (p, fp,a(x)) extends to a diffeomorphism from P × X into P × X for every a ∈ -→ A and:

sup a∈ - → A ||(p, x) ∈ X × P → (fp,a -fp,a)(x)||C r < ϑ .
The family ( Fp)p is a ϑ-U-perturbation when it satisfies assumption (U).

For ϑ-U-perturbations with small ϑ, we will show that for any α ∈ ← -A , the points

fp,α -1 a • . . . • fp,α -n •••α -1 a(0) converge to πp,a(α) ∈ X s.t. p → πp,a(α) is a C r -map C r -close to p → πp,a(α).
We will consider also parameterized families of ϑ-perturbations ( Ft,p)p:

F := (( Ft,p)p)t∈T .
Here t varies in T := (-1, 1) τ with τ > 0 and (t, p, x) → ft,p,a(x) is C r for every a ∈ -→ A . When each ( Ft,p)p is a ϑ-U-perturbation, we say that F is a parameterized families of ϑ-U-perturbations. When ϑ is small enough, we will denote by πt,p,a(α) the limit point corresponding to any α ∈ ← -A and Kt,p,a := πt,p,a( ← -A ), and then the C r -maps p → πt,p,a(α) will be C r -close to p → πp,a(α), uniformly in (t, α). We will set conditions under which Kt,p,a has positive Lebesgue measure for a.e. t ∈ T .

Strategy and Organization of the paper

The strategy will be to focus on the dynamics restricted to the local stable manifolds, which allows us to reduce the dynamics to that one of a C r -skew-product.

Hence we forget for some time families of endomorphisms and we work with families of (pre-)C r -skew-products. We say that such a family of (pre-)C r -skew-products (Fp)p∈P satisfying assumption (U) satisfies also the Transversality assumption (T) when the following is satisfied:

(T): There exists C > 0 such that for every sequences a ∈ -→ A and α, β ∈ ← -A satisfying α-1 = β-1, and for every r > 0, we have:

Leb d {p ∈ P : ||πp,a(α) -πp,a(β)|| < r} ≤ Cr N ,
and moreover for every small ϑ > 0 and every family F of ϑ-U-perturbations, for any t ∈ T , a ∈ -→ A , α, β ∈ ← -A s.t. α-1 = β-1 and r > 0 we have:

Leb d {p ∈ P : ||πt,p,a(α) -πt,p,a(β)|| < r} ≤ Cr N .
The main technical result to prove Theorem B is the following. It sets conditions under which a given family of skew-products intersects its fibers into a set of positive measure, up to perturbations. Here is the strategy to prove Theorem C. For every parameter p and a-fiber, we define a probability measure νp,a supported on the limit set Kp,a. To show that Kp,a has positive measure, it is enough to show that νp,a is absolutely continuous relatively to the N -dimensional Lebesgue measure, and so to prove that its density is finite almost everywhere. We compute the integral of the density relatively to the parameter and the phase space. The trick is to use the Fubini Theorem to integer first relatively to p. The finiteness of the integral is implied by the transversality assumption (T) and the inequality ∆(p) > N . The same method will give the same results for families of ϑ-U-perturbations, with additional integration relatively to t.

To prove Theorem B, we go back to C r -families (Fp)p of endomorphisms and we restrict the dynamics to the local stable manifolds W a p , which are tagged in exponent by infinite forward sequences a in letters in an alphabet A. Since dim E s p = 1, we are led to study families (Fp)p of C r -skew-products acting on fibers which are segments.

We then look at the action induced by (Fp)p on s-jets (with s ≤ r -2) and this gives a new family of C 2 -skew-products (Gp 0 )p 0 acting on fibers of dimension δ d,s . This new family satisfies assumption (U) (the assumption dim E s p = 1 is used here), its similarity dimension is larger than δ d,s by ( ). We extend (Fp)p into a larger C r -family of endomorphisms so that the associated extended family of C 2 -skewproducts acting on s-jets satisfies moreover the transversality assumption (T).

To conclude, we pick a family (Γt,p)t,p of parallel segments Γt,p close to a local stable manifold W a p and s.t. the set of s-jets at any p0 ∈ P of the projection of Γt,p on a fixed direction transversal to Γt,p when varying t has positive measure. The local unstable set intersects each segment Γt,p in a set which is the limit set of a ϑ-Uperturbation with small ϑ, at every parameter p. We then apply the second part of Theorem C to get positive sets of s-jets for the intersection points between the local unstable set and Γt,p for a.e. t at a.e. p0: in other terms we get positive sets of s-jets in the direction of Γt,p for these values of t and p0. To conclude, we apply the Fubini Theorem to find positive sets of bidimensional s-jets for points inside local unstable manifolds at a.e. p0. The same extension scheme works for (Gp)p close to (Fp)p, which proves that (Kp)p is an almost C r,s -parablender.

Finally, Theorem A is an immediate consequence of Theorem B together with Remark 6.0.4, by taking the constant family (F) p∈(-1,1) and the order s of the jets equal to 0 (remark in particular that δ1,0 = 1).

In Section 4, we study a model given by families of IFS of affine maps on an interval. We simplify the proof of Simon, Solomyak and Urbański [SSU] in this context and introduce the strategy of the proof of Theorem C here. In Section 5, we prove Theorem C. In each fiber, the behavior of the dynamics looks like the model. Finally, we prove Theorem B in Section 6.

Model: IFS of affine maps on the interval

Setting and results

In this Section, we simplify the proof of a result of Simon, Solomyak and Urbański about IFS on an interval. This can be seen as a model for the behavior of the dynamics inside the fibers of a skew-product, as we will see in Section 5.

Let us fix X := [-1, 1] and P := (-1, 1). We consider families (Ψp)p∈P , where, for every p ∈ P, the IFS Ψp is a finite family Ψp = (ψ a p )a∈A of affine contractions ψ a p : X → X s.t. ψ a p (X) X. The absolute value of the linear coefficient of ψ a p is denoted by Λp,a. We suppose that for every a ∈ A, the map ψ a p depends continuously on p. In fact, we even suppose that the affine contraction ψ a p is still defined for p in some open neighborhood of [-1, 1] and still depends continuously on p.

For every p ∈ P and α

= (α-n, • • • , α-1) ∈ A * , we denote by ψ α p = ψ α -1 p • • • • • ψ α -n p
the corresponding composition, which is an affine contraction of the segment X into its own interior. When α = e, the map ψ e p is simply the identity. The absolute value of the linear coefficient of ψ α p is denoted by Λp,α.

By continuity of the derivative of ψ a p relatively to p ∈ P, there exist 0 < γ < γ < 1 s.t. γ < Λp,a < γ for every p ∈ P and a ∈ A. Then Λp,α ≤ γ |α| for every α ∈ A * . If we now take α ∈ ← -A , the sequence of points ψ α |n p (0) tends to a point πp(α). This defines for every p ∈ P a C 0 -map πp :

← -A → X. Since this convergence is uniform in p, the map p → πp(α) is C 2 for any α. Moreover the map p → πp is continuous, the set of C 0 -maps from ← -A to R being endowed with the uniform C 0 -norm. We set:

Kp := πp( ← - A ) .
We also suppose that the following assumption (T aff ) is satisfied by (Ψp)p∈P .

(T aff ): There exists C > 0 s.t. for every α, β ∈ ← -A satisfying α-1 = β-1, we have:

Leb1{p ∈ P : |πp(α) -πp(β)| < r} ≤ Cr for any r > 0 .
It is immediate that for any p ∈ P, there exists exactly one number ∆(p) ≥ 0 s.t.:

a∈A Λ ∆(p) p,a = 1 .
This is the similarity dimension of the IFS Ψp. We are now in position to state the following result, which is a direct consequence of Theorem 3.1 of [SSU]:

Theorem D. (Simon, Solomyak, Urbański) Let (Ψp)p∈P be a family of IFS of affine contractions satisfying (T aff ) and ∆(p) > 1 for any p ∈ P. Then it holds:

Leb1(Kp) > 0 for Leb1 a.e. p ∈ P .

Proof of Theorem D

Proof of Theorem D. Here is the strategy. Let p0 ∈ P such that ∆(p0) > 1 + for a small > 0. To prove the result, it is enough to show that there exists δ > 0 s.t. B := (p0 -δ, p0 + δ) is included in P and Leb1(Kp) > 0 for Leb1 a.e. p ∈ B.

We define a probability measure µ on ← -A by setting µ[α] = Λ ∆(p 0 ) p 0 ,α for every cylinder defined by α ∈ A * . For any p ∈ P, let νp be the pushforward of µ by πp, which is supported on Kp. To conclude, it is enough to show that there exists δ > 0 s.t. for Leb1 a.e. p ∈ B, the measure νp is absolutely continuous relatively to Leb1. We set

D(νp, x) := lim inf r→0 νp(x -r, x + r) 2r
for every p ∈ P and x ∈ R, which is the lower density of the measure νp at x.

Lemma 4.2.1. The map (p, x) ∈ P × R → D(νp, x) is Borel measurable.
Proof. Since p ∈ P → πp is continuous, it is also the case for p ∈ P → νp (the set of probability measures being endowed with the weak- * topology). Using this and since x ∈ R → νp(x -r, x + r) is Borel measurable for every p ∈ P and r > 0, the map (p, x) ∈ P × R → νp(x -r, x + r) is Borel measurable for every r > 0. Since r → νp(x -r, x + r) is monotonic and r → 2r continuous, the lower limit D(νp, x) does not change if r is restricted to positive rationals. Thus the measurability of (p, x) ∈ P × R → D(νp, x) reduces to that of the lower limit of countably many measurable maps.

We prove below:

Proposition 4.2.2. There exists δ > 0 s.t. B ⊂ P and the following is finite:

I := p∈B x∈R D(νp, x)dνpdLeb1 < +∞ .
This is enough to show that for Leb1 a.e. p ∈ B, νp is absolutely continuous relatively to Leb1. Indeed, then, for Leb1 a.e. p ∈ B, we will have D(νp, x) < +∞ for νp a.e.

x ∈ R and we apply the following result from the third item of Lemma 2.12 in [Ma]. This concludes the proof of Theorem D.

Proof of Proposition 4.2.2. For δ small enough, the interval B := (p0 -δ, p0 + δ) is included in P. If necessary, we reduce δ so that Λ 1+ /2 p 1 ,a ≤ Λp 2 ,a for every p1, p2 ∈ B and a ∈ A. In particular, this implies the following:

∀p1, p2 ∈ B, ∀α ∈ A * , Λ 1+ 2 p 1 ,α ≤ Λp 2 ,α .
(3)

The strategy is to bound I by a new integral which will be easily shown to be finite using (T aff ), for this specific choice of δ. First, by Fatou's lemma, it holds:

I ≤ lim inf r→0 1 2r p∈B x∈R νp(x -r, x + r)dνpdLeb1 . ( 4
)
We can write νp(x -r, x + r) = y∈R 1 {|x-y|<r} dνp as the integral of the indicator function 1 {|x-y|<r} , equal to 1 if |x -y| < r, and 0 if not. Using this and then the definition of νp as the pushforward of µ by πp, we have:

x∈R νp(x -r, x + r)dνp = (α,β)∈ ← - A × ← - A 1 {|πp(α)-πp(β)|<r} dµ × µ , (5)
where 1 {|πp(α)-πp(β)|<r} is equal to 1 if |πp(α) -πp(β)| < r and 0 if not. Then, we inject Eq. ( 5) into Eq. ( 4) and use Fubini's Theorem to reverse the order of integration:

I ≤ lim inf r→0 1 2r (α,β)∈ ← - A × ← - A Leb1{p ∈ B : |πp(α) -πp(β)| < r}dµ × µ . ( 6
)
We are going to write the latter integral as a sum whose terms are all easier to bound. For every finite word ρ ∈ A * , we denote by Cρ the set of pairs (α,

β) ∈ ← - A × ← - A such that α -|ρ| • • • α-1 = β -|ρ| • • • β-1 = ρ but α -|ρ|-1 = β -|ρ|-1 . We notice that ← - A × ← - A = n≥0 ρ∈A n
Cρ and so by Eq. ( 6) we have:

I ≤ lim inf r→0 1 2r n≥0 ρ∈A n (α,β)∈Cρ Leb1{p ∈ B : |πp(α) -πp(β)| < r}dµ × µ . (7)
We show below that a consequence of the transversality assumption (T aff ) is:

Lemma 4.2.4. For every n ≥ 0, ρ ∈ A n and (α, β) ∈ Cρ, we have:

Leb1{p ∈ B : |πp(α) -πp(β)| < r} r • Λ -1-/2 p 0 ,ρ .
We can inject the bound of Lemma 4.2.4 into Eq. ( 7):

I n≥0 ρ∈A n (α,β)∈Cρ Λ -1-/2 p 0 ,ρ dµ × µ . ( 8
)
We use the equality µ[ρ] = Λ ∆(p 0 ) p 0 ,ρ , the inequality (1 + 2 )/∆(p0) < (1 + 2 )/(1 + ) < 1 -3 and finally the inequality γ < Λp,a < γ to get:

Λ -1-/2 p 0 ,ρ µ[ρ] -1-/2 ∆(p 0 ) ≤ µ[ρ] -(1-3 ) γ n 3 • µ[ρ] -1 . ( 9
)
We now inject this bound into Eq. ( 8) to find:

I n≥0 γ n 3 ρ∈A n (µ × µ)(Cρ) µ[ρ] ≤ n≥0 γ n 3 ρ∈A n µ[ρ] = n≥0 γ n 3 < +∞ ,
where we used the inequality

(µ × µ)(Cρ) ≤ µ[ρ] 2 (coming from Cρ ⊂ [ρ]
2 ) to prove the second inequality. This concludes the proof of Proposition 4.2.2.

Proof of Lemma 4.2.4. For every p ∈ P, n ≥ 0, ρ ∈ A n and (α, β) ∈ Cρ, it holds:

|πp(α) -πp(β)| = Λp,ρ • |πp(σ n (α)) -πp(σ n (β))| . ( 10
)
Indeed, the points πp(α) and πp(β) are the respective images of πp(σ n (α)) and πp(σ n (β)) by the map ψ ρ p which is an affine contraction, and the absolute value of the linear coefficient of ψ ρ p is Λp,ρ. Thus, using Eq. ( 3), it holds:

Leb1{p ∈ B : |πp(α) -πp(β)| < r} = Leb1{p ∈ B : |πp(σ n (α)) -πp(σ n (β))| < r Λp,ρ } Leb1{p ∈ B : |πp(α)-πp(β)| < r} ≤ Leb1{p ∈ B : |πp(σ n (α))-πp(σ n (β))| < r Λ 1+ 2 p 0 ,ρ

}

To conclude, by (T aff ) and since B ⊂ P the right-hand term of the latter is smaller than

C • r • Λ -1-/2 p 0 ,ρ .
Example. Let us give a simple example of application of Theorem D. Let n ≥ 2 be an integer. We set A := {0, 1, 2, . . . , n}, X := [0, 1] and

P := (1/n, 1 -1/n). Let c < 1/n be a real number close to 1/n. For a ∈ A, we put ψ a p (x) := cx + 1 2 (1/n -c) + a/n if 0 ≤ a < n and ψ n p (x) := cx + p if a = n. Condition (T aff
) is clearly satisfied when n is large. Moreover trivial computations show that the similarity dimension is ∆(p) = -log(n + 1)/log(c) > 1 for any p ∈ P. By Theorem D, Kp has positive one-dimensional Lebesgue measure for a.e. p ∈ P.

The unipotent case: Proof of Theorem C

We now extend Theorem D to the case of families of fiberwise unipotent skewproducts. The fibers are indexed by a ∈ -→ A and the dynamics on each fiber will look like the one of the model previously introduced. Here are some differences:

• We will not restrain ourselves to fibers of dimension 1 and we will not suppose that the dynamics on each fiber is conformal but we will suppose that its differentials are unipotent with contracting eigenvalue (assumption (U)).

• We will need distortion results (Lemmas 5.2.1, 5.2.2, 5.2.3 and 5.2.4) since the dynamics will not supposed to be affine this time.

Notations and immediate facts

We adopt from now the following formalism in order to prove Theorem C. Let (Fp)p∈P be a family of C r -skew-products satisfying (U) and (T). We recall that there exist open neighborhoods X and P of X and P in R N and R d s.t. each map (p, x) → (p, fp,a(x)) extends to a diffeomorphism from X × P into X × P and the map Fp is a (N -dimensional) C r -skew-product for every p ∈ P . We set:

∀p ∈ P , a ∈ -→ A , a ∈ A, x ∈ X , ψ a p,a (x) := fp,aa(x) (11)
and notice that ψ a p,a : X → X is a C 2 -map depending continuously on (p, a). The C 2 -norm of ψ a p,a on X is then bounded independently of p ∈ P, a ∈ -→ A and a ∈ A. We now define for any p ∈ P , a ∈ -→ A , n > 0 and α = (α-n,

• • • , α-1) ∈ A * : ∀x ∈ X , ψ α p,a (x) := ψ α -1 p,a • • • • • ψ α -n p,α -n+1 •••α -1 a (x) = fp,α -1 a • • • • • fp,α -n •••α -1 a(x) .
In particular, assumption (U) implies that for every p ∈ P , a ∈ -→ A , α ∈ A * and x ∈ X , the differential Dψ α p,a (x) is unipotent inferior and thus has a unique eigenvalue. This motivates the definition of the following contraction rate: Definition 5.1.1. For any p ∈ P , a ∈ -→ A , α ∈ A * and x ∈ X , let λp,a,α(x) be the absolute value of the unique eigenvalue of the differential Dψ α p,a (x) and:

Λp,a,α := maxx∈X λp,a,α(x) .

(12)

For any p ∈ P , a ∈ -→ A , α = (α-n, . . . , α-1) ∈ A * and x ∈ X , we then have:

λp,a,α(x) = n k=1 λp,a k ,α -k ψ α -k-1 p,a k+1 • • • • • ψ α -n p,an (x) with a k := α |k-1 a . ( 13
)
By continuity of Dψ a p,a (x) relatively to p ∈ P, a ∈ -→ A and x ∈ X and by compactness of P, -→ A and X, there exist 0 < γ < γ < 1 s.t. for any p, a, a, it holds:

∀x ∈ X, γ < λp,a,a(x) < γ . ( 14
)
In particular for every α ∈ A * it holds:

γ |α| < Λp,a,α < γ |α| . ( 15
)
We will need later the following, whose proof is in the Appendix:

Lemma 5.1.2. There exists a real polynomial P positive on R+ s.t. for any p ∈ P,

a ∈ -→ A , α ∈ A * , x ∈ X and (i, j) ∈ {1, • • • , N } 2 with i > j, the modulus of the (i, j) th coefficient of the differential Dψ α p,a (x) is smaller than P (|α|) • λp,a,α(x).
In particular, for any p ∈ P, a ∈ -→ A and α ∈ A * of length sufficiently large, the map ψ α p,a is a contraction. For p ∈ P, a ∈ -→ A and α ∈ ← -A , the diameter of ψ α |n p,a (X) is then small when n is large. Thus the sequence of points ψ α |n p,a (0) converges to πp,a(α) ∈ X. This defines for every p ∈ P and a ∈ -→ A a C 0 -map πp,a : ← -A → X. The map (p, a) ∈ P × -→ A → πp,a is then continuous, the set of C 0 -maps from ← -A to R N being endowed with the uniform C 0 -norm. We set:

Kp,a := πp,a( ← - A ) .
For any family F := (( Ft,p)p)t∈T of ϑ-U-perturbations with small ϑ > 0, the map

ψ α t,p,a := ft,p,α -1 a • • • • • ft,p,α -n •••α -1
a is also a contraction when |α| is large and so the sequence of points ψ α |n t,p,a (0) still converges to πt,p,a(α) ∈ X. This allows us to define a C 0 -map πt,p,a :

← -A → X for any t and p, and its limit set Kt,p,a := πt,p,a( ← -A ).

Lemma 5.1.3. The map p → πp,a(α) is C r for every a ∈ -→ A and α ∈ ← -A . Moreover, for every family F of ϑ-U-perturbations with small ϑ > 0, the map (t, p) → πt,p,a(α) is C r and p → πt,p,a(α) is C r -close to p → πp,a(α) uniformly in t ∈ T .

Proof. These maps are the respective uniform limits of the following C r -maps

p → ψ α |n p,a (0) and (t, p) → ψ α |n t,p,a (0) .
To conclude, we remark that these convergences are both exponential and that the maps fp,a and ft,p,a are (uniformly in t and a) C r -close when ϑ is small.

As an immediate consequence of Lemma 5.1.2, the pressure function Πp,a defined in Definition 3.1.3 is equal to:

Πp,a(s) = lim n→+∞ 1 n log α∈A n Λ s p,a,α for any s ≥ 0 . ( 16
)
Moreover this map satisfies the following nice properties:

Proposition 5.1.4. The map s ∈ R+ → Πp,a(s) ∈ R is well-defined, strictly decreasing, continuous, independent of a, Πp,a(0) > 0 and lims→+∞Πp,a(s) = -∞. In particular, it has exactly one zero denoted by ∆(p), depending continuously on p.

From now on we suppose that ∆(p) > N for any p ∈ P.

Distortion lemmas

We now state distortion results, whose proofs are given in the Appendix:

Lemma 5.

Λ p,a,α /D4 < Λt,p,a,α < D4Λ 1/ p,a,α
where Λt,p,a,α is the maximum among x ∈ X of the absolute value λt,p,a,α(x) of the unique eigenvalue of the differential Dψ α t,p,a (x).

Choice of a probability measure µ

We will first need the following result of Bowen ([Bo1] thm 1.4 P7 and its proof P19) about the existence of a Gibbs measure. We recall that σ is a full shift. We state the result in this case but it remains true for subshifts of finite type topologically mixing. We suppose that a parameter p0 has been fixed (the precise choice will be made in the next subsection). We fix an arbitrary a0 ∈ -→ A .

Theorem E (Bowen). Let φ :

← → A → R be a Hölder map with positive exponent (the set ← → A being endowed with the distance d∞). Then there exists a unique σ-invariant measure µ on ← → A s.t. for every A ∈ ← → A , it holds:

µ[A |n ] exp -Πn + n-1 k=0 φ(σ k (A))
where

Π = Π(φ) = limn→∞ 1 n logZn(φ), with
Zn(φ) := x∈A n exp(Sx) and Sx = sup{ n-1 k=0 φ(σ k (y)) : y ∈ [x]} .
Writing A = αa ∈ ← → A as the concatenation of α ∈ ← -A and a ∈ -→ A , we can apply the previous result with the map

φ : A ∈ ← → A → ∆(p0) • logλp 0 ,a,α -1 πp 0 ,α -1 a(σ(α)) .
We show in the Appendix:

Lemma 5.3.1. The map φ is Hölder with positive exponent.

Using Lemma 5.2.1 and 5.2.3, we note that Π = Π(φ) coincides with Πp 0 (∆(p0)) and thus vanishes by definition of ∆(p0) (see Proposition 5.1.4). Moreover, by Eq. ( 13), for any

A ∈ ← → A the sum n-1 k=0 φ(σ k (A)) is equal to: ∆(p0) • log n k=1 λp 0 ,a k ,α -k πp 0 ,a k+1 (σ k (α)) = ∆(p0) • logλp 0 ,a,α |n πp 0 ,α |n a(σ n (α))
with a k := α |k-1 a. By Theorem E, this gives us a σ-invariant measure µ on ← → A such that for every A = αa ∈ ← → A , we have:

µ[A |n ] λ ∆(p 0 ) p 0 ,a,α |n πp 0 ,α |n a(σ n (α)) when n → +∞ .
Using successively Lemmas 5.2.1 and 5.2.3, for any A = αa ∈ ← → A , it holds:

µ[A |n ] Λ ∆(p 0 ) p 0 ,a,α |n Λ ∆(p 0 ) p 0 ,a 0 ,α |n when n → +∞ . ( 17
)
We then define a σ-invariant probability measure on ← -A , still denoted µ, by giving to each cylinder in ← -A the same measure than the corresponding one in ← → A . Then:

µ[ρ] Λ ∆(p 0 ) p 0 ,a 0 ,ρ when ρ ∈ A n and n → +∞ . ( 18
)
Remark 5.3.2. We will not need the σ-invariance property of µ in the following but only the estimation from Eq. (18).

Proof of Theorem C

The strategy is the same as for the proof of Theorem D. Let us consider p0 ∈ P and a ∈ -→ A . We have ∆(p0) > N + , where := 1 2 (min p∈P ∆(p) -N ) > 0. To prove the result, we show that there exists δ > 0 s.t. the d-dimensional ball B of center p0 of radius δ is included in P with LebN (Kp,a) > 0 and LebN ( Kt,p,a) > 0 for Leb d a.e. p ∈ B and Lebτ a.e. t ∈ T , for every family F of ϑ-U-perturbations with small ϑ.

We endow

← -A with the probability measure µ defined in Subsection 5.3. For any p ∈ P and t ∈ T , let νp,a and νt,p,a be the images of µ by the maps πp,a and πt,p,a.

As Proposition 4.2.2 implies Theorem D, Theorem C is a consequence of: Proposition 5.4.1. There exists δ > 0 s.t. the ball B of center p0 and radius δ is included in P and the two following integrals are finite:

I := p∈B x∈R N lim inf r→0 νp,a(x + B(r)) cN r N dνp,adLeb d < +∞ , I := p∈B t∈T x∈R N lim inf r→0 νt,p,a(x + B(r)) cN r N dνt,p,adLebτ dLeb d < +∞ ,
for any family F of ϑ-U-perturbations with small ϑ, where B(r) ⊂ R N is the ball of center 0 and radius r and the constant cN is defined by cN r N := LebN (B(r)).

Proof. Let us take a small δ s.t. B ⊂ P. The radius δ will be reduced one time so that I and I are finite. We first begin by bounding I. The proof is similar to the one of Proposition 4.2.2: we begin by using Fatou's Lemma, the definition of νp,a and the Fubini-Tonelli Theorem to find the following bound:

I ≤ lim inf r→0 1 cN r N (α,β)∈ ← - A × ← - A Leb d {p ∈ B : ||πp,a(α) -πp,a(β)|| < r}dµ × µ .
We write

← - A × ← - A = n≥0 ρ∈A n Cρ where Cρ is the set of pairs (α, β) ∈ ← - A × ← - A s.t. α -|ρ| • • • α-1 = β -|ρ| • • • β-1 = ρ but α -|ρ|-1 = β -|ρ|-1 . Thus I is smaller than: lim inf r→0 1 cN r N n≥0 ρ∈A n (α,β)∈Cρ Leb d {p ∈ B : ||πp,a(α) -πp,a(β)|| < r}dµ × µ . (19)
We show below that a consequence of the transversality assumption (T) is: Lemma 5.4.2. We fix η := -logγ 2N + and reduce δ if necessary s.t. δ < δ(η) (where δ(η) is defined in Lemma 5.2.2). Then for any n ≥ 0, ρ ∈ A n , (α, β) ∈ Cρ, we have:

Leb d {p ∈ B : ||πp,a(α) -πp,a(β)|| < r} r N • Λ -N -2 /3 p 0 ,a 0 ,ρ .
For any family F of ϑ-U-perturbations of (Fp)p with small ϑ, for any t ∈ T , we have:

Leb d {p ∈ B : ||πt,p,a(α) -πt,p,a(β)|| < r} r N • Λ -N -2 /3 p 0 ,a 0 ,ρ .
Notice that when the similarity dimension is close to the dimension N of the fibers (and so small), we need to work with a ball of small radius η. We can inject the first bound of Lemma 5.4.2 into Eq. ( 19):

I n≥0 ρ∈A n (α,β)∈Cρ Λ -N -2 /3 p 0 ,a 0 ,ρ dµ × µ . ( 20
)
We use successively Eq. ( 18), the inequality (N + 2 3 )/∆(p0) < (N + 2 3 )/(N + ) < 1 -4N and finally Eq. ( 15) (which gives µ[ρ] γ n ) to get:

Λ -N -2 /3 p 0 ,a 0 ,ρ µ[ρ] -N -2 /3 ∆(p 0 ) ≤ µ[ρ] -(1-4N ) γ n 4N • µ[ρ] -1 . ( 21
)
We now inject this bound into Eq. ( 20) to find:

I n≥0 γ n 4N ρ∈A n (µ × µ)(Cρ) µ[ρ] ≤ n≥0 γ n 4N ρ∈A n µ[ρ] = n≥0 γ n 4N < +∞ ,
where we used the inequality (µ × µ)(Cρ) ≤ µ[ρ] 2 (coming from Cρ ⊂ [ρ] 2 ) to prove the second inequality. To bound I for every family F of ϑ-U-perturbations of (Fp)p with small ϑ, we just remark that the same proof works when ϑ is small enough, with an additional integration relatively to t ∈ T .

Proof of Lemma 5.4.2. Let us begin with the following distortion lemma:

Lemma 5.4.3. There exists D5 > 0 s.t. for every p1, p2 ∈ P, a1, a2 ∈ -→ A and ρ ∈ A * , the following holds true:

||p1 -p2|| < δ =⇒ Λ 1+ 2N p 1 ,a 1 ,ρ ≤ D5 • Λp 2 ,a 2 ,ρ . Proof. If ||p1 -p2|| < δ, then ||p1 -p2|| < δ(η). By Lemma 5.2.2, it holds: Λ 1+ 0 p 1 ,a 1 ,ρ ≤ D 1+ 0 2 e |ρ|η(1+ 0 ) Λ 1+ 0 p 2 ,a 1 ,ρ ≤ D 1+ 0 2 e |ρ|η(1+ 0 ) γ |ρ| 0 Λp 2 ,a 1 ,ρ ,
with 0 := /(2N ). By Lemma 5.2.3, Λp 2 ,a 1 ,ρ Λp 2 ,a 2 ,ρ when |ρ| → +∞, with bounds independent of p2. The result follows since, by definition of η, it holds: e |ρ|η(1+ 0 ) γ |ρ| 0 = e |ρ|(η(1+ 0 )+ 0 logγ) with η(1 + 0) + 0logγ = 0 . Lemma 5.4.4. There exists a real polynomial R positive on R+ s.t. for every p ∈ P, a ∈ -→ A , n ≥ 0, ρ ∈ A n and (α, β) ∈ Cρ, it holds:

||πp,a(α) -πp,a(β)|| ≥ Λp,a,ρ R(n) • ||πp,ρa(σ n (α)) -πp,ρa(σ n (β))|| . ( 22
)
Moreover for any > 1, for every family F of ϑ-U-perturbations of (Fp)p with ϑ > 0 small enough we have for every t ∈ T , p ∈ P, a ∈ -→ A , n ≥ 0, ρ ∈ A n and (α, β) ∈ Cρ:

||πt,p,a(α) -πt,p,a(β)|| ≥ Λ p,a,ρ R(n) • ||πt,p,ρa(σ n (α)) -πt,p,ρa(σ n (β))|| . ( 23
)
The proof is in the Appendix. Given a ∈ -→ A , n ≥ 0, ρ ∈ A n , (α, β) ∈ Cρ, using Eq. ( 22) and then Lemma 5.4.3 together with the fact that B is the ball of center p0 and radius δ, it holds:

Leb d {p ∈ B : ||πp,a(α)-πp,a(β)|| < r} ≤ Leb d {p ∈ B : ||πp,ρa(σ n (α))-πp,ρa(σ n (β))|| < R(n)r Λp,a,ρ } ≤
To conclude, by (T) and since B ⊂ P the latter is smaller than:

r N • Q(n) • Λ -N -/2 p 0 ,a 0 ,ρ with Q(n) := C • D N 5 • (R(n)) N .
The result follows since Λp 0 ,a 0 ,ρ decreases exponentially with n, and /2 < 2 /3. The proof of the second item is similar, by taking close to 1 in Eq. ( 23).

Jets: Proof of Theorem B

We now prove Theorem B. The strategy is to study the dynamics of the family (Fp)p inside the local stable manifolds to reduce the problem to the dynamics of a family (Fp)p of C r -skew-products (Step 1) with one-dimensional fibers, from which we construct a family of C 2 -skew-products (Gp 0 )p 0 acting on s-jets (Step 2). Then we extend the latter one into a larger family (Gq 0 )q 0 to satisfy the transversality assumption (T) (Step 3). Finally, we look at the intersection between the unstable set and a family of curves all close to a stable manifold. In each curve, this intersection is equal to the limit set of a perturbation of the skew-product. We then apply successively Theorem C and the Fubini Theorem to conclude to a set of jets of positive measure at a.e. parameter, which gives the parablender property (Step 4).

Step 1: Dynamically defined family of skew-products. We first need to define local stable and unstable manifolds. Let us fix a small ε > 0 and an arbitrary parameter in P, taken arbitrarily equal to 0 for simplicity. We recall that K0 is a hyperbolic basic set for F0. Up to a change of metric on the stable (resp. unstable) bundles of K0, we suppose that DF0 strictly contracts (resp. expands) the stable (resp. unstable) bundle by a factor λ < 1 (resp. 1/λ) uniform over z ∈ K0.

It has been shown by Qian and Zhang (see section 4 of [QZ]) that the limit inverse ← → K0 can be endowed with a map [•] defined on a subset of ← → K0 × ← → K0 with values in ← → K0 so that for every sufficiently closed orbits x, y ∈ ← → K0, the orbit z = [x, y] is well-defined and the 0-coordinate projection π(z) ∈ K0 of z is the intersection of the local unstable manifold of x and the local stable manifold of y0. This map endows ← → K0 with a structure of Smale space (see Ruelle [Ru], Chapter 7, for the definition of a Smale space). This implies that ← → K0 admits Markov partitions of arbitrarily small diameter (see again Ruelle [Ru], Chapter 7, for this result and the definition and properties of a Markov partition for a Smale space). We pick such a partition of ← → K0 by a finite number of compact rectangles R1, . . . , RM of diameter small compared to ε. Up to reducing ε, we can suppose that F0 is a local diffeomorphism when restricted to the 0-coordinate projection Ri := π(Ri) ⊂ K0 of any rectangle Ri.

The topological entropy of F0|K0 is larger than δ d,s • | log m(DF0)| by ( ). Then for large we can pick N ε, orbits xa = (x i a ) 0≤i≤ -1 of length (with a ∈ A := {1, 2, . . . , N ε, }) under F0 which are ( , ε)-separated (in the sense of Bowen) with:

( ) logN ε, > δ d,s • | log m(DF0)| .
In particular the cardinality N ε, of A is at least 2. We can extend each of these orbits xa of length into an infinite orbit in ← → K0, still denoted xa. Up to slightly perturbating xa, we can suppose that for every 0 ≤ k ≤ -1, the orbit ← → F0 k (xa) is in the interior of some rectangle Ri that we denote R a,k . In particular

x k a = F k 0 (x 0 a ) is in R a,k = π(R a,k
). Since the rectangles Ri are in finite number M independent of , up to modifying N ε, by a multiplicative constant (independent of large), we can suppose that all orbits xa (a ∈ A) begin in the same Ri and end in the same Rj, with ( ) still true. For any a ∈ A, let us define:

Ra := Ra,0 ∩ F -1 0 (Ra,1) ∩ • • • ∩ F -( -1) 0 (R a, -1 ) .
which is not empty by assumption. Since the -orbits (x i a ) 0≤i≤ -1 (with a ∈ A) under F0 are ( , ε)-separated and since the diameter of each rectangle of the partition is small compared to ε, the sets Ra (a ∈ A) are pairwise disjoint compact subsets of Ri having their images by F 0 included in Rj. Up to adding a constant (independent of large) to , using the properties of Markov partitions, we can suppose moreover that Ri and Rj are equal. We denote this set by R in the following and thus the sets Ra are non empty pairwise disjoint compact subsets of R with images under F 0 included in R. This does not change the number N ε, of such sets Ra and so inequality ( ) is still true. For every non empty finite word

β = β0 • • • βp ∈ A p+1 , we now set R β := R β 0 ∩ F - 0 (R β 1 ) ∩ • • • ∩ F -p 0 (R βp ) .
which is a non empty compact set by the properties of Markov partitions, and the sets R β among β ∈ A p+1 are pairwise disjoint subsets of R for a fixed value of p.

Finally, for a ∈ -→ A , we define:

Ra := n>0 Ra |n
which is also a non empty compact subset of R, included in K0, and the sets Ra (a ∈ -→ A ) are pairwise disjoint. Finally, we notice that since the partition admits a continuation in a neighborhood of 0 in P, the sets Ra, R β and Ra admit continuations Rp,a, R p,β and Rp,a with the same dynamical properties when p varies in a neighborhood of 0 in the parameter space. Up to taking a finite covering of P by such neighborhoods and extending a finite number of times (Fp)p in Steps 2-3-4, we can suppose that Rp,a, R p,β and Rp,a vary continuously in a neighborhood of P, that we can suppose equal to P up to reducing it, and inequality ( ) is satisfied in P.

Let us take a small ε > 0. For any infinite forward sequence a ∈ -→ A and any p ∈ P , we notice that all the points in Rp,a are asymptotic (using the hyperbolicity) and thus belong to a same stable manifold W of Kp (which is one-dimensional). We define W a p as the ε -neighborhood in W of the maximal arc of W bounded by points of Rp,a. We parameterize W a p with X := [-1, 1] via a C r -map s a p s.t. (s a p )p depends Hölder in the C r-1 -topology on a by Remark 7.0.3.

Since Fp is a local diffeomorphism and since W a p is injectively immersed, up to decreasing ε , the restriction of F p to each W a p is a diffeomorphism satisfying:

∀p ∈ P , ∀a ∈ -→ A , F p (W a p ) ⊂ W σ(a) p := s σ(a) p ((-1, 1))
by hyperbolicity. We define the C r -diffeomorphism fp,a := (s σ(a) p

) -1 • F p • s a p on a small neighborhood X of X independent of (p, a). Its image is s.t. fp,a(X ) X. The differential DF p strictly contracts (resp. expands) E s p,z (resp. E u p,z ) by a factor λ < λ < 1 (resp. 1/λ) uniform over p ∈ P and z ∈ Kp. Up to modifying F p outside a neighborhood of Kp (this operation does not modify the local stable/unstable sets of Kp in a neighborhood of Kp nor jets of points inside it), we can suppose that F p contracts strictly each W a p by λ and so each map fp,a contracts by < λ. Moreover the map (p, x) → fp,a(x) is C r . It extends on P × X , up to reducing P and X . Easy computations show that the map a → ((p, x) → fp,a(x)) is Hölder for the C r-1 -topology and continuous for the C r -topology. We set

Fp : (a, x) ∈ -→ A × X → (σ(a), fp,a(x)) ∈ -→ A × X ,
and (Fp)p is a family of fiberwise λ-contracting C r -skew-products satisfying the preliminary assumptions of Theorem C. We now show: Lemma 6.0.1. The similarity dimension ∆(p) is larger than δ d,s for any p ∈ P.

Proof. The fact that the similarity dimension does not depend on the fiber is a consequence of Proposition 5.1.4. Moreover for a fixed fiber a, there are N ε, contractions fp,a which contract by at most m(DFp) . Thus the result follows by the definition of the similarity dimension and inequality ( ).

Using backward sequences instead of forward sequences, we can define similarly families (W α p )p of local unstable manifolds of Kp parameterized by maps s α p s.t. the map F p restricted to some subset of each W σ(α)

p := s σ(α) p ((-1, 1)) is a diffeomorphism onto W α
p which expands strictly by 1/λ. Moreover the local unstable manifold W α p intersects the local stable manifold W a p at a unique point. Remark 6.0.2. For any C r -family (Fq)q∈Q of endomorphisms which extends (Fp)p∈P (by this, we mean that Q is a neighborhood of P), by hyperbolic continuation, we can extend (Kp)p∈P into the continuation (Kq)q∈Q of a hyperbolic basic set of stable dimension 1, up to reducing Q. Thus we can also extend the local stable and unstable manifolds (using Theorem F in the Appendix) and so the family (Fp)p into a family (Fq)q of fiberwise λ-contracting C r -skew-products.

In the following, we prove Theorem B in the case = 1. The proof is the same when > 1, with a variant in the proof of Proposition 6.0.3 described in Remark 6.0.6.

Step 2: Maps acting on jets. From (Fp)p, we define a family of C 2 -skew-products (Gp 0 )p 0 acting on s-jets at any p0. We begin by performing this Step for d = 1 for the sake of simplicity concerning the notations, and then treat the general case d ≥ 2.

For any p0 ∈ P and a ∈ -→ A , we define from (fp,a)p the following map gp 0 ,a acting on s-jets: ,a(x) and to apply the multidimensional Faà di Bruno formula (see for example Theorem 2.1 in [CS]).

gp 0 ,a : (xp,
We recall that by assumption we have 0 < |Dfp 0 (xp 0 )| < λ < 1. Then for any p0 ∈ P and a ∈ -→ A , the map gp 0 ,a has inferior unipotent differentials with eigenvalues uniformly bounded between 0 and 1 in modulus.

We recall that δ1,s = s + 1. We define a set of s-jets Y , identified with a subset of R s+1 as follows (the term of degree i corresponding to the i th -coordinate). We set:

Y := X × [-R1, R1] × • • • × [-Rs, Rs] .
We can choose R1 large compared to the diameter of X and Ri+1 large compared to Ri for 1 ≤ i ≤ s-1. Since its differentials are unipotent with non zero contracting diagonal coefficients, an immediate induction shows that gp 0 ,a is a C 2 -diffeomorphism from a small neighborhood Y of Y onto gp 0 ,a(Y )

Y . Up to rescaling, we can suppose that Y := [-1, 1] s+1 . The map a → ((p, x) → fp,a(x)) is Hölder for the C r-1 -topology and continuous for the C r -topology and we have s ≤ r -2. This implies that the map a → ((p0, y) → gp 0 ,a(y)) is Hölder for the C 1 -topology and continuous for the C 2 -topology. To summarize, we just prove that (Gp 0 )p 0 , with

Gp 0 : (a, y) ∈ -→ A × Y → (σ(a), gp 0 ,a(y)) ∈ -→ A × Y ,
is a family of C 2 -skew-products satisfying the preliminary assumptions of Theorem C, (U) and ∆(p0) > s + 1 = δ1,s for any p0 ∈ P.

As already said, we get the same result with d ≥ 2 but with painful notations. Indeed, it is enough to remark that the action of (fp,a)p on jets of multiorder (k1, . . . , k d ) with i ki ≤ s only depends on jets of the same multiorder, with a linear coefficient Dfp 0 (xp 0 ), and on jets of multiorder (k 1 , . . . , k d ) with i k i < i ki.

In particular, its differentials still satisfy assumption (U).

Step 3: Extending the family. We now extend (Gp 0 )p 0 to satisfy the transversality assumption (T) inside a larger family. But we have to ensure that this extension comes from an extension of (Fp)p, itself coming from an extension of (Fp)p.

Proposition 6.0.3. There exists a family of C 2 -skew-products (Gq 0 )q 0 ∈Q:

Gq 0 : (a, y) ∈ -→ A × Y → (σ(a), gq 0 ,a(y)) ∈ -→ A × Y ,
with Q := P × (-1, 1) m for some m > 0, satisfying the preliminary assumptions of Theorem C, (U), (T), ∆(q0) > δ d,s for q0 ∈ Q and:

G (p 0 ,0) = Gp 0 and g (p 0 ,0),a = gp 0 ,a for every p0 ∈ P and a ∈ -→ A .

Moreover there exists a C r -family (Fq)q∈Q of local diffeomorphisms extending (Fp)p∈P s.t. if (Fq)q = (fq,a)q is its associated family of C r -skew-products, we have f (p 0 ,0),a = fp 0 ,a and gq 0 ,a is the map acting on the s-jets at p0 derived from (fq,a)q (here the jets at p0 are taken varying p for fixed q 0 where q0 := (p0, q 0 )). The family (Fq)q∈Q is of the form Fq = F (p,q ) = Fp + Σ p,q where (Σ p,q ) (p,q ) is a C r -family s.t. Σp,0 = 0.

We now finish the proof of Theorem B. We postpone the proof of Proposition 6.0.3 after it since it is technical.

Step 4: Conclusion. Let us pick the family (Gq 0 )q 0 of C 2 -skew-products satisfying the preliminary assumptions of Theorem C, (U), (T) and ∆(q0) > δ d,s for every q0 ∈ Q given by Proposition 6.0.3. We will pick well-chosen ϑ-U-perturbations and apply the second part of Theorem C. We will conclude by using the Fubini Theorem.

We pick the continuation (kq)q of a point kq ∈ Kq at the intersection of local stable and unstable manifolds W a q and W α q for arbitrary a ∈ -→ A and α ∈ ← -A . Up to working locally in the parameter space Q, we can pick a C r -family (Γt,q)t,q of segments of same direction u Γ (these segments remain all parallel when varying t and q), which do not intersect W s (Kq) and which intersect (W α q )t,q in a curve (zt,q)t,q C 0 -close to (kq)t,q. Moreover we choose (Γt,q)t,q s.t. the v Γ -coordinate X (q, t) in the basis (u Γ , v Γ ) of Γt,q, for a fixed direction v Γ transversal to Γt,q, is of the form

X (q, t) := |i|≤s tip i for t = (t1, • • • , t δ d,s ) ∈ T := (-1, 1) δ d,s
which does not depend on q but only on p for q = (p, q ). In particular, for any q0 ∈ Q, the map t ∈ T → J s q 0 X (q, t) sends diffeomorphically T to a non empty set of s-jets in p (and thus a set of s-jets in p of positive δ d,s -dimensional measure).

By the parametric inclination Lemma 7.0.4 (in the Appendix), up to taking an inverse iterate, for any t ∈ T , we can suppose that (Γt,q)q is (uniformly in t ∈ T ) close to the continuation (W a q )q of a local stable manifold for some a ∈ -→ A . We can parameterize it by the segment X with a C r -family of charts C r -close to (s a q )q. Still iterating backwards, for every a ∈ A, there is a family of submanifolds (Γ a t,q )q close to (W aa q )q (uniformly in t ∈ T ) s.t. Γ a t,q is sent by Fq into the interior of Γt,q. We can parameterize (Γ a t,q )q by X with a family of charts close to (s aa q )q and in these charts the restriction of Fq from Γ a t,q into Γt,q defines a family of maps ( ft,q,aa)q C r -close to (fq,aa)q. Iterating backwards by induction, we can define for every α ∈ A * families of curves (Γ α t,q )q which are (uniformly in α) C r -close to (W αa q )q by Lemma 7.0.4 and s.t. each Γ aα t,q is sent into the interior of Γ α t,q . This defines families of maps ( ft,q,αa)q C r -close to (fq,αa)q (uniformly in t ∈ T and α ∈ A * ). We set ft,q,b := f q,b when b is not of the form αa with α a non empty word. When looking at the action of the families of maps ( ft,q,a)q on s-jets (w.r.t. p) for fixed values of t and a, one obtains a family of maps (gt,q 0 ,a)q 0 from Y into itself C 2 -close to (gq 0 ,a)q 0 (uniformly in t and a). This defines a family (indexed by t) of ϑ-U-perturbations of the family (Gq 0 )q 0 of C 2 -skew-products, with arbitrarily small ϑ.

By the second part of Theorem C, we have Leb δ d,s ( Kt,q 0 ,a) > 0 for Leb d+m a.e. q0 ∈ Q and Leb δ d,s a.e. t ∈ T , where the limit set Kt,q 0 ,a is formed by jets at p0 taken while varying p for fixed q 0 where q0 := (p0, q 0 ). By the Fubini Theorem, for a.e. q 0 we have: Leb δ d,s ( Kt,(p 0 ,q 0 ),a ) > 0 for Leb d a.e. p0 ∈ P and for Leb δ d,s a.e. t ∈ T .

We fix such a q 0 . We consider ( Fp)p := (F (p,q 0 ) )p and Γt,p := Γ t,(p,q 0 ) . We notice that Kt,(p 0 ,q 0 ),a is (in the charts) the set of the s-jets at p0 of the intersection points between the local unstable set of Kp and Γt,p. Moreover the set of s-jets at any p0 of the v Γ -coordinate of Γt,p in the basis (u Γ , v Γ ) when varying t in T has positive δ d,s -dimensional Lebesgue measure. Since we have a positive set of one-dimensional s-jets in the direction of u Γ for a.e. t ∈ T , we just have to use the Fubini Theorem to conclude to a set of bidimensional s-jets of positive measure for a.e. p0 ∈ P.

The same proof works for every family (Gp)p which is C r -close to (Fp)p with the extension (Gq)q∈Q given by Gq = Gp + Σ p,q where (Σ p,q ) (p,q ) is the C r -family given by Proposition 6.0.3. Indeed, the preliminary conditions of Theorem C, (U), ∆(q0) > δ d,s are open conditions and the extension in Proposition 6.0.3 in order to get (T) works for nearby families with the same additive perturbation since having a relative positive speed is an open property. Thus we can apply Theorem C to the family of C 2 -skew-products derived from (Gq)q. This achieves the proof that (Kp)p is an almost C r,s -parablender and thus shows Theorem B. Remark 6.0.4. When the family (Fp)p is of the form (F)p with F independent of p and when the order s of the jets is equal to 0, the extension (Fq)q satisfying the conclusions of Theorem B can be taken of the form (F + Σ q ) (p,q ) with (Σ q ) q independent of p. This will help to prove Theorem A from Theorem B.

Proof of Proposition 6.0.3. The proof is divided in two Steps. The goal and main difficulty of the extension is about satisfying (T): we want to extend the family (Fp)p∈P by adding finitely many new parameters, which will give a positive relative speed to pairs of limit points with different combinatorics, inside each fiber. We set

Ω := {(a, α, β) ∈ -→ A × ← - A × ← - A : α-1 = β-1}
and by introducing new parameters, we will give a positive relative speed to the points coded by α and β inside the fiber coded by a for any (a, α, β) ∈ Ω, and this will give the transversality assumption (T) inside the extended family.

We begin by choosing a covering of Ω by small products of cylinders. The reason why we are going to work with a covering by small products of cylinders is that this will allow us to control precisely the relative movement of the two limit points. We will then extend iteratively the original family (Fp)p∈P by adding new parameters for each set of the covering.

We first pick an arbitrary number h > 0. The extended family (Fq)q∈Q will be taken s.t. the families (F (p,q ) )p∈P are (uniformly in q ) h-C r -close to (Fp)p∈P . We pick an integer M large enough so that λ M /(1 -λ) is small.

Step 1: Working locally. Let us pick any (a, α, β) ∈ Ω. If a is periodic of minimal period p ∈ A * , since α-1 = β-1, we can choose β s.t. β-1 is different from the last letter of p.

Fact 1. The sequences σ k (a) where 0 ≤ k ≤ M are all distinct from β-1a.

Fact 2. The sequences β |k a where 2 ≤ k ≤ M are all distinct from β-1a.

Proof of Facts 1 and 2. If a is not periodic, this is immediate. If a is periodic, this is due to the fact that β-1 is different from the last letter of its minimal period p.

From now, we distinguish two disjoint Cases which cover all possible tuples in Ω: We take these cylinders so that their lengths are large compared to M .

We notice that the compact set Ω is covered by the union of the (open) products of cylinders [ρa] × [ρα] × [ρ β ] associated to any (a, α, β) ∈ Ω. We can find then a finite covering of Ω by such products.

Step 2: Extension. We now construct an extension of (Fp)p obtained by extending successively a finite number of times the family, adding at each step δ d,s parameters corresponding to a product of cylinders

[ρ] × [ρ ] × [ρ ] ⊂ Ω in the finite covering defined in Step 1. Here [ρ] is a cylinder in -→ A and [ρ ] and [ρ ] are cylinders in ← - A s.t. ρ -1 = ρ -1 .
The δ d,s parameters are intended to move the s-jet of the limit point associated to α relatively to the one corresponding to β, inside the fiber encoded by a, where (a, α, β) is any tuple in

[ρ] × [ρ ] × [ρ ].
The union of the points of Rp,a on the local stable manifolds W a p for a ∈ [ρ -1 ρ] is disjoint from the union of the points of Rp,a on the local stable manifolds W a p for a / ∈ [ρ -1 ρ] at every parameter p ∈ P. We then pick a C r -family (hp)p of bump functions hp equal to h in a neighborhood of the first ones and equal to 0 in a neighborhood of the second ones for every p ∈ P. We also pick a C r -family of maps v s p s.t. v s p (z) is close to the stable direction of Kp at Fp(z) for any point z ∈ Kp.

We now extend the family by setting for every q := (p, q ) with p ∈ P and q = (qi)i ∈ R δ d,s small:

Fq(z) = Fp(z) + hp(z) • i qi • p i • v s p (z) ∀z ∈ R 2 , ( 24
)
where we sum over i = (i1, . . . , i d ) s.t.

k i k ≤ s with p i = p i 1 1 • • • p i d d .
This defines a C r -family of endomorphisms (Fq)q. For small values of q , let us say for q in some open neighborhood of P × {0}, these are still local diffeomorphisms. Moreover, the family of hyperbolic basic sets admits a continuation as a family (Kq)q (see the Appendix).

By Remark 6.0.2, we can define the associated family of C r -skew-products (Fq)q which extends (Fp)p and then the associated map (Gq 0 )q 0 acting on the s-jets derived from (Fq)q, defined as in Step 2 of the proof of Theorem B (let us recall that the jets are taken while varying only p for fixed q if we set q := (p, q )). In particular, (Gq 0 )q 0 is a family of C 2 -skew-products satisfying the preliminary assumptions of Theorem C and also (U). By Proposition 5.1.4, the map q → ∆(q) is continuous. Thus, up to restricting the parameter space, we have ∆(q0) > δ d,s for any q0.

We now prove that (Gq 0 )q 0 satisfies the property (T) restricted to any (a, α, β) ∈

[ρ] × [ρ ] × [ρ ]
. More precisely we show below the following technical lemma:

Lemma 6.0.5. Up to reducing Q, for every a ∈ [ρ], α ∈ [ρ ], β ∈ [ρ ]
, p0 ∈ P and r > 0, with q := (p, q ), the set of q 0 s.t. J s p 0 πq,a(α) and J s p 0 πq,a(β) are r-close is of Lebesgue measure dominated by r δ d,s , with an independent constant. Moreover for every family of ϑ-perturbations of (Gq 0 )q 0 with ϑ small enough, t ∈ T , a ∈ [ρ], α ∈ [ρ ], β ∈ [ρ ], p0 ∈ P and r > 0, the set of q 0 s.t. J s p 0 πt,q,a(α) and J s p 0 πt,q,a(β) are r-close is of Lebesgue measure dominated by r δ d,s , with the same constant.

We recall that the s-jets at p0 are taken while varying p around p0 for a fixed value of q equal to q 0 . In particular, by the Fubini Theorem, the set of (p0, q 0 ) s.t. J s p 0 πq,a(α) and J s p 0 πq,a(β) are r-close is also of measure dominated by r δ d,s and the same holds for ϑ-perturbations. The proof of Lemma 6.0.5 is below. We first finish the proof of Proposition 6.0.3. We extend iteratively the initial family (Fp)p a finite number of times by the same method. At each step, we extend by adding δ d,s new parameters corresponding to a new product of cylinders [ρ] × [ρ ] × [ρ ] in the finite covering of Ω defined in Step 1. The adaptation of Lemma 6.0.5 is straightforward. This proves property (T) and so (Gq 0 )q 0 ∈Q is a family of C 2 -skew-products, with Q := P × (-1, 1) m for some m > 0 (up to rescaling). This concludes the proof.

Proof of Lemma 6.0.5. We first prove the result for the Case 1 in Step 1 of the proof of Proposition 6.0.3. Let Πq,a(α) and Πq,a(β) be the points of the phase space R 2 equal to πq,a(α) and πq,a(β) in the parametrization of W a q . Both belong to W a q .

We pick local coordinates in a neighborhood of Π0,a(α) centered at Π0,a(α) with a basis given by R • e s 0 + R • e u 0 , where we denoted by R • e s 0 and R • e u 0 the stable and unstable directions of ← → K0 at Π0,a(α). We write:

J s p 0 Πq,a(α) =: J s p 0 Πq,a(α) • e s 0 + J u p 0 Πq,a(α) • e u 0 .
Take care that the s on the left hand term is the order of the jet and the s on the right hand term just means stable . We can proceed similarly for Π0,a(β).

1. Easy case: 0-jets. We first perform the proof for 0-jets to show the general idea. We fix the parameter p0 = 0. We begin by studying Πq,a(α), more precisely the variations of the 0-jet of Π (0,q ),a (α), i.e. Π (0,q ),a (α) itself, when moving q . In order to do this, we set Π0(q ) := Π (0,q ),a (α)

and denote by Π k (q ) its preimage by F k (0,q ) on W α |k a (0,q ) (this is the point equal to π (0,q ),α |k a (σ k (α)) in the parametrization of the stable manifold). We pick local coordinates centered at each P k := Π k (0) with a basis given by the corresponding preimages R • e s k and R • e u k of R • e s 0 and R • e u 0 by F k 0 for k > 0. These are the stable and unstable directions of ← → K0 at P k . In the decomposition R • e s k + R • e u k , we write

Π k (q ) =: Π s k (q ) • e s k + Π u k (q ) • e u k .
In the coordinates given by P k+1 + R • e s k+1 + R • e u k+1 and P k + R • e s k + R • e u k with P k+1 = P k = (0, 0), the map F0 restricted to a neighborhood of P k+1 = (0, 0) sends P k+1 = (0, 0) to P k = (0, 0) and is C 1 -close to its differential which is diagonal. In particular, the (1, 1)-coefficient is a real number λ k s.t. |λ k | < λ < 1.

By hyperbolic continuation, there exists B > 0 independent of k and M s.t.:

C k := dΠ s k dq (0) < B .
We recall that Π k+1 (q ) is sent onto Π k (q ) by F (0,q ) . Moreover F (0,q ) (Π k+1 (q )) is the sum of F0(Π k+1 (q )) and a term Σ k (q ) = Σ s k (q ) • e s k + Σ u k (q ) • e u k coming from Eq. ( 24). In particular, we notice that we have:

dΣ s k dq (0) < 2h .
We have:

dΠ s k dq (0) = λ k • dΠ s k+1 dq (0) + dΣ s k dq (0) . C k < λ k • C k+1 + 2h .
We also notice that Σ k (q ) is equal to zero when 0 ≤ k ≤ M -1 by the first item of Case 1. This gives for every M > M :

C0 < B • λ M + 2h • (λ M -1 + • • • + λ M ) .
By taking M large, we get:

C0 ≤ 2h λ M 1 -λ ,
which is small compared to h by assumption. We finally get that the derivative of Π (0,q ),a (α) = Π0(q ) at q = 0 in the stable direction R • e s 0 is small compared to h: d dq Π s (0,q ),a (α) is small compared to h .

The same holds true when replacing Π (0,q ),a (α) by the preimage Π 1 (q ) of Π (0,q ),a (β) by F (0,q ) on W β -1 a (0,q ) , using the second item of Case 1. By Eq. ( 24) and since e s 0 is close to v s 0 (Π 1 (0)), the derivative d dq Π s (0,q ),a (β) is then close to h. Thus:

d dq Π s (0,q )
,a (α) -Π s (0,q ),a (β) is bounded away from (0, 0) .

In particular, up to reducing Q, the set of q 0 s.t. J 0 0 Πq,a(α) and J 0 0 Πq,a(β) are rclose is of Lebesgue measure dominated by r with an independent constant. We can proceed the same way for any p0, and the domination constant is independent. Integrating using the Fubini Theorem, the set (p0, q 0 ) s.t. J 0 p 0 Πq,a(α) and J 0 p 0 Πq,a(β) are r-close is of Lebesgue measure dominated by r, and thus it is also the case for J 0 p 0 πq,a(α) and J 0 p 0 πq,a(β). The statement about ϑ-perturbations follows easily with the same arguments when ϑ is small.

General

Case: s-jets. Now we go to the general but more difficult case of s-jets. This time, we vary also the parameter p. We fix p0 = 0. We investigate the differentials of J s 0 Πq,a(α) and J s 0 Πq,a(β) when derivating relatively to q .

We keep the same coordinates for each k: we keep expressing the point Π k (p, q ) (depending also on p this time) in the coordinates P k + R • e s k + R • e u k (independent of (p, q ) small). Let J s 0 Π k (q ) and J u 0 Π k (q ) be the components in this basis of the s-jet of Π k (p, q ) at p0 = 0 for q fixed. Note that by hyperbolic continuation, there exists B > 0 independent of k and M s.t.:

C k := dJ s 0 Π k dq (0) < B .
For q fixed the family (Π k (p, q ))p is the image of (Π k+1 (p, q ))p by the family of maps (F (p,q ) )p. The family (F (p,q ) (Π k+1 (p, q )))p is the sum of (Fp(Π k+1 (p, q )))p and of the family (Σ p,k )p of perturbations coming from Eq. ( 24). In particular:

dJ s 0 Σ k,p dq (0) 2δ d,s h .
Again Σ k,p (q ) is equal to zero when 0 ≤ k ≤ M -1 by the second item of Case 1. However this time the action of (Fp)p on s-jets is more complicated than on 0-jets. For each k, we write the Taylor expansion of Fp in P k+1 of order s. Then we replace each of its coefficients by its s-jet in p, and the variables by J s 0 Π k+1 (q ) and J u 0 Π k+1 (q ), and we expand this expression. This shows that

J s 0 Π k (q ) = M k • J s 0 Π k+1 (q ) + M k • J u 0 Π k+1 (q ) + J s 0 Σ k,p (q ) .
The term J s 0 Σ k,p (q ) has its derivative at q = 0 bounded by 2δ d,s h, and its coefficients in p i depends only on qi. The matrix M k is inferior triangular with all its diagonal coefficients equal to λ k with |λ k | < λ < 1. On the other hand, the matrix M k is inferior triangular with all its diagonal coefficients equal to 0. Iterating, using the first item of Step 1 and then taking the derivative at q = 0, we see that

d dq J s 0 Π (p,q ),a (α)
is inferior triangular with all its diagonal coefficients bounded by

2δ d,s h λ M 1 -λ ,
which is small compared to h by assumption on M . Thus for every i = (i1, . . . , i d ) s.t. k i k ≤ s, the coordinate in p i of the stable component of the s-jet J s 0 Πq,a(α) has small derivative compared to h while moving the parameter qi around 0.

The same holds for the preimage Π 1 (p, q ) of Πq,a(β) by Fq. By Eq. ( 24) and since R • e s 0 is close to R • v s 0 (Π 1 (0, 0)), for every i = (i1, . . . , i d ) s.t. k i k ≤ s, the coordinate in p i of the stable component of the s-jet J s 0 Πq,a(β) has a derivative close to h (up to a non zero independent multiplicative constant) while moving the parameter qi. Thus the coordinate in p i of J s 0 Πq,a(α) -J s 0 Πq,a(β) has a non zero derivative while moving qi around 0. The same still holds for other values of p0. Then, up to reducing Q, the set of (p0, q 0 ) s.t. J s p 0 Πq,a(α) and J s p 0 Πq,a(β) are r-close is of Lebesgue measure dominated by r δ d,s , and thus it is also the case for J s p 0 πq,a(α) and J s p 0 πq,a(β). This gives the result, using the Fubini Theorem. The statement about ϑ-perturbations follows easily with similar arguments. This ends the proof in Case 1.

In Case 2, the proof is more simple. First take the original sequences α, β and a of Case 2 around which the cylinders [ρα], [ρ β ] and [ρa] are centered. The coordinate in p i of the stable component of the s-jet J s 0 Πq,a(β) has still a non zero derivative while moving the parameter qi (this only needs the second item of Case 2 and not the first item of Case 1 not present in Case 2). On the other hand, Πq,a(α) is the image of Πq,a(β) by some iterate of Fq restricted on the M first images of W a q since a is periodic of period p and the last letters of α are of the form β-1 • p f for some f > 0. These local stable manifolds do not depend on q by the first item of Case 2. Moreover the action of (Fq)q restricted to these stable manifolds on stable components of jets is inferior triangular with diagonal coefficients between 0 and 1. Thus the coordinate in p i of J s 0 Πq,a(α) -J s 0 Πq,a(β) has again a non zero derivative while moving qi. Since the lengths of the cylinders are large, this remains true for any (α, β, a) in [ρα]×[ρ β ]×[ρa]. We conclude as in Case 1, which ends the proof. Remark 6.0.6. When = 1, to give a relative movement inside the a-fiber to the points encoded by α and β, we considered the preimages of these two points by Fp (respectively on W α -1 a p and W β -1 a p

). The second one was not periodic and distinct from the first one. We then perturbed Fp in a neighborhood of this second preimage.

In the case when > 1, we shall look at the respective successive preimages of these two points by Fp (respectively on W α -1 a p and W β -1 a p and their -1 successive images by Fp) and take the first pair of preimages which are distinct. One of them is not periodic, and we perform the same perturbation as before in a neighborhood of this point. Then the proof is the same with the same distinction in two cases whether the M successive preimages of the other preimage intersect this neighborhood or not.

Appendix

Proofs of Intermediate Lemmas 5.1.2, 5.3.1 and 5.4.4

Proof of Lemma 5.1.2. Let us fix p ∈ P, a ∈ -→ A , x ∈ X and α = (α-n, • • • , α-1) ∈ A n for some n > 0. We notice that the differential Dψ α p,a (x) can be written as the product of n factors:

Dψ α p,a (x) = n k=1 Dψ α -k p,a k ψ α -k-1 p,a k+1 • • • • • ψ α -n p,an (x) with a k := α |k-1 a (25)
By (U), each of these n factors is unipotent inferior and then can be written as a sum of N terms:

Dψ α -k p,a k ψ α -k-1 p,a k+1 • • • • • ψ α -n p,an (x) = M0 + M1 + • • • + MN-1 . ( 26
)
Here M0 is a diagonal matrix which has all its diagonal coefficients of absolute value equal to the coefficient λp,a k ,α -k ψ

α -k-1 p,a k+1 • • • • • ψ α -n
p,an (x) . On the other side, for every 1 ≤ k ≤ N -1, all the coefficients of M k are equal to 0 except possibly on the k th (small) diagonal line below the (great) diagonal line.

We can write each of the n factors of Eq. ( 25) as in Eq. ( 26) and then expand Dψ α p,a (x) as a sum of N n factors of n terms. Among them, any such product with ≥ N matrices having zero coefficients on and above the great diagonal line vanishes. Thus, between the N n terms whose sum equals Dψ α p,a (x), there are at most

P (n) := n N -1 • (N -1) N -1 + n N -2 • (N -1) N -2 + • • • n 1 • (N -1) + 1 (27)
which are non zero, with P polynomial. Each of these ≤ P

(n) terms M1 • • • Mn is a product of n factors M k .
Let us consider such a product M1 • • • Mn. At most N -1 of the M k have all their coefficients equal to 0 except possibly on one of the small diagonal lines below the (great) diagonal line. The ≥ n -(N -1) other factors are all diagonal matrices. Each coefficient of the resulting product M1 • • • Mn is then either zero or equal to the product of n non zero coefficients c k , with c k a non zero coefficient of

M k . If M k is diagonal, |c k | is equal to λp,a k ,α -k ψ α -k-1 p,a k+1 • • • • • ψ α -n
p,an (x) . If not, |c k | is bounded by some independent constant C1 since for every a ∈ A, the C 1 -bounded map ψ a p,a depends continuously on p ∈ P and a ∈ -→ A . By Eq. ( 13), we have:

λp,a,α(x) = n k=1 λp,a k ,α -k ψ α -k-1 p,a k+1 • • • • • ψ α -n p,an (x) with a k := α |k-1 a . ( 28
)
Moreover, each of the coefficients λp,a k ,α -k ψ

α -k-1 p,a k+1 •• • ••ψ α -n p,an (x) is larger than γ . Thus the resulting coefficient of M1 • • • Mn is smaller than λp,a,α(x)•(γ ) -N +1 •C N -1 1 in modulus. Thus any coefficient of Dψ α p,a (x) is bounded by P (n) • λp,a,α(x), where P (n) := P (n) • (γ ) -N -1 • C N -1 1 is a positive polynomial on R+.
Proof of Lemma 5.4.4. For every p ∈ P, a ∈ -→ A , n ≥ 0, ρ ∈ A n and (α, β) ∈ Cρ, the points πp,a(α) and πp,a(β) are the respective images of πp,ρa(σ n (α)) and πp,ρa(σ n (β)) by ψ ρ p,a . Let us denote by v = (v1, • • • , vN ) the vector v := πp,ρa(σ n (α))-πp,ρa(σ n (β)). We denote by j ∈ {1, • • • , N } the maximal index s.t. |vj| > 2N • P (n) • |vi| for every i < j, where the polynomial P was defined in Lemma 5.1.2. Using this, it follows:

|vj| ≥ F • (2N P (n)) -N • ||v|| (29)
for some positive constant F (depending only on N ). The segment between the two points πp,ρa(σ n (α)) and πp,ρa(σ n (β)) is fully included in X since X is convex. Let χ : [0, 1] → R be the C 1 -map which sends x ∈ [0, 1] to the j th coordinate of ψ ρ p,a (πp,ρa(σ n (β)) + xv). By the mean value equality, there exists x ∈ (0, 1) s.t.:

χ(1) -χ(0) = χ (x) = j i=1 aj,i • vi (30)
where aj,i is the coefficient of index (j, i) of the differential Dψ ρ p,a (y) with y := πp,ρa(σ n (β)) + xv. The right-hand equality is due to the fact that Dψ ρ p,a (y) is unipotent inferior by (U). We notice that |aj,j| = λp,a,ρ(y). By Lemma 5.1.2, |aj,i| is smaller than P (n) • λp,a,ρ(y). By Eq. ( 30), we then have:

|χ(1)-χ(0)| ≥ λp,a,ρ(y)•|vj|- i<j P (n)•λp,a,ρ(y)• |vj| 2N • P (n) ≥ λp,a,ρ(y) • |vj| 2 . (31)
Noticing that ||πp,a(α) -πp,a(β)|| ≥ |χ(1) -χ(0)| and injecting Eq. ( 29) in Eq. ( 31), it then holds:

||πp,a(α) -πp,a(β)|| ≥ F 2 • (2N P (n)) -N • λp,a,ρ(y) • ||πp,ρa(σ n (α)) -πp,ρa(σ n (β))|| . (32) By Lemma 5.2.1 and noting R(n) := 2D 1 F • (2N P (n)) N which is positive on R+, we have: ||πp,a(α) -πp,a(β)|| ≥ Λp,a,ρ R(n) • ||πp,ρa(σ n (α)) -πp,ρa(σ n (β))|| . ( 33
)
The proof of the second item is similar and we apply Lemma 5.2.4 to conclude.

Proof of Lemma 5.3.1. We notice that the coefficient λp,a,α -1 πp,α -1 a(σ(α)) is positive and uniformly distant from 0 and +∞. Since log is C 1 on ]0, +∞[, we just have to show that the following map is Hölder with positive exponent on its domain:

(α, a) → λp,a,α -1 πp,α -1 a(σ(α)) .

Let us recall that the latter is the (1, 1) coefficient of Dψ

α -1
p,a (πp,α -1 a(σ(α))) (up to the sign). By assumption the map a ∈ -→ A → Dfp,a is Hölder for the C 0 -topology and so it is enough to show that the map (α, a) ∈ ← -A × -→ A → πp,a(α) ∈ X is itself Hölder. By hyperbolicity, the map α ∈ ← -A → πp,a(α) ∈ X is Hölder for any a ∈ -→ A , with exponent and constant independent of a. Thus it is enough to show that the map a ∈ -→ A → πp,a(α) ∈ X is Hölder for any α ∈ ← -A , with independent constants. But using again both the hyperbolicity and that a → fp,a and a → Dfp,a are Hölder for the C 0 -topology, we see that the maps a → ψ αn p,a (0) are Hölder, with exponent and constant independent of p, α and n. But this sequence converges uniformly to the map a → πp,a(α), which concludes the proof.

Proof of Distortion Lemmas 5.2.1, 5.2.2, 5.2.3 and 5.2.4

Proofs of Lemma 5.2.1. The Lemma will follow easily from the two following: Sublemma 7.0.1. There exists A > 0 s.t. for any p ∈ P, a ∈ -→ A , a ∈ A, it holds:

1 -A|x -y| ≤ λp,a,a(x) λp,a,a(y) ≤ 1 + A|x -y| ∀ x, y ∈ X .
Proof. The non zero number λp,a,a(x) is the (1, 1) coefficient of the differential Dψ a p,a (x) (up to the sign). Since the maps ψ a p,a are uniformly (in p, a and a) C 2 bounded, denoting by à a uniform bound of the second differential of ψ a p,a on X among p ∈ P, a ∈ -→ A , a ∈ A, the coefficient λp,a,a(x) is between λp,a,a(y) -Ã|x -y| and λp,a,a(y) + Ã|x -y|. We notice that γ < λp,a,a(y). Denoting A := Ã/γ and taking the quotient, we get the desired inequality. Sublemma 7.0.2. There exists A > 0 such that for every p ∈ P, a ∈ -→ A , n ≥ 0, α ∈ A n , x, y ∈ X, the points ψ α p,a (x) and ψ α p,a (y) are A • γ n/2 distant.

Proof. It is an immediate consequence of the inequality Λp,a,α < γ |α| and of Lemma 5.1.2 that the diameter of ψ α p,a (X) is dominated by γ |α|/2 .

We can now conclude. Using Eq. ( 13) we write both λp,a,α(x) and λp,a,α(y) as products of n factors and thus their quotients as:

λp,a,α(x) λp,a,α(y) = n k=1 λp,a k ,α -k ψ α -k-1 p,a k+1 • • • • • ψ α -n p,an (x) λp,a k ,α -k ψ α -k-1 p,a k+1 • • • • • ψ α -n p,an (y)
where we set again a k := α |k-1 a ∈ -→ A . Using the two Sublemmas, the previous quotient is between n k=1 (1-AA •γ n/2 ) and n k=1 (1+AA •γ n/2 ). Since 0 < γ < 1, the infinite products (1 ± AA • γ n/2 ) converge and their limits are respectively in (0, 1) and (1, +∞), which concludes the proof.

Proof of Lemma 5.2.2. Let us fix η > 0. By Sublemmas (7.0.1) and (7.0.2), there exist n0 ∈ N s.t. for every p ∈ P, a ∈ -→ A , n > n0, α ∈ A n , a ∈ A and x, y ∈ ψ α p,a (X), we have e -η < λp,a,a(x)/λp,a,a(y) < e η . We recall that the map ψ a p,a depends continuously in the C 2 -norm on a ∈ -→ A and p in P (both compact sets). Thus there exists δ(η) > 0 such that for every a ∈ -→ A and p1, p2 ∈ P with ||p1 -p2|| < δ(η), we have e -η < λp 1 ,a,a(x)/λp 2 ,a,a(y) < e η for every n > n0, α ∈ A n , a ∈ A, x ∈ ψ α p 1 ,a (X) and y ∈ ψ α p 2 ,a (X). We denote by D2 > 0 the maximum of the quotients λp 1 ,a,α(x)/λp 2 ,a,α(y) among p1, p2 ∈ P, a ∈ -→ A , n ≤ n0, α ∈ A n and x, y ∈ X. We conclude by noticing that for every n > n0 and x, y ∈ X, we have:

λp 1 ,a,α(x) λp 2 ,a,α(y) = λp 1 ,a,•••α -n 0 •••α -1 (x) λp 2 ,a,α -n 0 •••α -1 (y) • n k=n 0 +1 λp 1 ,a k ,α -k ψ α -k-1 p,a k+1 • • • • • ψ α -n p,an (x) λp 2 ,a k ,α -k ψ α -k-1 p,a k+1 • • • • • ψ α -n
p,an (y) . Proof of Lemma 5.2.3. Let us fix p ∈ P. We saw in the proof of Lemma 5.3.1 that (a, α) → λp,a,α -1 πp,α -1 a(σ(α))

is Hölder with positive exponent on its domain. We now fix an arbitrary β ∈ ← -A . Let us take a, a ∈ -→ A . By Eq. ( 13) and proceeding as in the proof of Lemma 5.2.1, we see that the quotient λp,a,α(πp,αa(β)) λ p,a ,α (π p,αa (β))

is bounded between 0 and +∞, with constants independent of p, a, a and α. To conclude, we just have to apply Lemma 5.2.1.

Proof of Lemma 5.2.4. By Sublemma (7.0.2), there exists n0 ∈ N s.t. for every p ∈ P, n > n0, ρ ∈ A n , a ∈ A and x, y ∈ ψ ρ p,a (X), we have λp,a,a(y) < λp,a,a(x) < λp,a,a(y) 1/ .

Then up to taking ϑ-perturbations for small ϑ, for every t ∈ T , p ∈ P, a ∈ -→ A , n > n0, ρ ∈ A n , a ∈ A, x ∈ ψ ρ t,p,a (X) and y ∈ ψ ρ p,a (X), we have λp,a,a(y) < λt,p,a,a(x) < λp,a,a(y) 1/ .

Let D4 > 0 be the maximum of λt,p,a,ρ(x)/λ p,a,ρ (y) and λ 1/ p,a,ρ (y)/ λt,p,a,ρ(x) among t ∈ T , a ∈ -→ A , p ∈ P, n ≤ n0, ρ ∈ A n , x, y ∈ X. We conclude as for Lemma 5.2.2.

Pressure function: Proof of Proposition 5.1.4

For simplicity, we fix p ∈ P and denote for every s ≥ 0: Zn(s) := α∈A n Λ s p,a,α > 0 .

For every s ≥ 0, the sequence n ∈ N+ → logZn(s) is subadditive and so the limit Πp,a(s) = limn→+∞ 1 n logZn(s) exists and is finite by Fekete's Lemma. We notice that Πp,a(0) is equal to the topological entropy of the shift σ which is positive since A has at least two letters. An immediate consequence of Lemma 5.2.3 is that the pressure Πp,a only depends on p. We denote it by Πp. We notice that s ∈ R+ → Zn(s) is log convex. Thus the map s ∈ R+ → logZn(s) is convex. The limit map s → Πp(s) is then also convex and thus continuous. We remark that for s, s ≥ 0, it holds: (34), this implies that Πp(s ) ≤ Π p (s) ≤ Πp(s/ ) for every s ≥ 0 and p ∈ U p, and so ∆(p)/ ≤ ∆(p ) ≤ ∆(p) for every p ∈ U p, . This proves the continuity of p → ∆(p).

Zn(s + s ) = α∈A n Λ s+s p,

Hyperbolicity theory

We recall here some background on hyperbolic compact sets of C r -endomorphisms. This subsection is mainly taken from Section 1 of the article [START_REF] Berger | Generic family with robustly infinitely many sinks[END_REF] of Berger and from Appendix C of the article [BB] of Berger and the author.

Let M be a manifold. A subset K ⊂ M is left invariant by a C 1 -endomorphim F from M into M if F(K) = K. When F is a diffeomorphism, the invariant compact set K ⊂ M is hyperbolic if there exists a DF-invariant splitting T M|K = E s E u so that E s is contracted by DF and E u is expanded by DF:

∃λ < 1, C > 0, ∀k ∈ K, ∀n ≥ 0, ||DF n |E s k || ≤ Cλ n and ||(DF n |E u k ) -1 || ≤ Cλ n .
When F is a local diffeomorphism, we shall study the inverse limit ← → K F of K: ← → K F := {(ki)i ∈ K Z : F(ki) = ki+1, ∀i ∈ Z} .

It is a compact space for the topology induced by the product one of K Z . The dynamics induced by F on ← → K F is the shift ← → F and is invertible. Let π : ← → K F → K be the zero-coordinate projection. Let π * T M be the bundle over ← → K F whose fiber at k is T π(k) M. The map DF acts canonically on π * T M as ← → F on the basis and as the linear map D π(k) F on the fiber of k ∈ ← → K F .

The Actually the definition of hyperbolicity for local diffeomorphisms is consistent with the definition of hyperbolicity for diffeomorphisms when the dynamics is invertible.

Here is a useful result about structural stability:

Theorem (Przytycki). Let K be a hyperbolic set for a C 1 -local diffeomorphism F of M. Then for every C 1 -local diffeomorphism F which is C 1 -close to F, there exists a continuous map i F : ← → K F → M which is C 0 -close to π and so that:

1. i F • ← → F = F • i F , 2. K F := i F ( ← → K F ) is hyperbolic for F .
Let us also recall the definition of a stable manifold in this context. For every k ∈ K and η > 0, we define the stable manifold and local stable manifold of k by: Remark 7.0.3. An immediate adaptation of the proof of Theorem C.5 in Appendix C of [BB] actually shows that the families (W s η (kp; Fp))p∈P and (W u η (k p ; Fp)p∈P depend Hölder for the C r-1 -topology on respectively k0 ∈ K0 and k 0 ∈ ← → K F 0 .

We will need the following parametric inclination lemma: Proof. The proof is similar to the one of the parametric inclination Lemma C.6 of [BB] but for inverse iterations this time: one extends Fp on a neighborhood U of U it in such a way that Γp is included in the stable manifold of some saddle point. Then we apply Theorem F.

  hF > dim E s • | log m(DF)| with m(DF) := min z∈K,u∈E s z ,||u||=1

Theorem C .

 . Let (Fp)p∈P be a family of C r -skew-products satisfying (U), (T) and ∆(p) > N for any p ∈ P. Then for every a ∈ -→ A , we have: LebN (Kp,a) > 0 for Leb d a.e. p ∈ P , and for every family F of ϑ-U-perturbations of (Fp)p∈P with small ϑ, it holds: LebN ( Kt,p,a) > 0 for Leb d a.e. p ∈ P and Lebτ a.e. t ∈ T .

Case 1 :

 1 the sequences α |k a where 1 ≤ k ≤ M are all distinct from β-1a. In this case, we consider the two cylinders [ρα] and [ρ β ] of ← -A of length M defined by ρα := α |M and ρ β := β |M . We pick a small cylinder [ρa] of -→ A s.t. a ∈ [ρa] and: • The cylinders [α |k ρa] where 1 ≤ k ≤ M are all disjoint from [β-1ρa]. • The cylinders [β |k ρa] where 2 ≤ k ≤ M are all disjoint from [β-1ρa]. Case 2: a is periodic of minimal period p and there exists f > 0 s.t. the last letters of α are of the form β-1 • p f . We consider two small cylinders [ρα] and [ρ β ] of ← -A containing α and β, and a small cylinder [ρa] of -→ A s.t. a ∈ [ρa] and: • The cylinders σ k ([ρa]) where 0 ≤ k ≤ M are all disjoint from [β-1ρa]. • The cylinders [β |k ρa] where 2 ≤ k ≤ M are all disjoint from [β-1ρa].

W

  s (k; F) := {k ∈ M : d(F n (k), F n (k )) -→ n→+∞ 0} . W s η (k; F) := {k ∈ M : η > d(F n (k), F n (k )) -→ n→+∞ 0} .For k ∈ ← → K F and η > 0, the unstable manifold and local unstable manifold of k are:W u (k, F) = {k 0 ∈ M : ∃(k i )i<0 s.t. F(k i-1 ) = k i and d(kn, k n ) -→ n→-∞ 0} . W u η (k, F) = {k 0 ∈ M : ∃(k i )i<0 s.t. F(k i-1 ) = k i and η > d(kn, k n ) -→ n→-∞ 0} .These sets are properly embedded C r -manifolds. For simplicity, we do not denote F in the article when it is obvious: W s (k) for example. Let r ≥ 1 and let M be a manifold. Suppose that (Fp)p is a C r -family of local diffeomorphisms Fp of M leaving invariant the continuation of a compact hyperbolic set Kp. Then there exists η > 0 s.t. the families (W s η (kp; Fp))p∈P and (W u η (k p ; Fp))p∈P of C rsubmanifolds are of class C r and depend C 0 on respectively k0 ∈ K0 and k 0 ∈ ← → K F 0 .

  ∂pxp, . . . , ∂ s p xp) |p=p 0 → (fp,a(xp), ∂p(fp,a(xp)), . . . , ∂ s p (fp,a(xp))) |p=p 0 , when defined. Since (fp,a)p is a C r -family of maps with s ≤ r-2, the family (gp 0 ,a)p 0 is itself a C 2 -family of maps. It depends continuously on a. Notice that ∂p(fp,a(xp)) |p=p 0 = Dfp 0 (xp 0 ) • ∂p(xp) |p=p 0 + ∂p(fp(xp 0 )) |p=p 0 . We notice that ∂p(fp,a(xp)) |p=p 0 has possibly a non zero derivative relatively to xp 0 (independent of ∂p(xp) |p=p 0 ), a derivative Dfp 0 (xp 0 ) relatively to ∂p(xp) |p=p 0 and derivatives equal to zero relatively to each ∂ j Dfp 0 (xp 0 ) relatively to ∂ k p (xp) |p=p 0 and derivatives equal to zero relatively to each ∂ j p (xp) |p=p 0 for k < j ≤ s. To see this, it is enough to write the map p → fp,a(xp) as the composition of p → (p, xp) and (p, x) → fp

p (xp) |p=p 0 for 2 ≤ j ≤ s. When 1 < k ≤ s, the term ∂ k p (fp,a(xp)) |p=p 0 has possibly a non zero derivative relatively to ∂ j p (xp) |p=p 0 for 0 ≤ j ≤ k -1 (independent of ∂ k p (xp) |p=p 0 ), a derivative

  a,α ≤ • γ ns and then Πp(s + s ) ≤ Πp(s) + s • logγ. Since logγ < 0, the map s ∈ R+ → Πp(s) is strictly decreasing. Tending s → +∞, we see that Πp(s) tends to -∞. Finally, by the intermediate value theorem, the map s ∈ R+ → Πp(s) has a unique zero ∆(p). It remains to prove the continuity of p → ∆(p). By Eq. (16), we have: By Lemma 5.4.3 (whose proof does not need the continuity of p → ∆(p)), we see that for every p and > 1, there exists a neighborhood U p, of p and a constant D5 > 0 s.t. for every p ∈ U p, and α ∈ A * , the term Λ p ,a,α is bounded between Λ p,a,α /D5 and D5Λ

	α∈A n p,a,α Πp(s) = lim Λ s n→+∞ 1 α∈A n n log Λ s p,a,α for any s ≥ 0 .	(34)

1/

p,a,α . Injecting this in Eq

Lemma 7.0.4.

  Let r ≥ 1 and U R m . Suppose that (Fp)p is a C r -family of local diffeomorphisms Fp of U leaving invariant a compact hyperbolic set Kp. Let k = (k i 0 )i ∈ ← → K F 0 and (Γp)p be a C r -family of manifolds of the same dimension as W s η (k p ; Fp). Suppose that Γp does not intersect the stable set of Kp and (Γp)p intersects transversally (W u η (k p ; Fp))p at a C r -family of points (zp)p. Then for any > 0 and n large there is a submanifold Γ n p C r -close to W s η (k -n p ; Fp), whose image by F n p is in a -neighborhood of zp in Γp and s.t. (Γ n p )p is C r -close to (W s η (k -n

p ; Fp))p.