Blake Barker
email: bhbarker@indiana.edu

Mathew A Johnson

Pascal Noble

L Miguel Rodrigues
email: rodrigues@math.univ-lyon1.fr

Kevin Zumbrun
email: kzumbrun@indiana.edu

Stability of periodic Kuramoto-Sivashinsky waves

Keywords: periodic waves, Kuramoto-Sivashinsky equations, modulational stability. 2000 MR Subject Classification: 35B35

In this note, we announce a general result resolving the long-standing question of nonlinear modulational stability, or stability with respect to localized perturbations, of periodic travelingwave solutions of the generalized Kuramoto-Sivashinski equation, establishing that spectral modulational stability, defined in the standard way, implies nonlinear modulational stability with sharp rates of decay. The approach extends readily to other second-and higher-order parabolic equations, for example, the Cahn Hilliard equation or more general thin film models.

Introduction

In this note, we describe recent results obtained using techniques developed in [START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF][START_REF] Johnson | Nonlinear stability of viscous roll waves, to appear[END_REF][START_REF] Barker | Witham averaged equations and modulational stability of periodic solutions of hyperbolic-parabolic balance laws[END_REF] on linear and nonlinear stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky (gKS) equation (1.1)

u t + γ∂ 4 x u + ∂ 3 x u + δ∂ 2 x u + ∂ x (u 2 /2) = 0, δ > 0,
a canonical model for pattern formation in one spatial dimension that has been used to describe, variously, plasma instabilities, flame front propagation, turbulence in reaction-diffusion systems, and thin film flow down an incline [S1, KT, CD, DSS, PSU].

More generally, we consider (taking without loss of generality γ = 1) an equation of the form (1.2)

u t + ∂ 4 x u + ∂ 3 x u + δ∂ 2 x u + ∂ x f (u) = 0,
f ∈ C 2 , δ not necessarily positive. Our methods apply also, with slight modifications to accomodate quasilinear form (see [JZN]) to the Cahn Hilliard equation and other fourth-order models for thin film flow as discussed for example in [BMSZ]. Indeed, the argument, and results, extend to arbitrary 2 2r-order parabolic systems, so is essentially completely general for the diffusive case. As shown in [JZ], they can apply also to mixed-order and relaxation type systems in some cases as well.

It has been known since 1976, almost since the introduction of the model (1.1) in 1975 [START_REF] Kuramoto | On the formation of dissipative structures in reaction-diffusion systems[END_REF][START_REF] Sivashinsky | Nonlinear analysis of hydrodynamic instability in laminar flame. I. Derivation of basic equations[END_REF], that there exist spectrally stable bands of solutions in parameter space; see for example the numerical studies in [CKTR, FST]. Moreover, numerical time-evolution experiments described for example in [CD] suggest that these waves are nonlinearly stable as well, serving as attractors in the chaotic dynamics of (gKS). However, up to now this conjecture had not been rigorously verified.

Here, we announce the result, resolving this open question, that spectral modulational stability, defined in the standard sense of the modulational stability literature,1 implies linear and nonlinear modulational stability. Our analysis gives at the same time new understanding even at the formal level of Whitham averaged equations. Further details, along with numerical investigations of existence and spectral stability, will be given in [START_REF] Barker | Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation[END_REF].

The traveling-wave equation

Substituting u = ū(x -ct) into (1.2), we find that the traveling-wave equation is -cu

+ u + u + δu + f (u) = 0, or, integrating once in x, (2.1) -cu + u + u + δu + f (u) = q,
where q ∈ R is a constant of integration. Written as a first-order system in (u, u , u ), this is

(2.2)   u u u   =   u u c -u -δu -f (u) + q   .
It follows that periodic solutions of (1.2) correspond to values (X, c, q, b) ∈ R 6 , where X, c, and q denote period, speed, and constant of integration, and b = (b 1 , b 2 , b 3 ) denotes the values of (u, u , u ) at x = 0, such that the values of (u, u , u ) at x = X of the solution of (2.2) are equal to the initial values (b 1 , b 2 , b 3 ).

Following [JZ], we assume: ) The map H : R 6 → R 3 taking (X, c, q, b) → (u, u , u )(X, c, q, b; X) -b is full rank at ( X, c, q, b), where (u, u , u )(•; •) is the solution operator of (2.2).

(H1) f ∈ C K+1 , K ≥ 4. ( H2
By the Implicit Function Theorem, conditions (H1)-(H2) imply that the set of periodic solutions in the vicinity of Ū form a smooth 3-dimensional manifold (counting translations)2

(2.3) { Ū β (x -α -c(β)t)}, with α ∈ R, β ∈ B ⊂ R 2 .
3 Bloch decomposition and spectral stability conditions

In co-moving coordinates, the linearized equation about ū reads

(3.1) v t = Lv := (c -a)v x -v xxxx -v xxx -δv xx , a := df (ū),
and the eigenvalue equation as

Lv := -v xxxx -v xxx -δv xx + (c -av x = λv.
Following [G], we define the one-parameter family of Bloch operators

(3.2) L ξ := e -iξx Le iξx , ξ ∈ [-π, π)
operating on the class of L 2 periodic functions on [0, X]; the (L 2 ) spectrum of L is equal to the union of the spectra of all L ξ with ξ real with associated eigenfunctions w(x, ξ, λ) := e iξx q(x, ξ, λ), where q, periodic, is an eigenfunction of L ξ . By standard considerations, the spectra of L ξ consist of the union of countably many continuous surfaces λ j (ξ); see, e.g., [G].

Without loss of generality taking X = 1, recall now the Bloch representation

(3.3) u(x) = 1 2π π -π e iξ•x û(ξ, x)dξ
of an L 2 function u, where û(ξ, x) := k e 2πikx û(ξ + 2πk) are periodic functions of period X = 1, û(•) denoting with slight abuse of notation the Fourier transform of u in x. By Parseval's identity, the Bloch transform u

(x) → û(ξ, x) is an isometry in L 2 : u L 2 (x) = π -π 1 0 |û(ξ, x)| 2 dx dξ = û L 2 (ξ;L 2 (x)) , where L 2 (x) is taken on [0, 1] and L 2 (ξ) on [-π, π]. More generally, for q ≤ 2 ≤ p, 1 p + 1 q = 1
, there holds the generalized Hausdorff-Young's inequality [JZ] (3.4)

u L p (x) ≤ û L q (ξ;L p (x)) .
The Bloch transform diagonalizes the periodic-coefficient operator L, yielding the inverse Bloch transform representation

(3.5) e Lt u 0 = 1 2π π -π e iξ•x e L ξ t û0 (ξ, x)dξ .
Following [JZ], we assume along with (H1)-(H2) the strong spectral stability conditions: (D1) σ(L ξ ) ⊂ {Reλ < 0} for ξ = 0. (D2) Reσ(L ξ ) ≤ -θ|ξ| 2 , θ > 0, for ξ ∈ R and |ξ| sufficiently small. (D3) λ = 0 is an eigenvalue of L 0 of multiplicity 2.3 Assumptions (H1)-( H2) and (D1)-(D3) imply [START_REF] Johnson | Nonlinear stability of periodic traveling waves of viscous conservation laws in the generic case[END_REF][START_REF] Barker | Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation[END_REF] that there exist 2 smooth eigenvalues

(3.6) λ j (ξ) = -ia j ξ + o(|ξ|), j = 1, 2
of L ξ bifurcating from λ = 0 at ξ = 0. Following [JZ], we make the further nondegeneracy hypotheses: (H3) The coefficients a j in (3.6) are distinct.4 (H4) The eigenvalue 0 of L 0 is nonsemisimple, i.e., dim ker L 0 = 1.

Spectral stability and the Whitham averaged equations

As noted in [Se, JZ], coefficients a j in (3.6) are characteristics of the 2 × 2 Whitham averaged system (4.1)

M t + F x = 0, ω t + (cω) x = 0
formally governing large-time (∼ small frequency) behavior, evaluated at the values c, ω of the background wave ū, where M (c, ω) is the mean of u over one period of the periodic wave with speed c and frequency ω = 1/X, and F (c, ω) is the mean of f (u). Here, ω ∼ ψ x , c ∼ -ψ t /ψ x , where ψ denotes phase in the modulation approximation (4.2) u(x, t) ≈ ū(ψ(x, t)).

In the context of (1.1), thanks to the Galillean invariance x → x -ct, u → u + c, (4.1) reduces to

(4.3) c t + (H(ω) -m(ω)c) x = 0, ω t + (cω) x = 0,
where m(ω) denotes the mean over one period of u for a zero-speed wave of frequency ω, and H(ω) the mean of u 2 /2, and in the classical situation ε = 0 considered in [FST], to c t + (H(ω)) x = 0, ω t + (cω) x = 0, which linearized about background values c = 0, ω = ω 0 , yields a wave equation

(4.4) ψ tt + ω 0 dH(ω 0 )ψ xx = 0
so long as dH(ω 0 ) < 0. Indeed, by odd symmetry, we may conclude in this case that the second-order corrections b j in the further expansion

(4.5) λ j (ξ) = ia j ξ -b j ξ 2 • • •
of (3.6) are equal, hence λ j (ξ) agree to second order with the dispersion relations of the viscoelastic wave equation

(4.6) ψ tt + ω 0 dH(ω 0 )ψ xx = d(ω 0 )ψ txx , d = 2b 1 = 2b 2 .
This recovers the formal prediction of "viscoelastic behavior" of modulated waves carried out in [FST] and elsewhere, or "bouncing" behavior of individual periodic cells. Put more concretely, (4.6) predicts that the maxima of a perturbed periodic solution should behave approximately like point masses connected by viscoelastic springs. However, we emphasize that such qualitative behaviorin particular, the fact that the modulation equation is of second order-does not derive only from Gallilean or other invariance of the underlying equations, as might be suggested by early literature on the subject, but rather from the more general structure of conservative (i.e., divergence) form [Se, JZ].5 Indeed, for any choice of f , λ j (ξ) may be seen to agree to second order with the dispersion relation for an appropriate diffusive correction of (4.1), a generalized viscoelastic wave equation. See [START_REF] Noble | Whitham's modulation equations for shallow flows[END_REF][START_REF] Noble | Modulational stability of periodic waves of the generalized Kuramoto-Sivashinsky equation[END_REF] for further discussion of Whitham averaged equations and their derivation.

Linear estimates

The main difficulty in obtaining linear estimates is that, by (D3) and (H4), the zero eigenspace of L 0 has an associated 2 × 2 Jordan block. This means that e L0t is not only neutral but grows as O(t). Viewed from the Bloch perspective, it means that the eigenprojections of L ξ blow up as ξ → 0. Performing a careful spectral perturbation analysis, separating out the singular part of the eigendecomposition of e L ξ t in (3.5), and applying (3.4), as in Lemma 2.1, Prop. 3.3,and Prop. 3.4 of [JZ], we obtain the following detailed description of linearized behavior.

Proposition 5.1. Under assumptions (H1)-( H4) and ( D1)-(D3), the Green function G(x, t; y) of (3.1) decomposes as (5.1) G(x, t; y) = ū (x)e(x, t; y) + G(x, t; y),

where, for some C > 0 and all t > 0, 1 (5.4) e(x, t; y) ≡ 0 for 0 ≤ t ≤ 1, and for all t > 0, 1 ≤ q ≤ 2 ≤ p ≤ ∞, 0 ≤ j, l, j + l ≤ K, and 1 ≤ r ≤ 4,

≤ q ≤ 2 ≤ p ≤ ∞ and 1 ≤ r ≤ 4, ∞ -∞ G(•, t; y)f (y)dy L p (R) ≤ Ct -1 4 ( 1 2 -1 p ) (1 + t) -1 4 ( 2 q -1 2 -1 p ) f L q ∩L 2 (5.2) ∞ -∞ ∂ r y G(•, t; y)f (y)dy L p (R) ≤ Ct -1 4 ( 1 2 -1 p )-r 4 (1 + t) -1 4 ( 2 q -1 2 -1 p )-1 2 + r 4 f L q ∩L 2 (5.3) ∞ -∞ ∂ t G(•, t; y)f (y)dy L p (R) ≤ Ct -1 4 ( 1 2 -1 p )-1 (1 + t) -1 4 ( 2 q -1 2 -1 p )+ 1 2 f L q ∩L 2 ,
(5.5) ∞ -∞ ∂ j x ∂ l t e(•, t; y)f (y)dy L p (R) ≤ C (1 + t) -1 2 ( 1 q -1 p )-(j+k) 2 f L q (R) ∞ -∞ ∂ j x ∂ l t ∂ r y e(•, t; y)f (y)dy L p (R) ≤ C (1 + t) 1 2 -1 2 ( 1 q -1 p )-(j+k) 2 f L q (R) .
Moreover, for some constants p j , and a j and b j as in (4.5),

(5.6) e(•, t; y) -

2 j=1 p j errfn • -y -a j t 4b j t , t L p ≤ Ct -1 2 (1-1 p ) , t ≥ 1.
Defining ψ := -e and noting that ū(x)+ψ(x, t)ū (x) ∼ ū(x+ψ(x, t)), we see from (5.1)-(5.6) that linearized behavior indeed agrees to lowest order with modulation by a phase function ψ satisfying a generalized viscoelastic wave equation obtained by diffusive correction of (4.1), consisting of a first-order hyperbolic-parabolic system in ψ x , ψ t .

This observation not only generalizes the second-order scalar description (4.6) obtained in the special case ε = 0 [FST], but gives also new information even in that case. For, note that the formal modulation description can be a bit misleading as regards the assumption of initial data. In particular, from the description (4.6), one might be tempted to conclude, tacitly assuming ψ t | t=0 ≡ 0, that the linear response to a compactly supported perturbation would consist of the D'Alembertian picture of two approximately compactly supported wave forms in ψ moving in opposite directions, diffusing slowly at Gaussian rate. Yet, the explicit bound (5.6) shows that this is rather a description of the derivative ψ x !6

Indeed, as described further in Section 1.2, [JZN], it is the variables ψ x , ψ t that are primary, rather than ψ, ψ t as suggested by (4.6), and it is these variables that are most closely related to the initial perturbation (u -ū)| t=0 . Thus, our analysis gives not only technical verification of existing observations, but also new intuition regarding the nature of modulated behavior.

Nonlinear stability

Using the linear bounds of Prop. 5.1 together with nonlinear cancellation estimates as in [JZ, JZN], we obtain, finally, our main result describing nonlinear behavior under localized perturbations. Theorem 6.1. Assuming (H1)-( H4) and ( D1)-(D3), let ū = (τ , ū) be a traveling-wave solution of (1.2). Then, for some C > 0 and ψ ∈ W 2,∞ (x, t),

(6.1) ũ -ū(• -ψ -ct) L p (t) ≤ C(1 + t) -1 2 (1-1/p) ũ -ū L 1 ∩H K | t=0 , ũ -ū(• -ψ -ct) H K (t) ≤ C(1 + t) -1 4 ũ -ū L 1 ∩H K | t=0 , (ψ t , ψ x ) W K+1,p ≤ C(1 + t) -1 2 (1-1/p) ũ -ū L 1 ∩H K | t=0 , and
(6.2) ũ -ū(• -ct) L ∞ (t), ψ(t) L ∞ ≤ C ũ -ū L 1 ∩H K | t=0
for all t ≥ 0, p ≥ 2, for solutions ũ of (1.2) with ũ -ū L 1 ∩H K | t=0 sufficiently small. In particular,

ū is nonlinearly bounded L 1 ∩ H K → L ∞ stable.
Similarly as in the discussion of linear behavior, we note that Theorem 6.1 asserts asymptotic L 1 ∩ H K → L p convergence of ũ toward the modulated wave ū(x -ψ(x, t)), but only bounded L 1 ∩ H K → L ∞ stability about ū(x), a quite different picture from that suggested at first sight by (4.6).

Application to Kuramoto-Sivashinsky

We conclude our discussion by displaying some representative traveling wave orbits and their associated spectrum, for the case ε = 0.2, computed respectively using MATLAB and the spectral Galerkin package SpectrUW. These indicate, similarly as in the ε = 0 case studied in [CKTR, FST], the existence of a band of spectrally stable periodic traveling waves. For related studies, and an animation of the spectral evolution, see http://www.math.indiana.edu/gallery/TravelingWave.phtml. For more detailed numerical verification using the periodic Evans function [G], see [START_REF] Barker | Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation[END_REF]. Here ε = 0.2, δ = 1, γ = 1. Running left to right in the top row, we display the evolution of the periodic orbits in the three-dimensional phase portrait as X is increased, with frames directly below depicting the spectrum of the corresponding linearized operator about the wave. Similarly as in the well-known computations [CKTR, FST] in the ε = 0 case, we see the familiar picture of initial instability (first frame) on an interval [X Hopf , X * ) from the Hopf bifurcation value X = X Hopf up to a special value X * , transitioning to spectral stability (second frame) for X within a stable band (X * , X * ), then back to instability (third and final frame) for X > X * . Orbits were computed with MATLAB; spectra were computed with the SpectrUW package developed at University of Washington [CDKK].

Figure 1 :

 1 Figure1: Evolution of spectra as period X increases with c = 0 held constant. Here ε = 0.2, δ = 1, γ = 1. Running left to right in the top row, we display the evolution of the periodic orbits in the three-dimensional phase portrait as X is increased, with frames directly below depicting the spectrum of the corresponding linearized operator about the wave. Similarly as in the well-known computations[CKTR, FST] in the ε = 0 case, we see the familiar picture of initial instability (first frame) on an interval [X Hopf , X * ) from the Hopf bifurcation value X = X Hopf up to a special value X * , transitioning to spectral stability (second frame) for X within a stable band (X * , X * ), then back to instability (third and final frame) for X > X * . Orbits were computed with MATLAB; spectra were computed with the SpectrUW package developed at University of Washington[CDKK].

In particular, the sense verified in[CKTR, FST]

That α, β may be chosen to enter in this specific way follows through a second application of the Implicit Function Theorem from the fact that translation in x generates one-dimensional fibers foliating the three-dimensional manifold of periodic solutions, along which wave form Ū (•) and wave speed c are constant.

The zero eigenspace of L 0 , corresponding to variations along the 3-dimensional manifold of periodic solutions in directions for which period does not change[JZ], is at least 2-dimensional by (H2).

Equivalent to strict hyperbolicity of the formal Whitham averaged system (4.1).

As discussed further in[Z], conservation of mass lies outside the usual Noetherian formulation.

More generally, the explicit description (5.6) shows that the principle, noncompactly supported, response ū e, corresponding to the part of the second-order wave solution associated with initial data ψt| t=0 , may be neglected roughly speaking when the integral of the initial perturbation is much smaller than its L 1 norm; in the case of purely modulational initial perturbations h(x)ū (x), this corresponds to the slowly-varying wave form case |h ū| << |hū |.

Research of B.B. was partially supported under NSF grants no. DMS- 0300487 and DMS-0801745. Research of M.J. was partially supported by an NSF Postdoctoral Fellowship under NSF grant DMS-0902192. ; noble@math.univ-lyon1.fr: Research of P.N. was partially supported by the French ANR Project no. ANR-09-JCJC-0103-01. Stay of M.R. in Bloomington was supported by Frency ANR project no. ANR-09-JCJC-0103-01. Research of K.Z. was partially supported under NSF grant no. DMS- 0300487.


Lorenz Knots

Pierre Dehornoy
email: pierre.dehornoy@ujf-grenoble.fr

Eko Hironaka

Ruth Kellerhals

SMALL DILATATION HOMEOMORPHISMS AS MONODROMIES OF

We exhibit low-dilatation families of surface homeomorphisms among monodromies of Lorenz knots.

. The goal of this note is to exhibit an additional construction, that comes from Lorenz knots, that is, periodic orbits of the Lorenz vector field.

This text contains few new results. Most of the content comes from the article [Deh14] where we investigated the homological dilatation of Lorenz knots. The interest here is (i) to restrict our attention to subfamilies of Lorenz knots for which a stronger statement can be obtained with much less technical efforts, (ii) to notice that what was proven for homological dilatation in [Deh14] can also be proven for geometrical dilatation.

Introduction

It is known since Thurston [START_REF] Fathi | Travaux de Thurston sur les surface[END_REF][START_REF] Thurston | On the geometry and dynamics of diffeomorphisms of surfaces[END_REF] that every homeomorphism of a surface is isotopic to either a periodic homeomorphism, or to a pseudo-Anosov one, or to a reducible one. A pseudo-Anosov homeomorphism of a surface S is a homeomorphism h such that S admits two transverse measured foliations, called stable and unstable and usually denoted by (F s , µ s ), and (F u , µ u ), that are invariant under h, and such that there exists a positive real λ(h), called the geometrical dilatation of h, such that µ s and µ u are uniformly multiplied by λ(h) -1 and λ(h) under h respectively. Another property of h is that all closed curves on S are stretched at speed λ(h): for any auxiliary metric on S and for any closed curve γ on S , we have lim n→∞ log( h n (γ) )/n = log(λ(h)). The most standard example is given by the action of a hyperbolic matrix in SL 2 (Z) on the torus R 2 /Z 2 . In this case the invariant foliations are given by the two eigendirections of the matrix and the dilatation is the largest eigenvalue. On surfaces of higher genus, the foliations have prong-type singularities (see Figure 1.1). A reducible homeomorphism is one that admits invariant curves. Such curves divide the surface into elementary pieces where the dynamics is either periodic or of pseudo-Anosov type.

This decomposition result of Thurston can be compared with the fact that every polynomial is either cycloctomic, or has positive Mahler measure, or is reducible. The dilatation factor for a pseudo-Anosov Figure 1.1. The stable and unstable foliations of a pseudo-Anosov homeomorphism (in red and green respectively). On the top left around a regular point, on the top right around a 3-prong singularity, and on the bottom around a boundary component with several singularities. homeomorphism, or better its normalized version λ(h) |χ(S )| , is a natural counterpart of the Mahler measure. In particular, on a given surface, it is easy to find homeomorphisms with arbitrarily high dilatation (for example by iterating a fixed homeomorphism), but low-dilatation homeomorphisms are harder to construct. For g ≥ 1 and s ≥ 0, one usually defines δ g,s as the infimum of the dilatation of a pseudo-Anosov homeomorphism on a closed surface of genus g with s boundary components. It is a priori not clear whether δ g,s is 1 or larger, and in the latter case whether it is a minimum or not. D. Penner showed [START_REF] Robert | Bounds on least dilatations[END_REF] that δ g,s is actually a minimum larger than 1, and that there exist two positive constansts c 1 , c 2 such that for all g, one has c 1 g ≤ log δ g,0 ≤ c 2 g (similar results hold for s ≥ 1). The optimal value of c 1 is not known, and the only known values of δ g,0 are δ 1,0 and δ 2,0 [START_REF] Cho | The minimal dilatation of a genus-two surface[END_REF]. It also follows from the work of Penner that, for any positive D and for a fixed surface, only finitely many mapping classes have a dilatation smaller than D. It is then natural to study what happens when the genus tends to infinity.

A family (h n , S n ) n∈N , where S n is a closed orientable surface and h n a pseudo-Anosov homeomorphism of S n , is said to be of low-dilatation if the sequence log(λ(h n ))|χ(S n )| is bounded.

Low-dilatation families are well understood in the context of 3-manifolds, where a theorem of B. Farb, C. Leininger and D. Margalit [START_REF] Farb | Small dilatation pseudo-Anosov homeomorphisms and 3manifolds[END_REF] states that the punctured suspensions of a low-dilatation family live in some fibered faces of the Thurston norm ball of a finite number of 3-manifolds. However it is still unknown how different homeomorphisms having the same suspensions are related.

Question 1.1 (Farb-Leininger-Margalit [START_REF] Farb | Small dilatation pseudo-Anosov homeomorphisms and 3manifolds[END_REF]). Given a positive number D, what can be said about the dynamics of those homeomorphisms with normalized dilatation smaller than D (i.e., those satisfying log(λ(h))|χ(S )| ≤ log D)? Are they all obtained by some stabilization of the elements of a finite list? For example, E. Hironaka showed [START_REF] Hironaka | On hyperbolic perturbations of algebraic links and small Mahler measure[END_REF] that the polynomials of smallest Mahler measure in degrees 2, 4, 6, 8, and 10 all arise as dilatations of monodromies of fibered links obtained by Hopf or trefoil plumbings on some torus links, so that the monodromies are small perturbations of some periodic surface homeomorphisms.

What we do here is to exhibit low-dilatation families of dynamical origin, by considering certain subfamilies of the set of Lorenz knots, that is, periodic orbits of the (geometric model of the) Lorenz flow. These knots are fibered, so that they give rise to surface homeomorphisms, most of which are of pseudo-Anosov type. Our statement here is a variant of Theorem A of [START_REF] Dehornoy | On the zeroes of the Alexander polynomial of a Lorenz knot[END_REF], where we restrict our attention to subfamilies of Lorenz knots for which we obtain better bounds on the dilatation. Denote by Lorenz b,k the set of Lorenz knots described by a hanging Young diagram (see later) made of a rectangle of width b at the bottom of which is attached a diagram with at most k cells (see Figure 1.2). For these families of Lorenz knots, Question 1.1 has a positive answer: the monodromies act like periodic homeomorphisms on a huge part of the surface (corresponding to the rectangular part of the associated Young diagram), and the non-periodicity is concentrated in a part of the surface of bounded size (corresponding to the additional k cells). Indeed, the rectangular part of the diagram corresponds exactly to a torus link, which is known to have periodic monodromy.

Lorenz knots as iterated Murasugi sums

Lorenz knots are defined as periodic orbits of the (geometric) Lorenz flow. They have been introduced and first studied by J. Birman and R. Williams [START_REF] Birman | Knotted Periodic Orbits in Dynamical Systems I: Lorenz System[END_REF]. We refer to the original article or to [START_REF] Dehornoy | Les noeuds de Lorenz[END_REF] for more details. Let us just mention that Lorenz knots form a family that contains all torus knots and is stable under cabling, so that it also contains all algebraic knots. Also, Lorenz knots are fibered, so that to each of them is canonically associated its monodromy, a homeomorphism of the genus-minimizing spanning surface. As Lorenz knots can be considered as perturbations of torus knots, it is natural to investigate the dilatation of the monodromies of those Lorenz knots which are hyperbolic.

2.a. Young diagrams, Lorenz knots, and canonical spanning surfaces. There are several ways of enumerating Lorenz knots and links. The most convenient from our point of view is using Young diagrams (introduced in this context in [START_REF] Dehornoy | Les noeuds de Lorenz[END_REF]). The procedure is shown on Figure 2.3.

Starting from a Young diagram D, one puts its bottom-left corner on top (we call this hanging position). Then, by desingularizing evering intersection point into a positive braid crossing, one associates a braid, called a Lorenz braid and denoted by β(D). Its closure forms a Lorenz link, that we denote by K(D). All Lorenz links can be obtained in this way. Now, to the closure of every braid is associated a canonical spanning surface, obtained by gluing a disc behind every strand and a ribbon at every crossing. Applying this construction to β(D) yields a canonical spanning surface for K(D), that we denote by S (D). One can check that the Euler characteristics of S (D) is the number of cells of D, hence denoted by χ(D).

Figure 2.4. To every Lorenz braid β(D) (on the left), one associates a link K(D) and a canonical spanning surface S (D) (in the middle). This surface can actually be immersed into the plane (following S. Baader [START_REF] Baader | Trefoil plumbing[END_REF], on the right). In this representation, the correspondance between elementary curves (in green) and cells of the diagram is straightforward.

2.b. Monodromy. In this section, we describe an inductive construction of the surface S (K) for every Lorenz knot K, called the Murasugi sum. This procedure ensures that K is a fibered knot with fiber S (K), and yields a decomposition of the associated monodromy h(K) as an explicit product of Dehn twists.

By construction, for every cell c of a Young diagram D a simple close curve on S (D) that winds once around c is canonically associated. We call it a elementary curve and denote it by γ(c) (see Figure 2 The order is from right to left, and in every column from bottom to top.

Note that if K(D) is a multi-component link, the fiber surface may not be unique, as well as the monodromy. However, if K(D) is a knot, we have uniqueness of the fiber surface and of the monodromy homeomorphism.

We will only sketch the proof of Proposition 2.1 and refer to the survey [START_REF] Dehornoy | Les noeuds de Lorenz[END_REF] for more details. The starting point is the 2-component Hopf link, which is the Lorenz link associated to the Young diagram with one cell only. The 2-component Hopf link is known to be fibered, the fiber surface being a twisted annulus that we call the Hopf annulus, and the monodromy being a right-handed Dehn twist. The action of the monodromy on the annulus (on the right, seen on an abstract annulus) is a Dehn twist on the green curve: the red segment is sent to the purple one.

The induction step for proving Proposition 2.1 is done using the Murasugi sum of surfaces [START_REF] Murasugi | On the genus of the alternating knot, I[END_REF]. This is an operation that takes two surfaces with boundary S 1 , S 2 , depends on a choice of a 2n-gon in each of them, and associates a new surface with boundary S 1 ♯S 2 that contains S 1 and S 2 as subsurfaces (see Figure 2.7). This operation preserves the fibered character, in the following sense: if S 1 , S 2 are two fibered surfaces in S 3 with monodromies h 1 , h 2 , then the Murasugi sum S 1 ♯S 2 , where S 1 is glued on top, is fibered with monodromy h 1 • h 2 (see the proof of D. Gabai [START_REF] Gabai | The Murasugi sum is a natural geometric operation[END_REF] or an expanded version in [START_REF] Dehornoy | Les noeuds de Lorenz[END_REF]).

In particular, Murasugi gluing a Hopf annulus to a fibered surface yields another fibered surface. In this way, starting from the canonical Seifert surface associated to a hanging Young diagram D, we obtain that the surface associated to the diagram D ′ obtained from D by adding a cell on the bottom-right border of D is also fibered. Moreover, the monodromy associated to a Hopf annulus is a right-handed Dehn twist so that the monodromy associated to the surface S (D) is a product of Dehn twists along the cores of the glued Hopf annuli. The order of the product is determined by the order of the gluing. The latter needs to preserve the respective positions of the Hopf annuli, namely one should glue first an annulus that is on top of another one. The order given on Figure 2.5 obeys this constaint. This completes the (sketch of) proof of Proposition 2.1. 2.c. Action of the monodromy on elementary curves. The dilatation of a pseudo-Anosov homeomorphism can be read on its action on curves. So, in order to bound the dilatation, one should bound the stretching of curves under the homeomorphism. Cutting the canonical surface S (D) associated to a diagram D along all elementary curves reduces S (D) to a neighborhood of its boundary, so that elementary curves contain all the information on S (D). In particular for Lorenz knots, it is enough to estimate the stretching of elementary curves under h(D) is order to control the dilatation of h(D). Now come the two key observations. For some orientation reason, the second observation works only when considering h -1 (D) instead of h(D). Therefore we consider the inverse of the monodromy, which makes little difference.

We say that a cell c of a hanging Young diagram is internal if there is a cell, say c ′ , in North-West position with respect to c (see Figure 2.8). Otherwise it is called external. The proof is displayed on Figure 2.9, where the successive images of the curve γ(d) under consecutive Dehn twists are depicted (see also [START_REF] Baader | Trefoil plumbing[END_REF]).

In order to fully control h(D) -1 , we need to know what happens to external cells when iterating (backwards) the monodromy. For a general Lorenz link, the behaviour is rather hard to control (this is the reason of the heavy computations in [START_REF] Dehornoy | On the zeroes of the Alexander polynomial of a Lorenz knot[END_REF]). However, if we suppose that the diagram we are considering lies in Lorenz b,k , things become simpler. In particular the image of an elementary curve corresponding to an external cell is not so simple, but its second image is.

For D a diagram in Lorenz b,k , we call mixing zone of D the set of those k cells that are outside the main rectangle of D (see Figure 1.2). We also assume that we have an auxiliary metric on S for which all elementary curves have length at most 1.

Lemma 2.3. Assume that D is a Young diagram in Lorenz b,k , and that h(D) is the monodromy associated to the canonical surface S (D). Let c be an external cell of D. Then h(D) -2 (γ(c)) is a curve of length at most k that lies entirely in the mixing zone.

The proof is depicted on Figure 2.10. The idea is that, with arguments similar to the proof of Lemma 2.2, one can describe the curve h(D) -1 (γ(c)): it is the concatenation of one external curve, and many internal curves. When iterating h(D) -1 once more, the different contributions cancel, except in the mixing zone. 3: an external cell c and its image under h(D) -2 . On the left, h(D) -1 (γ(c)) is a curve that turns positively around blocks of orange cells and negatively around blocks of green (and blue) cells. On the center left, the image under h(D) -1 of the curve h(D) -1 (γ(c)), except the part that winds around the blue cell. On the center right, the image under h(D) -1 of the part of the curve h(D) -1 (γ(c)) that winds around the blue cell. On the right, the concatenation of those two parts is the curve h(D) -2 (γ(c)), it is a curve that only winds around some cells of the mixing zone of D.

Let c be an arbitrary cell in the mixing zone on S (D). By Lemma 2.2, the l(D) first images of the curve γ(c) under h(D) -1 all correspond to internal cells, hence have length one. After a few more iteration, the image is then an elementary external curve, and after two more iterations, it is a curve in the mixing zone of length at most k. Then the process goes on: the next l iterations yield a curve of length at most k. Summarizing, the length of h(D) -n (γ(c)) grows by a factor at most k every l(D) steps. Therefore, the growth rate of γ(c) is bounded by log k/l(D). Now, the same argument works for any cell, not just in the mixing zone, except that the initial dilatation arises earlier. But this does not change the growth rate, hence h(D) -1 asymptotically streches all curves on S (D) by a factor at most log k/l(D).

Finally, an elementary computation shows that the Euler characteristics of S (D) is b•l(D)+k, therefore the dilatation is smaller than b log k/(|χ(D)|k).

Figure 1

 1 Figure 1.2. A diagram coding a knot in Lorenz b,k , for b = 6 and k = 11. The mixing zone (in yellow) corresponds to those additional k cells.

Figure 2

 2 Figure 2.3. To every hanging Young diagram D (on the left), one associates a braid β(D) (on the right) whose closure is the Lorenz link K(D).

  Proposition 2.1. Let D be a Young diagram. Then the Lorenz link K(D) is fibered with fiber S (D), and its monodromy h(D) is the product of all Dehn twists around all elementary curves of S (D), in the order prescribed on Figure2.5.

Figure 2

 2 Figure 2.5. Decomposition of the monodromy h(D) as the product of the Dehn twists around the χ(D) elementary curves on S (D) (that is, those curves that turns once around the cells of D).The order is from right to left, and in every column from bottom to top.

Figure 2

 2 Figure 2.6. A Hopf annulus in S 3 (on the left), with the associated elementary curve (in green).

Figure 2

 2 Figure 2.7. Two examples of Murasugi sums of canonical surfaces associated to positive braids. Observe that in the second example the left summand is glued on top.

Lemma 2. 2 .

 2 Assume that D is a Young diagram, L(D) is the associated Lorenz link, S (D) is the canonical Seifert surface for L(D), and h(D) is the associated monodromy. Let d be an internal cell of D and a be the cell in NW position with respect to c. Then we have h(D) -1 (γ(d)) = γ(a).

Figure 2 . 8 .

 28 Figure 2.8. An internal cell and its image under the (inverse of the) monodromy.

Figure 2

 2 Figure 2.9. Proof of Lemma 2.2. The image of the elementary curve γ(d) associated to an internal cell d under the inverse of the monodromy h(D) -1 : first the Dehn twists associated to cells that are distant from d do not modify γ(d).Then it is changed by the Dehn twist around a into a curve that encircles both a and d. This curve is then unchanged (in particular it does not intersect the two blue curves γ(b) and γ(c) on the second picture). Finally it is changed by the Dehn twist around d into γ(a). Subsequent twists to not modify it any more.

Figure 2 .

 2 Figure 2.10. Proof of Lemma 2.3: an external cell c and its image under h(D) -2 . On the

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