Classification of discrete weak KAM solutions on linearly repetitive quasi-periodic sets

In discrete schemes, weak KAM solutions may be interpreted as approximations of correctors for some Hamilton-Jacobi equations in the periodic setting. It is known that correctors may not exist in the almost periodic setting. We show the existence of discrete weak KAM solutions for non-degenerate and weakly twist interactions in general. Furthermore, assuming equivariance with respect to a linearly repetitive quasi-periodic set, we completely classify all possible types of weak KAM solutions.Comment: 44 pages, 1 figur

Explicit bounds for separation between Oseledets subspaces

We consider a two-sided sequence of bounded operators in a Banach space which are not necessarily injective and satisfy two properties (SVG) and (FI). The singular value gap (SVG) property says that two successive singular values of the cocycle at some index $d$ admit a uniform exponential gap; the fast invertibility (FI) property says that the cocycle is uniformly invertible on the fastest $d$-dimensional direction. We prove the existence of a uniform equivariant splitting of the Banach space into a fast space of dimension $d$ and a slow space of co-dimension $d$. We compute an explicit constant lower bound on the angle between these two spaces using solely the constants defining the properties (SVG) and (FI). We extend the results obtained in the finite-dimensional case for bijective operators and the results obtained by Blumenthal and Morris in the infinite-dimensional case for injective norm-continuous cocycles, in the direction that the operators are not required to be globally injective, that no dynamical system is involved, and no compactness of the underlying system or smoothness of the cocycle is required. Moreover, we give quantitative estimates of the angle between the fast and slow spaces that are new even in the case of finite-dimensional bijective operators in Hilbert spaces

Sub-actions for Anosov diffeomorphisms

We show a positive Livciz type theorem for C2 Anosov diffeomor-phisms f on a compact boundaryless manifold M and Hölder observ-ables A. Given A:M → R, α-Hölder, we show there exist V:M → R, β-Hölder, β < α and a probability measure µ, f-invariant such that A ≤ V ◦ f − V

Stabilité de l'accessibilité des difféomorphismes partiellement hyperboliques

Soit M une variété Riemannienne lisse, compacte et connexe. Un difféomorphisme f est dit partiellement hyperbolique dès que le fibré tangent de M admet une décomposition de Whithney, invariante sous Faction de la différentielle de f, en des sous fibrés instable, stable et central. La différentielle de f est respectivement uniformément dilatante et contractante sur les composantes instable et stable. Son comportement dans la direction centrale est uniformément strictement moins dilatant (resp. contractant) que dans la direction instable (resp. stable). Le théorème d'Hadamard-Perron implique que le sous-fibré instable (resp. stable) s'intègre en un unique feuilletage continu de M par des feuilles lisses dit feuilletage instable (resp. stable) de f. Un difféomorphisme f partiellement hyperbolique est dit accessible dès que toute paire de points de M peut être reliée par une concaténation finie de feuilles instable et stables. Nous étudions des critères permettant d'obtenir l'accessibilité et la stabilité de cette propriété pour de petites perturbations de f. Nous avons démontré que l'accessibilité est une propriété stable si le sous-fibré central est de dimension un. Ce théorème est la conséquence d'une généralisation d'un résultat de Joseph F. Plante (1972) : si f un difféomorphisme partiellement hyperbolique dont les feuilletages instable et stable sont jointement intégrables alors la somme des sous-fibrés instable et stable s'intègre en un feuilletage continu par des feuilles lisses et sous feuilleté par les feuilletages instable et stable de f.Let M be a smooth, compact and connected Riemannian manifold. Partially hyperbolic diffeomorphisms of M are a generalization of the Anosov diffeomorphisms : they have uniformly contracting and expanding directions with non complementary dimensions. By the Hadamard-Perron theorem, these directions are uniquely integrable : the foliations obtained are respecfively the unstable and stable foliations. Such diffeomorphism is accessible if every pair of points in M can be connected by a path which is piecewise in stable or an unstable leaf. We have studied necessary conditions for partially hyperbolic diffeomorphisms to be accessible and stably accessible under small perturbations of the diffeomorphism. We prove that accessibility is stable for partially hyperbolic diffeomorphism for which the Whitney sum of the unstable and stable directions is of codimension one. This theorem is a consequence of generalization of a result of Joseph F. Plante (1972) : if the unstable and stable foliations of a partially hyperbolic diffeomorphism are jointly integrable then the Whitney sum of the unstable and stable directions is uniquely integrable, and the integral foliation is subfoliated by the the unstable and stable foliations.ORSAY-PARIS 11-BU Sciences (914712101) / SudocSudocFranceF

Codage du flot géodésique sur les surfaces hyperboliques de volume fini

Cette thèse traite de l étude des objets reliés au codage de Bowen-Series du flot géodésiquepour des surfaces hyperboliques de volume fini. On démontre d abord que le billard géodésiqueassocié à domaine fondamental even corners d un groupe fuchsien cofini est conjuguéà une bijection du tore, appelée codage étendu, dont l un des facteurs est la transformationde Bowen-Series. L intérêt principal de cette conjugaison est qu elle ne fait toujours intervenirqu un nombre fini d objets. On retrouve ensuite des résultats classiques sur le codage deBowen-Series : il est orbite-équivalent au groupe, ses points périodiques sont denses, et ses orbitespériodiques sont en bijection avec les classes d équivalence d hyperboliques primitifs dugroupe ; ce qui permet finalement de relier sa fonction zeta de Ruelle à la fonction zeta de Selberg.Les preuves de ces résultats s appuient sur un lemme combinatoire qui abstrait la propriétéd orbite-équivalence à des familles de relations qui peuvent être définies sur tout ensemble surlequel agit le groupe. Il est aussi possible de conjuguer le codage étendu à un sous-shift detype fini, sauf pour un ensemble dénombrable de points. Enfin, on prouve que les distributionspropres pour la valeur propre 1 de l opérateur de transfert sont les distributions de Helgason defonctions propres du laplacien sur la surface, puis que l on peut associer à toute telle distributionpropre une fonction propre non triviale de l opérateur de transfert et que ce procédé admet uninverse dans certains cas.This thesis focuses on the study of the objects linked to the Bowen-Series coding of the geodesicflow for hyperbolic surfaces of finite volume. It is first proved that the geodesic billiardassociated with an even corners fundamental domain for a cofinite fuchsian group is conjugatedwith a bijection of the torus, called extended coding, one factor of which is the Bowen-Seriestransform. The sharpest property of that conjugacy is that it always only involves a finite numberof objects. Some classical results about the Bowen-Series coding are then rediscovered : itis orbit-equivalent with the group, its periodic points are dense, and its periodic orbits are inbijection with conjugacy classes of primitive hyperbolic isometries ; which eventually links itsRuelle zeta function to the Selberg zeta function. The proofs of those results use a combinatoriallemma that abstracts the orbit-equivalence property to families of relations that can be definedon every set on which the group acts. The extended coding is also proved to be conjugated witha subshift of finite type, except for a countable set of points. Finally, it is shown that eigendistributionsof the transfer operator for the eigenvalue 1 are the Helgason boundary values ofeigenfunction of laplacian on the surface, plus that one can associate to each such eigendistributiona non-trivial eigenfunction of the transfer operator and that this process has a reciprocalin some cases.BORDEAUX1-Bib.electronique (335229901) / SudocSudocFranceF

Description of some ground states by Puiseux technics

Let $(\Sigma^+_G, \sigma)$ be a one-sided transitive subshift of finite type, where symbols are given by a finite spin set $S$, and admissible transitions are represented by an irreducible directed graph $G\subset S\times S$. Let $H : \Sigma^+_G\to\mathbb{R}$ be a locally constant function (that corresponds with a local observable which makes finite-range interactions). Given $\beta > 0$, let $\mu_{\beta H}$ be the Gibbs-equilibrium probability measure associated with the observable $-\beta H$. It is known, by using abstract considerations, that $\{\mu_{\beta H}\}_{\beta>0}$ converges as $\beta \to + \infty$ to a $H$-minimizing probability measure $\mu_{\textrm{min}}^H$ called zero-temperature Gibbs measure. For weighted graphs with a small number of vertices, we describe here an algorithm (similar to the Puiseux algorithm) that gives the explicit form of $\mu_{\textrm{min}}^H$ on the set of ground-state configuration