GdT TESD Orsay, année 2021-22

Octobre 2021

"Discretized Anosov flows from a global viewpoint"

Abstract: The goal of the talk is to present a class of partially hyperbolic diffeomorphisms denoted as "discretized Anosov flows". We will see equivalent definitions for this notion and discuss about how this class comprises whole C1 connected components of the space of partially hyperbolic diffeomorphisms.

18 octobre: relâche

25 octobre: férié

Novembre 2021

8 novembre: Douglas Coates (Orsay)

"Decay of correlations, limit theorems and physical measures for interval maps with two neutral fixed points and critical points"

Abstract: We present a summary of recent and ongoing work joint with S.Luzzatto and M.Mubarak in which we consider a new family of interval maps. This family depends on four real parameters and is characterised by the presence of two intermittent fixed points and single discontinuity with either zero or unbounded derivative.
Through the construction of a Young tower we prove the existence of an absolutely continuous invariant measure which may be finite or infinite depending on the choice of parameters. In the finite measure case we establish polynomial decay of correlations, and show that either a central limit theorem or a stable limit will hold for a large class of observables dependent on their behavior at the fixed points. Moreover, in the infinite measure setting we establish the existence of physical measures by exploiting limit theorems that hold on the Young tower.

15 novembre: relâche

22 novembre: Frédéric Le Roux (Jussieu)

"Distorsion forte dans les groupes de transformations"

Résumé: Le groupe des homéomorphismes de la droite réelle possède la propriété suivante: toute partie dénombrable est contenue dans un sous-groupe engendré par 17 éléments.
Ceci répond à une question posée en 1935 par Schreier. On tentera de démontrer ce résultat, et d'explorer ce type de propriété dans divers groupes de transformations. (Travail en collaboration avec Kathryn Mann.)

29 novembre: relâche

Décembre 2021

13 décembre: Martin Leguil (Amiens)

"Mesures u-Gibbs et SRB des difféomorphismes d'Anosov du tore de dimension trois"

Résumé: Etant donné un système dynamique "chaotique", les mesures physiques et SRB jouent un rôle central dans la description de la statistique suivie par la plupart des orbites. Un angle d'attaque pour la compréhension de ces mesures consiste en l'étude d'une autre classe de mesures, a priori différentes, mais intimement liées aux mesures SRB: les mesures u-Gibbs. Dans un travail en commun avec Sébastien Alvarez, Davi Obata et Bruno Santiago, nous explorons les liens entre ces deux classes de mesures pour une famille de difféomorphismes d'Anosov du tore de dimension 3, et donnons des conditions géométriques sous lesquelles ces deux classes coïncident.

20 et 27 décembre: férié

Janvier 2022

17 janvier: Santiago Barbieri (Orsay & Univ. Rome 3)

"Genericity of effectively stable integrable Hamiltonian systems"

Abstract: Hamiltonian systems constitute an important class of dynamical systems. Those hamiltonian systems which are integrable in the sense of Arnold-Liouville possess an important property: their solutions can be written explicitly and the phase space is foliated by invariant tori carrying global quasi-periodic orbits. This kind of systems are exceptional but in applications it is not rare to see systems which are perturbations of integrable ones. A natural question is then to determine whether the stability of solutions is preserved for this latter type of systems. Kolmogorov-Arnold-Moser theory assures that, under generic hypotheses, a Cantor set of positive Lebesgue measure of invariant tori carrying quasi-periodic motions persists under a sufficiently small perturbation. On the other hand, instabilities may appear in the complementary of this set (Arnold diffusion). Moreover, a Theorem due to Nekhoroshev (1971-1977) shows that the solutions of a sufficiently regular integrable system verifying a transversality property known as "steepness" are stable over a long time under the effect of a suitably small perturbation. Nekhoroshev also showed (1973) that the steepness property is generic, both in measure and topologic sense, in the space of jets (Taylor polynomials) of sufficiently smooth functions. However, the proof of this result kept being poorly understood up to now and, surprisingly, the paper in which it is contained is hardly known, whereas the rest of the theory has been widely studied over the decades. Moreover, the definition of steepness is not constructive and no general rule to establish whether a given function is steep or not existed up to now, thus entailing a major problem in applications.
In this seminar, I will start by explaining the main ideas behind Nekhoroshev's proof of the genericity of steepness by making use of a more modern language. Indeed, the proof strongly relies on arguments of complex analysis and real algebraic geometry: the latter was much less developed than nowadays at the time that Nekhoroshev was writing, so that many passages appear to be quite obscure in the original article. Moreover, an important result of real algebraic geometry was buried in the proof and seems to have been proved again by Roytwarf and Yomdin in 1997 by making use of different arguments (and generalized in many directions by subsequent works of many authors). Finally, I will show how a deep understanding of the genericity of steepness allows to determine explicit algebraic criteria in the space of jets which make it possible to establish whether a given function is steep or not.
Joint work with Laurent Niederman.

24 et 31 janvier: relâche

Février 2022

14 février: Alexey Korepanov (LPSM, Jussieu)

"Probabilistic properties of the measure of maximal entropy for Sinai billiards"

Abstract: We know much about the physical measure for dispersing billiards, results like exponential mixing and Central Limit Theorem (CLT) are basic and standard, with a variety of different proofs. Much less is known about the measure of maximal entropy (MME): we have only very recently learned that it exists and is unique (Baladi-Demers 2020). I'll talk about (surprising!) mixing rates and CLT-type theorems for the MME. This is a work in progress, joint with Mark Demers.

21 et 28 février: férié

Mars 2022

28 mars: Katsutoshi Shinohara (Univ. Hitotsubashi, Japon)

"Constructing new kinds of aperiodic classes for C1-generic diffeomorphisms"

Abstract: A chain recurrence class is called aperiodic if it does not contain any periodic point. The research of aperiodic classes for C1-generic diffeomorphisms is important to understand the global structure of dynamical systems. In this talk, I would like to explain recent progress about the techniques of constructing examples of aperiodic classes, including ones such as (1) minimal but expansive, (2) minimal and supporting infinitely many ergodic measures, (3) non-transitive or (4) transitive but not minimal, for a class of C1-generic partially hyperbolic diffeomorphisms. This is a joint work with Christian Bonatti (CNRS, Univ. Bourgogne).

Avril 2022

"Periodic points on Veech surfaces"

Abstract: A non-square-tiled Veech surface has finitely many periodic points, i.e. points with finite orbit under the whole affine automorphism group. We first present an algorithm that, when given a non-square-tiled Veech surface as input, outputs its set of periodic points. We then describe upcoming work that uses this algorithm to prove that Prym eigenforms in the minimum stratum in genus 2, 3 and 4 do not have periodic points, except for the fixed points of the Prym involution. Part of the work in this talk is joint with Zawad Chowdhury, Samuel Everett and Destine Lee.

11 avril: relâche

18 et 25 avril: férié

Mai 2022

9, 16, 23 mai: relâche

30 mai: Maxence Phalempin (Brest)

"Théorème limite pour les auto-intersections des trajectoires du flot d'un gaz de Lorentz Z-périodique en horizon fini"

Résumé: Dans cet exposé, on étudie les trajectoires d'un flot pour un gaz de Lorentz Z-périodique en horizon fini, un système dynamique hyperbolique de mesure infinie issu d'un modèle introduit par H.Lorentz en 1905. Un tel système peut s'identifier à un flot spécial au dessus d'une Z-extension d'un billard de Sinai, pour laquelle D.Szasz et T.Varju ont développé un théorème limite local avec décorrélation. L'application de ce théorème associé à une "bonne" approximation des auto-intersections du flot permet de réduire l'étude de la trajectoire à la combinaison d'une marche aléatoire à une dimension sur la Z-extension et au choix aléatoire d'une phase dans un billard de Sinai au niveau local.

Juin 2022

13 juin: relâche

20 juin: Thierry de la Rue (Rouen)

"Questions autour de la transformation Pascal-adique"

Résumé: Les transformations adiques ont été introduites par Vershik dans les années 1980 comme des modèles combinatoires pour les constructions de type "couper-empiler" (cutting and stacking). Une telle transformation agit sur l'espace des chemins infinis dans un diagramme de Bratteli. Lorsque ce diagramme est le graphe de Pascal, on obtient ce que l'on appelle la transformation Pascal-adique, un exemple à la fois naturel à considérer et hautement non trivial à étudier.
Il est facile de décrire la famille des mesures invariantes ergodiques pour la transformation Pascal-adique: elle est paramétrée par un nombre réel p entre 0 et 1, la mesure associée à p correspondant simplement à tirer au hasard un chemin en effectuant des pas i.i.d., en allant vers la droite avec probabilité p et vers la gauche avec probabilité 1-p. Lorsqu'une telle mesure est fixée, on obtient un système dynamique mesuré ergodique et d'entropie nulle, mais dont on ignore pratiquement toutes les autres propriétés.
Dans cet exposé je voudrais discuter quelques conjectures classiques concernant cette famille de systèmes, en particulier autour de leurs propriétés de mélange, et de leur place dans le zoo des systèmes ergodiques d'entropie nulle.

27 juin: Mauricio Poletti (UFC, Brésil)

"Hölder continuity of the Lyapunov exponents for cocycles over hyperbolic maps"

Abstract: Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Hölder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Hölder continuous functions relative to the uniform topology.
This is a joint work with S.Klein and P.Duarte.