The remarkable Non-wandering domain theorem due to Sullivan leads to a complete classification of the dynamics for a rational function on its Fatou set. Up to now, the generalization of Sulllivan's theorem in high dimension focus on polynomial skew products. In the case we essentially need to study the semi-local theory, i.e. to study the Fatou set of polynomial skew products in a neighborhood of an invariant fiber which is attracting, parabolic or elliptic. In this talk I will overview the previous results on all these three kinds of polynomial skew products, and present a new theorem on the attracting case. The theorem states that there are no wandering domains for a polynomial skew product with an attracting invariant fiber when the multiplier is small.
Birkhoff averages (of an observable along orbits) are objects of interest when investigating statistical behaviour of a dynamical system. If there is a unique physical measure, the Birkhoff averages will converge, for a positive measure set of initial conditions, to the space average (i.e. the integral) of the observable, so the physical measure captures important statistical properties of the dynamical system. However, in the quadratic family, for example, physical measures do not always exist, and even when they do, they do not necessarily depend continuously on the parameter. In joint work with Alexey Korepanov, we examine what happens for finite time Birkhoff averages for nearby parameters.
Soit T une transformation de X dans X. Si x, y sont deux points de X,
(x,y) est un couple de Li-Yorke si la distance entre T^n(x) et T^n(y) a
une liminf nulle et une limsup strictement positive quand n tend vers
l'infini. Le système est chaotique au sens de Li-Yorke s'il existe un
ensemble S non dénombrable tel que tout couple de points distincts de S
est un couple de Li-Yorke. Il est connu que, pour les transformations de
l'intervalle ou du cercle, l'existence d'un couple de Li-Yorke suffit à
impliquer le chaos au sens de Li-Yorke. Nous montrons qu'on a le même
résultat pour les transformations de graphes topologiques (un graphe
topologique est un espace compact obtenu en recollant un nombre fini de
segments et de cercles). Ce résultat repose sur l'étude des ensembles
omega-limites pour les transformations de graphes topologiques d'entropie
nulle.
Travail en collaboration avec L'ubomír Snoha.
Avec T.Downarowicz nous avons introduit la notion de générateurs uniformes pour un système dynamique topologique discret $(X,T)$ :
ce sont des partitions boréliennes finies $P$ dont le diamètre des itérées $\bigvee_{k=-n}^nT^{-k}P$ tend vers $0$. Dans un travail
en cours je développe une théorie similaire pour les flots. Pour cela on plonge tout d'abord fidèlement le flot dans un flot spécial
au dessus d'un système zéro-dimensionel à l'aide d'une propriété de petits bords pour les flots. Puis on relie les propriétés
d'expansivité entropiques du flot suspendu avec celles du système discret sur la base. Enfin on représente le flot avec une fonction
toit à deux valeurs en adaptant la méthode de Rudolph.
Ce groupe de travail est dédié à l'étude des liens entre Courbure, Transport Optimal, et Probabilités (C-TOP !). Il a lieu à l'Institut Henri Poincaré (IHP) et fait partie des événements de l'ANR GeMeCoD. Page du groupe de travail.
Date | Orateur | Site | Titre |
---|---|---|---|
15/06/2017 - 14:00 |
ZVAVITCH Artem Université d'État de Kent États-Unis |
IHP 01 |
On the convexification effect of Minkowski summation |
We prove a large deviation result for return times of the orbits of a dynamical system in a r-neighbourhood of an initial point x. Our result may be seen as a dierentiable version of the work by Jain and Bansal who considered the return time of a stationary and ergodic process defined in a space of infinite sequences.
Let $M$ be a manifold with pinched negative sectional curvature. We show that, when $M$ is geometrically finite and the geodesic flow on $T^1M$ is topologically mixing, the set of mixing invariant measures is dense in the set $P(T^1M)$ of invariant probability measures. This implies that the set of weak-mixing measures which are invariant by the geodesic flow is a dense $G_\delta$ subset of $P(T^1M)$. We also show how to extend these results to geometrically infinite manifolds with cusps or with constant negative curvature.
A central question in dynamics is whether the topology of a system determines its geometry, whether the system is rigid. Under mild topological conditions rigidity holds in many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. We will discuss the case of circle maps with a flat interval. The class of maps with Fibonacci rotation numbers is a $C^1$ manifold which is foliated with co dimension three rigidity classes. Finally, we summarize the known non-rigidity phenomena in a conjecture which describes how topological classes are organized into rigidity classes.
We prove the existence of Sinai-Ruelle-Bowen measures for a class of $C^2$ partially hyperbolic diffeomorphisms via the method of random perturbations. This is a joint work with Y. Cao and Z. Mi.