Journée d’ouverture du Projet « Probabilités & Approximations Diophantiennes »

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Date/heure
Date(s) - 15/12/2015
10 h 00 min - 15 h 00 min

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JOURNÉE THÉMATIQUE

Intervenants :

– 10h : Paulo Varandas (UFBA, Brazil)
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*On the complexity and variational relation for semigroup actions

_ One of the main purposes of dynamical systems is to understand the behavior of the space of orbits of continuous group and semigroup actions on compact metric spaces. The most studied and well understood classes of such dynamical systems are ℤ-, ℕ- or ℝ-actions, which correspond to the dynamics of homeomorphisms, continuous endomorphisms or continuous flows, respectively. A notion of topological complexity for such dynamical systems has been proposed in the seventies and was very well studied by Goodwin, Bowen, Walters and Parry, among others, and proved a variational principle for the pressure.
Such strong relations between the topological and ergodic features of a dynamical system is still unavailable for general group actions. On the one hand, the theory is not unified since several notions of topological complexity have been proposed, and many of them depend on properties of the group action as commutativity or amenability. On the other hand, many group actions admit no common invariant measures and this notion should be replaced by a more flexible concept.
In this talk I will discuss a notion of topological entropy and pressure for finitely generated semigroup actions and discuss its regularity, relation with the exponential growth of periodic orbits, properties of the ζ function and relation with random dynamical systems in the setting of finitely generated semigroups of expanding maps.
These results are part of joint works with F. Rodrigues (UFRGS, Brazil) and M. Carvalho (U. Porto, Portugal).


– 11h : Alba Málaga-Sabogal (I2M, Marseille)
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*Mélange générique dans des billards polygonaux et plus…

_ Considérons des polygones dont les côtés sont parallèles à deux vecteurs fixés. Gutkin et Katok ont montré que pour toute direction fixée θ il existe un ensemble Gδ dense de polygones pour lesquels le billard est faiblement mélangeant en direction θ. Dans cet exposé, je vais présenter une généralisation de leur argument qui permet d’obtenir le mélange faible dans un ensemble Gδ-dense de mesure totale de directions pour un polygone générique. Je conclurai alors par une application à la multiple ergodicité du modèle du vent à arbre (Wind-Tree).
C’est un travail en collaboration avec Serge Troubetzkoy.


– 14h : Henk Bruin (University of Vienna, Austria)
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*Matching interval maps

_ A somewhat curious phenomenon, observed in e.g. the family of α-continued fraction maps, is that certain orbits will synchronize (called matching), and lead to a better form of invariant density and monotonicity properties of entropy than might a priori be expected. The same phenomenon occurs for families of piecewise linear interval maps, but especially the parameter space becomes very interesting, and I will present results on this, which are joint work with Carlo Carminati, Alessandro Profeti and Stefano Marmi (University of Pisa).


Organisateur : Serge Troubetzkoy


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