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Concentration is an important property of independent random variable, showing that any reasonable function of such variables does not vary a lot around its mean. Observables generated by the iteration of a chaotic enough dynamical system often share a lot of properties with independent random variables. In this survey talk, we discuss several situations where one can prove concentration for them, in uniformly or non-uniformly hyperbolic situations. We also explain why such a property is important to answer relevant geometric or dynamical questions.
concentration - martingales - dynamical systems - Young towers - uniform hyperbolicity - moment bounds[-]

Concentration is an important property of independent random variable, showing that any reasonable function of such variables does not vary a lot around its mean. Observables generated by the iteration of a chaotic enough dynamical system often share a lot of properties with independent random variables. In this survey talk, we discuss several situations where one can prove concentration for them, in uniformly or non-uniformly hyperbolic ...[+]

37A25 ; 37A50 ; 60F15

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37A25 ; 37A50 ; 60F17 ; 60G10

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We will consider (sub)shifts with complexity such that the difference from $n$ to $n+1$ is constant for all large $n$. The shifts that arise naturally from interval exchange transformations belong to this class. An interval exchange transformation on d intervals has at most $d/2$ ergodic probability measures. We look to establish the correct bound for shifts with constant complexity growth. To this end, we give our current bound and discuss further improvements when more assumptions are allowed. This is ongoing work with Michael Damron.[-]

We will consider (sub)shifts with complexity such that the difference from $n$ to $n+1$ is constant for all large $n$. The shifts that arise naturally from interval exchange transformations belong to this class. An interval exchange transformation on d intervals has at most $d/2$ ergodic probability measures. We look to establish the correct bound for shifts with constant complexity growth. To this end, we give our current bound and discuss ...[+]

37B10 ; 37A25 ; 68R15

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The classic mean ergodic theorem has been extended in numerous ways: multiple averages, polynomial iterates, weighted averages, along with combinations of these extensions. I will give an overview of these advances and the different techniques that have been used, focusing on convergence results and what can be said about the limits.

37A05 ; 37A25 ; 37A15

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The classic mean ergodic theorem has been extended in numerous ways: multiple averages, polynomial iterates, weighted averages, along with combinations of these extensions. I will give an overview of these advances and the different techniques that have been used, focusing on convergence results and what can be said about the limits.

37A05 ; 37A25 ; 37A15

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The classic mean ergodic theorem has been extended in numerous ways: multiple averages, polynomial iterates, weighted averages, along with combinations of these extensions. I will give an overview of these advances and the different techniques that have been used, focusing on convergence results and what can be said about the limits.

37A05 ; 37A25 ; 37A15

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For any countable group, and also for any locally compact second countable, compactly generated topological group, $G$, there exists a "universal" hypercyclic representation on a Hilbert space, in the sense that it simultaneously models every possible ergodic probability measure preserving free action of $G$. I will discuss the original proof of this theorem (a joint work with Benjy Weiss) and then, at the end of the talk, say some words about the development of this idea and its applications as expounded in a subsequent work of Sophie Grivaux.[-]

For any countable group, and also for any locally compact second countable, compactly generated topological group, $G$, there exists a "universal" hypercyclic representation on a Hilbert space, in the sense that it simultaneously models every possible ergodic probability measure preserving free action of $G$. I will discuss the original proof of this theorem (a joint work with Benjy Weiss) and then, at the end of the talk, say some words about ...[+]

37A15 ; 37A05 ; 37A25 ; 37A30 ; 47A16 ; 47A67 ; 47D03

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We will discuss old and recent results on topological and measurable dynamics of diagonal and unipotent flows on frame bundles and unit tangent bundles over hyperbolic manifolds. The first lectures will be a good introduction to the subject for young researchers.

37D40 ; 37A17 ; 37A25

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In this talk, we will prove the projective equidistribution of integral representations by quadratic norm forms in positive characteristic, with error terms, and deduce asymptotic counting results of these representations. We use the ergodic theory of lattice actions on Bruhat-Tits trees, and in particular the exponential decay of correlation of the geodesic flow on trees for Hölder variables coming from symbolic dynamics techniques.

20E08 ; 11J61 ; 37A25 ; 20G25 ; 37D40

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The behaviour of infinite translation surfaces is, in many regards, very different from the finite case. For example, the geodesic flow is often not recurrent or is not even defined for infinite time in a generic direction.
However, we show that if one focuses on a class of infinite translation surfaces that exclude the obvious counter-examples, one can adapted the proof of Kerckhoff, Masur, and Smillie and show that the geodesic flow is uniquely ergodic in almost every direction. We call this class of surface essentially finite.
(joint work with Anja Randecker).[-]

The behaviour of infinite translation surfaces is, in many regards, very different from the finite case. For example, the geodesic flow is often not recurrent or is not even defined for infinite time in a generic direction.
However, we show that if one focuses on a class of infinite translation surfaces that exclude the obvious counter-examples, one can adapted the proof of Kerckhoff, Masur, and Smillie and show that the geodesic flow is ...[+]

37D40 ; 51A40 ; 37A25

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Documents 37A25 17 résultats

Concentration properties of dynamical systems - Gouëzel, Sébastien (Auteur de la Conférence) | CIRM Nouveau

The almost sure invariance principle for beta-mixing measures - Haydn, Nicolai (Auteur de la Conférence) | CIRM Nouveau

Ergodic measures for subshifts with eventually constant growth - Fickenscher, Jon (Auteur de la Conférence) | CIRM H Nouveau

Multiple ergodic theorems: old and new - Lecture 1 - Kra, Bryna (Auteur de la Conférence) | CIRM H Nouveau

Multiple ergodic theorems: old and new - Lecture 2 - Kra, Bryna (Auteur de la Conférence) | CIRM H Nouveau

Multiple ergodic theorems: old and new - Lecture 3 - Kra, Bryna (Auteur de la Conférence) | CIRM H Nouveau

A universal hypercyclic representation - Glasner, Eli (Auteur de la Conférence) | CIRM H Nouveau

Dynamics on quotients of SL(2,C) by discrete subgroups - Lecture 2 - Schapira, Barbara (Auteur de la Conférence) | CIRM H Nouveau

Counting and equidistribution of integral representations by quadratic norm forms in positive characteristic? - Paulin, Frédéric (Auteur de la Conférence) | CIRM H Nouveau

Unique ergodicity of geodesic flow in an infinite translation surface - Rafi, Kasra (Auteur de la Conférence) | CIRM H Nouveau