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Documents Crovisier, Sylvain 39 résultats

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Orderability and groups of homeomorphisms of the circle - Mann, Kathryn (Auteur de la conférence) | CIRM H

Multi angle

As a counterpart to Deroin's minicourse, we discuss actions of groups on the circle in the C0 setting. Here, many dynamical properties of an action can be encoded by the algebraic data of a left-invariant circular order on the group. I will highlight rigidity and flexibility phenomena among group actions, and discuss new work with C. Rivas relating these to the natural topology on the space of circular orders on a group.

58D05 ; 37E30 ; 57S05 ; 20F60

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Transfer operators for Sinai billiards - lecture 3 - Baladi, Viviane (Auteur de la conférence) | CIRM H

Multi angle

We will discuss an approach to the statistical properties of two-dimensional dispersive billiards (mostly discrete-time) using transfer operators acting on anisotropic Banach spaces of distributions. The focus of this part will be our recent work with Mark Demers on the measure of maximal entropy but we will also survey previous results by Demers, Zhang, Liverani, etc on the SRB measure.

37D50 ; 37C30 ; 37B40

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Transfer operators for Anosov flows - lecture 3 - Tsuijii, Masato (Auteur de la conférence) | CIRM H

Multi angle

We present a functional-analytic approach to the study of transfer operators for Anosov flows. To study transfer operators, a basic idea in semi-classical analysis suggests to look at the action of the flow on the cotangent bundle. Though this idea is simple and intuitive (as we will explain in the lectures), we need some framework to make it work. In the lectures, we present such a framework based on a wave-packet transform.

37D20 ; 37C30

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Beyond Bowen specification property - lecture 1 - Climenhaga, Vaughn (Auteur de la conférence) | CIRM H

Multi angle

Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to specification" using a decomposition of the space of finite-length orbit segments, and then survey various applications, including factors of beta-shifts, derived-from-Anosov diffeomorphisms, and geodesic flows in non-positive curvature and beyond.[-]
Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to s...[+]

37D35 ; 37B10 ; 37B40

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Beyond Bowen specification property - lecture 2 - Climenhaga, Vaughn (Auteur de la conférence) | CIRM H

Multi angle

Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to specification" using a decomposition of the space of finite-length orbit segments, and then survey various applications, including factors of beta-shifts, derived-from-Anosov diffeomorphisms, and geodesic flows in non-positive curvature and beyond.[-]
Rufus Bowen introduced the specification property for uniformly hyperbolic dynamical systems and used it to establish uniqueness of equilibrium states, including the measure of maximal entropy. After reviewing Bowen's argument, we will present our recent work on extending Bowen's approach to non-uniformly hyperbolic systems. We will describe the general result, which makes precise the notion of "entropy (orpressure) of obstructions to s...[+]

37D35 ; 37B10 ; 37B40

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Beyond Bowen specification property - lecture 1 - Thompson, Daniel J. (Auteur de la conférence) | CIRM H

Multi angle

These lectures are a mostly self-contained sequel to Vaughn Climenhaga's talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen.[-]
These lectures are a mostly self-contained sequel to Vaughn Climenhaga's talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, ...[+]

37D35 ; 37D40 ; 37C40 ; 37D25

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Beyond Bowen specification property - lecture 2 - Thompson, Daniel J. (Auteur de la conférence) | CIRM H

Multi angle

These lectures are a mostly self-contained sequel to Vaughn Climenhaga's talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen.[-]
These lectures are a mostly self-contained sequel to Vaughn Climenhaga's talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, ...[+]

37D35 ; 37D40 ; 37C40 ; 37D25

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Beyond Bowen specification property - lecture 3 - Thompson, Daniel J. (Auteur de la conférence) | CIRM H

Multi angle

These lectures are a mostly self-contained sequel to Vaughn Climenhaga's talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, the question of when the “pressure gap” hypothesis can be verified becomes crucial. I will sketch our proof of the “entropy gap”, which is a new direct constructive proof of a result by Knieper. I will also describe new joint work with Ben Call, which shows that all the unique equilibrium states provided above have the Kolmogorov property. When the manifold has dimension at least 3, this is a new result even for the Knieper-Bowen-Margulis measure of maximal entropy. The common thread that links all of these arguments is that they rely on weak orbit specification properties in the spirit of Bowen.[-]
These lectures are a mostly self-contained sequel to Vaughn Climenhaga's talks in week 1. The focus of the week 2 lectures will be on uniqueness of equilibrium states for rank 1 geodesic flows, and their mixing properties. Burns, Climenhaga, Fisher and myself showed recently that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. I will discuss the proof of this result. With this result in hand, ...[+]

37D35 ; 37D40 ; 37C40 ; 37D25

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2y
These lectures will address the dynamics of vector fields or diffeomorphisms of compact manifolds. For the study of generic properties or for the construction of examples, it is often useful to be able to perturb a system. This generally leads to delicate problems: a local modification of the dynamics may cause a radical change in the behavior of the orbits. For the $C^1$-topology, various techniques have been developed which allow to perturb while controlling the dynamics: closing and connection of orbits, perturbation of the tangent dynamics... We derive various applications to the description of $C^1$-generic diffeomorphisms.[-]
These lectures will address the dynamics of vector fields or diffeomorphisms of compact manifolds. For the study of generic properties or for the construction of examples, it is often useful to be able to perturb a system. This generally leads to delicate problems: a local modification of the dynamics may cause a radical change in the behavior of the orbits. For the $C^1$-topology, various techniques have been developed which allow to perturb ...[+]

37C05 ; 37C29 ; 37Dxx

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Several recent papers on surface dynamics have used transverse foliations and maximal isotopies for homeomorphisms isotopic to the identity as a main tool in their work. In this mini-course we will introduce the basic concepts behind this tool and show a new way o deriving useful dynamical information by means of a forcing procedure. The applications involve ways of obtaining existence of non-contractible periodic points with consequences for rotation sets of toral homeomorphisms, exponential growth of periodic orbits and estimates on topological entropy of maps.[-]
Several recent papers on surface dynamics have used transverse foliations and maximal isotopies for homeomorphisms isotopic to the identity as a main tool in their work. In this mini-course we will introduce the basic concepts behind this tool and show a new way o deriving useful dynamical information by means of a forcing procedure. The applications involve ways of obtaining existence of non-contractible periodic points with consequences for ...[+]

37E30 ; 37E45

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