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Research talks;Dynamical Systems and Ordinary Differential Equations
We prove a couple of general conditional convergence results on ergodic averages for horocycle and
geodesic subgroups of any continuous $SL(2,\mathbb{R})$- action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $SL(2,\mathbb{R})$-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the “weak convergence” of push-forwards of horocycle measures under the geodesic flow and a new short proof of a theorem of Chaika and Eskin on Birkhoff genericity in almost all directions for the Teichmüller geodesic flow.
We prove a couple of general conditional convergence results on ergodic averages for horocycle and
geodesic subgroups of any continuous $SL(2,\mathbb{R})$- action on a locally compact space. These results are motivated by theorems of Eskin, Mirzakhani and Mohammadi on the $SL(2,\mathbb{R})$-action on the moduli space of Abelian differentials. By our argument we can derive from these theorems an improved version of the “weak convergence” of ...
37D40 ; 37C40 ; 37A17
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Research talks;Dynamical Systems and Ordinary Differential Equations
I will survey recent results on the generic properties of probability measures invariant by the geodesic flow defined on a nonpositively curved manifold. Such a flow is one of the early example of a non-uniformly hyperbolic system. I will talk about ergodicity and mixing both in the compact and noncompact setting, and ask some questions about the associated frame flow, which is partially hyperbolic.
37B10 ; 37D40 ; 34C28 ; 37C20 ; 37C40 ; 37D35
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Research schools
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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Research schools
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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Research schools
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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Research schools
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$ are equidistributed along measures of maximal entropy. - for C∞ maps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers.
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away ...
37C05 ; 37C40 ; 37D25
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Research schools
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$ are equidistributed along measures of maximal entropy. - for C∞ maps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers.
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away ...
37C05 ; 37C40 ; 37D25
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Research schools
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away from zero for $\delta \in]0,htop(f)[$ are equidistributed along measures of maximal entropy. - for C∞ maps the entropy is physically greater than or equal to the top Lyapunov exponents of the exterior powers.
Smooth parametrizations of semi-algebraic sets were introduced by Yomdin in order to bound the local volume growth in his proof of Shub’s entropy conjecture for C∞ maps. In this minicourse we will present some refinement of Yomdin’s theory which allows us to also control the distortion. We will give two new applications: - for any C∞ surface diffeomorphism f with positive entropy the saddle periodic points with Lyapunov exponents $\delta$-away ...
37C05 ; 37C40 ; 37D25
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Research schools
Works by Sarig and Benovadia have built symbolic dynamics for arbitrary diffeomorphisms of compact manifolds. This shows thatthere can be at most countably many ergodic hyperbolic equilibriummeasures for any Holder continuous or geometric potentials. We will explain how this yields uniqueness inside each homoclinic class of measures, i.e., of ergodic and hyperbolic measures that are homoclinically related. In some cases, further topological or geometric arguments can show global uniqueness.
This is a joint work with Sylvain Crovisier and Omri Sarig
Works by Sarig and Benovadia have built symbolic dynamics for arbitrary diffeomorphisms of compact manifolds. This shows thatthere can be at most countably many ergodic hyperbolic equilibriummeasures for any Holder continuous or geometric potentials. We will explain how this yields uniqueness inside each homoclinic class of measures, i.e., of ergodic and hyperbolic measures that are homoclinically related. In some cases, further topological or ...
37C40
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- 151 p.
ISBN 978-0-8218-2921-9
University lecture series , 0023
Localisation : Collection 1er étage
système dynamique # théorie ergodique # théorie ergodique continue # exposant de Lyapunov # système hyperbolique non-uniforme # variété stable # variété locale
37C40 ; 37D25 ; 34D08
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- 121 p.
ISBN 978-0-8218-3496-1
University lecture series , 0030
Localisation : Collection 1er étage
construction combinatoire # analyse combinatoire # théorie ergodique # dynamique # système dynamique # approximation # multiplicité spectrale # cohomologie
37A20 ; 37A25 ; 37C40 ; 37D20 ; 37D30
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- 122 p.
ISBN 978-3-03719-003-6
Zurich lectures in advanced mathematics
Localisation : Ouvrage RdC (PESI)
système dynamique # hyperbolicité partielle # ergodicité stable # système hyperbolique non-uniforme # exposant de Lyapunov # foliation # continuité # système dynamique continu
37-02 ; 37D25 ; 37C40 ; 37A30 ; 37D40
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- 138 p.
ISBN 978-3-540-40121-6
Springer monographs in mathematics
Localisation : Ouvrage RdC (MARG)
système dynamique # courbure négative # fonction zéta # opérateur de transfert # orbite périodique # système d'Anosov # flot hyperbolique
37A05 ; 35A10 ; 37B10 ; 37C10 ; 37C27 ; 37C30 ; 37C35 ; 37C40 ; 37D20 ; 37D35 ; 37D40
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- 339 p.
ISBN 978-0-8218-4274-4
Fields institute communications , 0051
Localisation : Collection 1er étage
système dynamique # théorie ergodique # ergodicité lisse # système hyperbolique # flots sur surface # méthode quasiconforme # théorie de Teichmüller # foliation # groupe de Kleinian # surface modulaire de Riemann
37C40 ; 37D25 ; 37D30 ; 37E35 ; 37F30 ; 37C85 ; 30F60 ; 30F40 ; 32G15
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- ix; 277 p.
ISBN 978-0-8218-9853-6
Graduate studies in mathematics , 0148
Localisation : Collection 1er étage
système dynamique # théorie ergodique # exposant de Lyapunov # dynamique topologique # hyperbolicité non-uniforme # flot géodésique
37D25 ; 37C40 ; 37-01
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- ix; 165 p.
ISBN 978-2-85629-778-0
Astérisque , 0358
Localisation : Périodique 1er étage
Cocycle abélien # équation cohomologique # invariant d'holonomie # principe d'invariance # cocycle linéaire # théorie de Livsic # exposant de Liapounoff # hyperbolicité partielle # rigidité # cocycle lisse
37A20 ; 37D25 ; 37D30 ; 37A50 ; 37C40
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