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Pants complexes of large surfaces were proved to be vigid by Margalit. We will consider convergence completions of curve and pants complexes and show that some weak four of rigidity holds for the latter. Some key tools come from the geometry of Deligne Mumford compactification of moduli spaces of curves with level structures.

57M10 ; 20F34 ; 14D23

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Satake duality is an isomorphism between the algebra of characters of a simple Lie group G and the Hecke algebra of the affine Lie group corresponding to the Langlands dual to G. We will suggest a (conjectural) generalisation of this isomorphism replacing the characters by functions on local systems on surfaces and Hecke algebra by the algebra on cells on local systems with values in an affine Lie group.

17B20 ; 30F60

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In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter.

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...

28A80 ; 37C45

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In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for explicit values of the parameter.

In the last few years ideas from additive combinatorics were applied to problems in fractal geometry and led to progress on some classical problems, particularly on the smoothness of Bernoulli convolutions and other self-similar measures. We will introduce some of these tools from additive combinatorics and present some of the main applications, including the smoothness of Bernoulli convolutions outside of a small set of exceptions, and for ...

28A80 ; 37C45

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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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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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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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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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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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

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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Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). Using these relations, we will explain how to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker.

Part II - Examples : In this part we will discuss a number of natural examples in which big mapping class groups and their subgroups arise. These include the inverse limit constructions of de Carvalho-Hall, the theory of finite depth (taut) foliations of 3-manifolds, the theory of “Artinization” of Thompson-like groups, two dimensional smooth dynamics, one dimensional complex dynamics (topology of the shift locus, Schottky spaces) and several other contexts. We will try to indicate how viewing these examples from the perspective of (big) mapping class groups is a worthwhile approach.

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of ...

37FXX ; 57Mxx

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Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). Using these relations, we will explain how to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker.

Part II - Examples : In this part we will discuss a number of natural examples in which big mapping class groups and their subgroups arise. These include the inverse limit constructions of de Carvalho-Hall, the theory of finite depth (taut) foliations of 3-manifolds, the theory of “Artinization” of Thompson-like groups, two dimensional smooth dynamics, one dimensional complex dynamics (topology of the shift locus, Schottky spaces) and several other contexts. We will try to indicate how viewing these examples from the perspective of (big) mapping class groups is a worthwhile approach.

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of ...

37FXX ; 57Mxx

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). Using these relations, we will explain how to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker.

Part II - Examples : In this part we will discuss a number of natural examples in which big mapping class groups and their subgroups arise. These include the inverse limit constructions of de Carvalho-Hall, the theory of finite depth (taut) foliations of 3-manifolds, the theory of “Artinization” of Thompson-like groups, two dimensional smooth dynamics, one dimensional complex dynamics (topology of the shift locus, Schottky spaces) and several other contexts. We will try to indicate how viewing these examples from the perspective of (big) mapping class groups is a worthwhile approach.

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of ...

37FXX ; 57Mxx

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). Using these relations, we will explain how to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker.

Part II - Examples : In this part we will discuss a number of natural examples in which big mapping class groups and their subgroups arise. These include the inverse limit constructions of de Carvalho-Hall, the theory of finite depth (taut) foliations of 3-manifolds, the theory of “Artinization” of Thompson-like groups, two dimensional smooth dynamics, one dimensional complex dynamics (topology of the shift locus, Schottky spaces) and several other contexts. We will try to indicate how viewing these examples from the perspective of (big) mapping class groups is a worthwhile approach.

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of ...

37FXX ; 57Mxx

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of the objects involved, and give some fruitful relations between the dynamics of the two actions. For example, we will see that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). Using these relations, we will explain how to construct non trivial quasimorphisms on subgroups of big mapping class groups. This includes joint work with Alden Walker.

Part II - Examples : In this part we will discuss a number of natural examples in which big mapping class groups and their subgroups arise. These include the inverse limit constructions of de Carvalho-Hall, the theory of finite depth (taut) foliations of 3-manifolds, the theory of “Artinization” of Thompson-like groups, two dimensional smooth dynamics, one dimensional complex dynamics (topology of the shift locus, Schottky spaces) and several other contexts. We will try to indicate how viewing these examples from the perspective of (big) mapping class groups is a worthwhile approach.

Part I - Theory : In the "theory" part of this mini-course, we will present recent objects and phenomena related to the study of big mapping class groups. In particular, we will define two faithful actions of some big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. We will describe some properties of ...

37FXX ; 57Mxx

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A dynamical system is called rigid if a weak form of equivalence with a nearby system, such as coincidence of some simple invariants, implies a strong form of equivalence. In this minicourse we will discuss smooth rigidity of hyperbolic dynamical systems and related geometric questions such as marked length spectrum rigidity of negatively curved manifolds. We will consider the following moduli: lengths of periodic orbits, spectra of Poincar\’e return maps of the periodic orbits, volume Lyapunov exponents. After a brief overview of some classical results we will focus on recent developments in rigidity of Anosov and partially hyperbolic systems as well as connections to geometric rigidity. The latter is based on joint work with B. Kalinin and V. Sadovskaya and with F. Rodriguez Hertz.

A dynamical system is called rigid if a weak form of equivalence with a nearby system, such as coincidence of some simple invariants, implies a strong form of equivalence. In this minicourse we will discuss smooth rigidity of hyperbolic dynamical systems and related geometric questions such as marked length spectrum rigidity of negatively curved manifolds. We will consider the following moduli: lengths of periodic orbits, spectra of Poincar\’e ...

37D20

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A dynamical system is called rigid if a weak form of equivalence with a nearby system, such as coincidence of some simple invariants, implies a strong form of equivalence. In this minicourse we will discuss smooth rigidity of hyperbolic dynamical systems and related geometric questions such as marked length spectrum rigidity of negatively curved manifolds. We will consider the following moduli: lengths of periodic orbits, spectra of Poincar\’e return maps of the periodic orbits, volume Lyapunov exponents. After a brief overview of some classical results we will focus on recent developments in rigidity of Anosov and partially hyperbolic systems as well as connections to geometric rigidity. The latter is based on joint work with B. Kalinin and V. Sadovskaya and with F. Rodriguez Hertz.

A dynamical system is called rigid if a weak form of equivalence with a nearby system, such as coincidence of some simple invariants, implies a strong form of equivalence. In this minicourse we will discuss smooth rigidity of hyperbolic dynamical systems and related geometric questions such as marked length spectrum rigidity of negatively curved manifolds. We will consider the following moduli: lengths of periodic orbits, spectra of Poincar\’e ...

37D20

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A dynamical system is called rigid if a weak form of equivalence with a nearby system, such as coincidence of some simple invariants, implies a strong form of equivalence. In this minicourse we will discuss smooth rigidity of hyperbolic dynamical systems and related geometric questions such as marked length spectrum rigidity of negatively curved manifolds. We will consider the following moduli: lengths of periodic orbits, spectra of Poincar\’e return maps of the periodic orbits, volume Lyapunov exponents. After a brief overview of some classical results we will focus on recent developments in rigidity of Anosov and partially hyperbolic systems as well as connections to geometric rigidity. The latter is based on joint work with B. Kalinin and V. Sadovskaya and with F. Rodriguez Hertz.

A dynamical system is called rigid if a weak form of equivalence with a nearby system, such as coincidence of some simple invariants, implies a strong form of equivalence. In this minicourse we will discuss smooth rigidity of hyperbolic dynamical systems and related geometric questions such as marked length spectrum rigidity of negatively curved manifolds. We will consider the following moduli: lengths of periodic orbits, spectra of Poincar\’e ...

37D20

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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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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