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# Documents  37D20 | enregistrements trouvés : 38

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## A microlocal toolbox for hyperbolic dynamics Dyatlov, Semyon | CIRM H

Post-edited

Research talks;Partial Differential Equations;Dynamical Systems and Ordinary Differential Equations;Geometry

I will discuss recent applications of microlocal analysis to the study of hyperbolic flows, including geodesic flows on negatively curved manifolds. The key idea is to view the equation $(X + \lambda)u = f$ , where $X$ is the generator of the flow, as a scattering problem. The role of spatial infinity is taken by the infinity in the frequency space. We will concentrate on the case of noncompact manifolds, featuring a delicate interplay between shift to higher frequencies and escaping in the physical space. I will show meromorphic continuation of the resolvent of $X$; the poles, known as Pollicott-Ruelle resonances, describe exponential decay of correlations. As an application, I will prove that the Ruelle zeta function continues meromorphically for flows on non-compact manifolds (the compact case, known as Smale's conjecture, was recently settled by Giulietti-Liverani- Pollicott and a simple microlocal proof was given by Zworski and the speaker). Joint work with Colin Guillarmou. I will discuss recent applications of microlocal analysis to the study of hyperbolic flows, including geodesic flows on negatively curved manifolds. The key idea is to view the equation $(X + \lambda)u = f$ , where $X$ is the generator of the flow, as a scattering problem. The role of spatial infinity is taken by the infinity in the frequency space. We will concentrate on the case of noncompact manifolds, featuring a delicate interplay between ...

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## Séminaire Bourbaki. Vol. 2008/2009: exposés 997-1011 | Société Mathématique de France 2010

Congrès

- x; 463 p.
ISBN 978-2-85629-291-4

Astérisque , 0332

Localisation : Périodique 1er étage

dimères planaires # surfaces aléatoires # courbes de Harnack # groupe de Cremona # application birationnelle # espace métrique hyperbolique # groupes discrets # géométrie convexe # représentations # groupes algébriques # structures géométriques localement homogènes # flots d'Anosov # groupes de Coxeter # espaces métriques hyperboliques # géométrie finslérienne # fusion # surface de Del Pezzo # fibré en coniques # borne de Minkowski # sous-groupe fini # Équation d'Euler incompressible # plongement isométrique non lisse # inclusion différentielle # intégration convexe # solutions faibles paradoxales # conjecture de Weinstein # équations de Seiberg-Witten # homologie de Seiberg-Witten-Floer # homologie de contact plongée # pincement # flot de Ricci # courbure isotrope # transformations de contact # groupes partiellement ordonnés # fonctions génératrices # courbes elliptiques # géométrie algébrique dérivée # champs de modules # corps de classes # schéma arithmétique # groupe fondamental abélianisé # zéro-cycles # marginales presque gaussiennes # corps convexes # mesures log-concaves # concentration # flots sur les espaces homogènes # classification des mesures invariantes # rigidité entropique # approximation diophantienne # sous-convexité # fonctions L # transport optimal # équations du type Monge-Ampère # estimations a priori # courbure et géométrie riemannienne # lieu de coupure # programme de Langlands # théorie de jauge # S-dualité # symétrie miroir # espace de modules de Hitchin # variété hyperbolique # variété de dimension 3 # volume dimères planaires # surfaces aléatoires # courbes de Harnack # groupe de Cremona # application birationnelle # espace métrique hyperbolique # groupes discrets # géométrie convexe # représentations # groupes algébriques # structures géométriques localement homogènes # flots d'Anosov # groupes de Coxeter # espaces métriques hyperboliques # géométrie finslérienne # fusion # surface de Del Pezzo # fibré en coniques # borne de Minkowski # sous-groupe ...

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## Geometric and probabilistic structures in dynamics:workshop on dynamical systems and related topics in honor of Michael Brin on the occasion of his 60th birthdayCollege Park # march 15-18, 2008 Burns, Keith ; Dolgopyat, Dmitry ; Pesin, Yakov | American Mathematical Society 2008

Congrès

- xv; 340 p.
ISBN 978-0-8218-4286-7

Contemporary mathematics , 0469

Localisation : Collection 1er étage

système dynamique # méthode probabiliste # géométrie riemannienne # biologie # dynamique symbolique # dynamique stochastique # dynamique aléatoire

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## Transfer operators for Anosov flows - lecture 3 Tsuijii, Masato | CIRM H

Multi angle

Research schools

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.

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## Transfer operators for Anosov flows - lecture 2 Tsuijii, Masato | CIRM H

Multi angle

Research schools

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.

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## Transfer operators for Anosov flows - lecture 1 Tsuijii, Masato | CIRM H

Multi angle

Research schools

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.

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## The asymptotic spectral gap of hyperbolic dynamics : tentative to improve the estimates Faure, Frédéric | CIRM H

Multi angle

Research talks;Partial Differential Equations;Dynamical Systems and Ordinary Differential Equations;Mathematical Physics

Hyperbolic (Anosov or Axiom A) flows have discrete Ruelle spectrum. For contact Anosov flows, e.g. geodesic flows, where a smooth contact one form is preserved, the trapped set is a smooth symplectic manifold, normally hyperbolic, and M. Tsujii, S. Nonnenmacher and M. Zworski, have given an estimate for the asymptotic spectral gap, i.e. that appears in the limit of high frequencies in the flow direction. We will propose a different approach that may improve this estimate. This will be presented on a simple toy model, partially expanding maps. Work with Tobias Weich. Hyperbolic (Anosov or Axiom A) flows have discrete Ruelle spectrum. For contact Anosov flows, e.g. geodesic flows, where a smooth contact one form is preserved, the trapped set is a smooth symplectic manifold, normally hyperbolic, and M. Tsujii, S. Nonnenmacher and M. Zworski, have given an estimate for the asymptotic spectral gap, i.e. that appears in the limit of high frequencies in the flow direction. We will propose a different approach that ...

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## Rigidity in rank one: dynamics and geometry - lecture 3 Gogolev, Andrey | CIRM H

Multi angle

Research schools

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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## Rigidity in rank one: dynamics and geometry - lecture 2 Gogolev, Andrey | CIRM H

Multi angle

Research schools

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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## Rigidity in rank one: dynamics and geometry - lecture 1 Gogolev, Andrey | CIRM H

Multi angle

Research schools

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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## ​​​​Mixing and the local central limit theorem for hyperbolic dynamical systems Nándori, Péter | CIRM H

Multi angle

Research talks

We present a convenient joint generalization of mixing and the local version of the central limit theorem (MLLT) for probability preserving dynamical systems. We verify that MLLT holds for several examples of hyperbolic systems by reviewing old results for maps and presenting new results for flows. Then we discuss applications such as proving various mixing properties of infinite measure preserving systems. Based on joint work with Dmitry Dolgopyat. We present a convenient joint generalization of mixing and the local version of the central limit theorem (MLLT) for probability preserving dynamical systems. We verify that MLLT holds for several examples of hyperbolic systems by reviewing old results for maps and presenting new results for flows. Then we discuss applications such as proving various mixing properties of infinite measure preserving systems. Based on joint work with Dmitry ...

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## Linear and fractional response: a survey Baladi, Viviane | CIRM H

Multi angle

Research schools

When a dynamical system admitting a natural (SRB) measure is perturbed, it is natural to ask how the SRB measure responds to the perturbation. In the tamest cases, this response is linear, and the derivative of the SRB measure with respect to the parameter can be expressed as a sum of decorrelations (involving the derivative of the system with respect to the parameter). In more subtle situations - for example, systems with bifurcations, or observables with singularities - the SRB measure may be a Hölder function of the parameter. This talk will present a panorama of results about linear and fractional response. When a dynamical system admitting a natural (SRB) measure is perturbed, it is natural to ask how the SRB measure responds to the perturbation. In the tamest cases, this response is linear, and the derivative of the SRB measure with respect to the parameter can be expressed as a sum of decorrelations (involving the derivative of the system with respect to the parameter). In more subtle situations - for example, systems with bifurcations, or ...

37D20

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## Fast slow systems with chaotic noise Kelly, David | CIRM H

Multi angle

Research talks;Dynamical Systems and Ordinary Differential Equations;Probability and Statistics

It has long been observed that multi-scale systems, particularly those in climatology, exhibit behavior typical of stochastic models, most notably in the unpredictability and statistical variability of events. This is often in spite of the fact that the underlying physical model is completely deterministic. One possible explanation for this stochastic behavior is deterministic chaotic effects. In fact, it has been well established that the statistical properties of chaotic systems can be well approximated by stochastic differential equations. In this talk, we focus on fast-slow ODEs, where the fast, chaotic variables are fed into the slow variables to yield a diffusion approximation. In particular we focus on the case where the chaotic noise is multidimensional and multiplicative. The tools from rough path theory prove useful in this difficult setting. It has long been observed that multi-scale systems, particularly those in climatology, exhibit behavior typical of stochastic models, most notably in the unpredictability and statistical variability of events. This is often in spite of the fact that the underlying physical model is completely deterministic. One possible explanation for this stochastic behavior is deterministic chaotic effects. In fact, it has been well established that the ...

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## Entropy and growth of periodic orbits for Anosov flows and their covers Sharp, Richard | CIRM H

Multi angle

Research schools

In this talk, we will discuss various growth rates associated to Anosov flows and their covers. The topological entropy of an Anosov flow on a compact manifold is realised as the exponential growth rate of its periodic orbits. If we pass to a regular cover of the manifold then we can consider a corresponding growth rate for the lifted flow. This growth is bounded above by the topological entropy but if the cover is infinite then the growth rate may be strictly smaller. For abelian covers, this phenomenon admits a precise description in terms of a variational principle. More recent work, joint with Rhiannon Dougall, considers more general infinite covers. In this talk, we will discuss various growth rates associated to Anosov flows and their covers. The topological entropy of an Anosov flow on a compact manifold is realised as the exponential growth rate of its periodic orbits. If we pass to a regular cover of the manifold then we can consider a corresponding growth rate for the lifted flow. This growth is bounded above by the topological entropy but if the cover is infinite then the growth rate ...

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## Emergence of the quantum wave equation in classical deterministic hyperbolic dynamics Faure, Frédéric | CIRM H

Multi angle

Research talks

In the 80’s, D. Ruelle, D. Bowen and others have introduced probabilistic and spectral methods in order to study deterministic chaos (”Ruelle resonances”). For a geodesic flow on a strictly negative curvature Riemannian manifold, following this approach and use of microlocal analysis, one obtains that long time fluctuations of classical probabilities are described by an effective quantum wave equation. This may be surprising because there is no added quantization procedure. We will discuss consequences for the zeros of dynamical zeta functions. This shows that the problematic of classical chaos and quantum chaos are closely related. Joint work with Masato Tsujii. In the 80’s, D. Ruelle, D. Bowen and others have introduced probabilistic and spectral methods in order to study deterministic chaos (”Ruelle resonances”). For a geodesic flow on a strictly negative curvature Riemannian manifold, following this approach and use of microlocal analysis, one obtains that long time fluctuations of classical probabilities are described by an effective quantum wave equation. This may be surprising because there is no ...

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## ​Diffusion limit for a slow-fast standard map De Simoi, Jacopo | CIRM H

Multi angle

Research talks

​Consider the map $(x, z) \mapsto (x + \epsilon^{-\alpha} \sin (2\pi x) + \epsilon^{-(1+\alpha)}z, z + \epsilon \sin(2\pi x))$, which is conjugate to the Chirikov standard map with a large parameter. For suitable $\alpha$, we obtain a central limit theorem for the slow variable $z$ for a (Lebesgue) random initial condition. The result is proved by conjugating to the Chirikov standard map and utilizing the formalism of standard pairs. Our techniques also yield for the Chirikov standard map a related limit theorem and a ''finite-time'' decay of correlations result.
This is joint work with Alex Blumenthal and Ke Zhang.
​Consider the map $(x, z) \mapsto (x + \epsilon^{-\alpha} \sin (2\pi x) + \epsilon^{-(1+\alpha)}z, z + \epsilon \sin(2\pi x))$, which is conjugate to the Chirikov standard map with a large parameter. For suitable $\alpha$, we obtain a central limit theorem for the slow variable $z$ for a (Lebesgue) random initial condition. The result is proved by conjugating to the Chirikov standard map and utilizing the formalism of standard pairs. Our ...

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## A perspective on the The Fibonacci trace map Damanik, David | CIRM H

Multi angle

Research talks;Analysis and its Applications;Dynamical Systems and Ordinary Differential Equations;Mathematical Physics

In this talk we explain how the Fibonacci trace map arises from the Fibonacci substitution and leads to a unified framework in which a variety of models can be studied. We discuss the associated foliations, hyperbolic sets, stable and unstable manifolds, and how the intersections of the stable manifolds with the model-dependent curve of initial conditions allow one to translate dynamical into spectral results.

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## A brief introduction to concentration inequalities Chazottes, Jean-René | CIRM H

Multi angle

Research schools

$Let (X,T)$ be a dynamical system preserving a probability measure $\mu$. A concentration inequality quantifies how small is the probability for $F(x,Tx,\ldots,T^{n-1}x)$ to deviate from $\int F(x,Tx,\ldots,T^{n-1}x) \mathrm{d}\mu(x)$ by an given amount $u$, where $F:X^n\to\mathbb{R}$ is supposed to be separately Lipschitz. The bound on that probability involves a constant $C$ depending only on the dynamical system (thus independent of $n$), and $\sum_{i=0}^{n-1} \mathrm{Lip}_i(F)^2$. In the best situation, the bound is $\exp(-C u^2/\sum_{i=0}^{n-1} \mathrm{Lip}_i(F)^2)$.
After explaining how to get such a bound for independent random variables, I will show how to prove it for a Gibbs measure on a shift of finite type with a Lipschitz potential, and present examples of functions $F$ to which one can apply the inequality. Finally, I will survey some results obtained for nonuniformly hyperbolic systems modeled by Young towers.
$Let (X,T)$ be a dynamical system preserving a probability measure $\mu$. A concentration inequality quantifies how small is the probability for $F(x,Tx,\ldots,T^{n-1}x)$ to deviate from $\int F(x,Tx,\ldots,T^{n-1}x) \mathrm{d}\mu(x)$ by an given amount $u$, where $F:X^n\to\mathbb{R}$ is supposed to be separately Lipschitz. The bound on that probability involves a constant $C$ depending only on the dynamical system (thus independent of $n$), ...

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## Thermodynamic formalism and applications to dimension theory Barreira, Luis | Birkhäuser 2011

Ouvrage

- xii; 297 p.
ISBN 978-3-0348-0205-5

Progress in mathematics , 0294

Localisation : Collection 1er étage

système dynamique # analyse multifractale # formalisme thermodynamique

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## The real fatou conjecture Graczyk, Jacek ; Swiatek, Grzegors | Princeton University Press 1998

Ouvrage

- 148 p.
ISBN 978-0-691-00258-3

Annals of mathematics studies , 0144

Localisation : Ouvrage RdC (GRAC)

conjecture réelle de Fatou # géodésique # hyperbolicité # polynôme quadratique # représentation # système dynamique # théorie ergodique

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