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- ix; 316 p.
ISBN 978-1-4704-2773-3

Contemporary mathematics , 0698

Localisation : Collection 1er étage

Nikolai Chernov # système dynamique # théorie ergodique # probabilité # mécanique statistique

11J70 ; 37A25 ; 37A35 ; 37A50 ; 37A60 ; 37C20 ; 37C29 ; 37D50 ; 37E10

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- 228 p.
ISBN 978-0-8218-4235-5

Contemporary mathematics , 0444

Localisation : Collection 1er étage

analyse de Fourier trigonométrique # développement # théorie de Riemann # espace HP # analyse harmonique # ondelette # système dynamique # théorie ergodique # théorie spectrale # opérateur de Markov # probabilités # processus stochastiques

00B25 ; 42-06 ; 42A63 ; 42B30 ; 42B35 ; 42C15 ; 42C40 ; 43-06 ; 37-06 ; 37A30 ; 37A50 ; 60-06 ; 60F05 ; 65D10

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- 145 p.
ISBN 978-0-8218-3869-3

Contemporary mathematics , 0430

Localisation : Collection 1er étage

théorie ergodique # système dynamique @ transformation conservant la mesure # équivalence d'orbite # transformation de Hilbert

28D05 ; 37A05 ; 37A20 ; 37A50 ; 47A35 ; 47A16 ; 60F15 ; 82C20 ; 60G50

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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 $n$), ...

37D20 ; 37D25 ; 37A50

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

I will discuss the simplest possible (non trivial) example of a fast-slow partially hyperbolic system with particular emphasis on the problem of establishing its statistical properties.

37A25 ; 37C30 ; 37D30 ; 37A50 ; 60F17

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Research talks;Dynamical Systems and Ordinary Differential Equations

We obtain results on mixing and rates of mixing for infinite measure semiflows and flows. The results on rates of mixing rely on operator renewal theory and a Dolgopyat-type estimate. The results on mixing hold more generally and are based on a deterministic (ie non iid) version of Erickson's continuous time strong renewal theorem.

37A25 ; 37A40 ; 37A50 ; 37D25

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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 ...

60H10 ; 37D20 ; 37D25 ; 37A50

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

The Poisson limit theorem which appeared in 1837 seems to be the first law of rare events in probability. Various generalizations of it and estimates of errors of Poisson approximations were obtained in probability and more recently this became a popular topic in dynamics in the form of study of asymptotics of numbers of arrivals at small (shrinking) sets by a stochastic process or by a dynamical system. I will describe recent results on Poisson and compound Poisson asymptotics in a nonconventional setup, i.e. for numbers of events of multiple returns to shrinking sets, namely, for numbers of combined events of the type $\left \{ \omega : \xi \left ( jn,\omega\right )\in \Gamma_N,j = 1,...,\ell \right \},n\leq N$ where $\xi \left ( k,\omega \right )$ is defined as a stochastic process from the beginning or it is built from a dynamical system by writing $\xi \left ( k,\omega \right )=T^k\omega .$ We obtain an essentially complete description of possible limiting behaviors of distributions of numbers of multiple recurrencies to shrinking cylinders for $\psi $-mixing shifts. Some possible extensions and related questions will be discussed, as well. Most of the results were obtained jointly with my student Ariel Rapaport and some of them are new even for the widely studied single (conventional) recurrencies case.
Keywords : Poisson limit theorems - nonconventional sums - multiple recurrence
The Poisson limit theorem which appeared in 1837 seems to be the first law of rare events in probability. Various generalizations of it and estimates of errors of Poisson approximations were obtained in probability and more recently this became a popular topic in dynamics in the form of study of asymptotics of numbers of arrivals at small (shrinking) sets by a stochastic process or by a dynamical system. I will describe recent results on Poisson ...

60F05 ; 37D35 ; 37A50

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Research talks;Dynamical Systems and Ordinary Differential Equations

37D40 ; 37A50 ; 37D10

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

We study law of rare events for random dynamical systems. We obtain an exponential law (with respect to the invariant measure of the skew-product) for super-polynomially mixing random dynamical systems.
For random subshifts of finite type, we analyze the distribution of hitting times with respect to the sample measures. We prove that with a superpolynomial decay of correlations one can get an exponential law for almost every point and with stronger mixing assumptions one can get a law of rare events depending on the extremal index for every point. (These are joint works with Benoit Saussol and Paulo Varandas, and Mike Todd).
We study law of rare events for random dynamical systems. We obtain an exponential law (with respect to the invariant measure of the skew-product) for super-polynomially mixing random dynamical systems.
For random subshifts of finite type, we analyze the distribution of hitting times with respect to the sample measures. We prove that with a superpolynomial decay of correlations one can get an exponential law for almost every point and with ...

37B20 ; 37A50 ; 37A25 ; 37Dxx

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

37A25 ; 37A50 ; 60F17 ; 60G10

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

We discuss various limit theorems for "nonconventional" sums of the form $\sum ^N_{n=1}F\left ( \xi \left ( n \right ),\xi \left ( 2n \right ),...,\xi \left ( \ell n \right ) \right )$ where $\xi \left ( n \right )$ is a stochastic process or a dynamical system. The motivation for this study comes, in particular, from many papers about nonconventional ergodic theorems appeared in the last 30 years. Such limit theorems describe multiple recurrence properties of corresponding stochastic processes and dynamical systems. Among our results are: central limit theorem, a.s. central limit theorem, local limit theorem, large deviations and averaging. Some multifractal type questions and open problems will be discussed, as well.
Keywords : limit theorems - nonconventional sums - multiple recurrence
We discuss various limit theorems for "nonconventional" sums of the form $\sum ^N_{n=1}F\left ( \xi \left ( n \right ),\xi \left ( 2n \right ),...,\xi \left ( \ell n \right ) \right )$ where $\xi \left ( n \right )$ is a stochastic process or a dynamical system. The motivation for this study comes, in particular, from many papers about nonconventional ergodic theorems appeared in the last 30 years. Such limit theorems describe multiple ...

60F05 ; 37D35 ; 37A50

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

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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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 ...

37A50 ; 37D50 ; 60F05 ; 37D20

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- xxiv; 487 p.
ISBN 978-3-642-54615-0

Mathématiques & applications , 0075

Localisation : Collection 1er étage

processus stochastique # théorie des probabilités # statistique bayésienne # traitement du signal # combinatoire énumérative # optimisation combinatoire # physique quantique

37A50 ; 46N30 ; 60H99 ; 60J20 ; 60J25 ; 60J60 ; 60J75 ; 62L20 ; 60-02 ; 60G05 ; 00A69

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- xxii; 266 p.
ISBN 978-2-85629-843-5

Astérisque , 0382

Localisation : Périodique 1er étage

forme modulaire de Hilbert # forme modulaire $\rho$-adique # forme modulaire surconvergente # représentation galoisienne # modularité # conjecture d'Artin # conjecture de Fontaine-Mazur

37A20 ; 37D25 ; 37D30 ; 37A50 ; 37C40

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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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- 230 p.
ISBN 978-3-540-34705-7

Theoretical and mathematical physics

Localisation : Ouvrage RdC (COLL)

système dynamique # système hyperbolique # shadowing # variétés invariantes # entropie # mesure ergodique # exposant de Lyapunov # mécanique statistique

37-01 ; 37A25 ; 37A30 ; 37A35 ; 37A50 ; 37D05 ; 37D20 ; 82-01

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- 266 p.
ISBN 978-0-521-83653-1

Cambridge tracts in mathematics , 0162

Localisation : Collection 1er étage

probabilité # processus de Levy # groupe de Lie # groupe de Lie semi-simple # comportement dynamique # flux stochastique

60G51 ; 17B99 ; 58J65 ; 43A80 ; 37H15 ; 58G32 ; 37A50

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- 144 p.
ISBN 978-3-540-42415-4

Lecture notes in mathematics , 1766

Localisation : Collection 1er étage

statistique # système dynamique # chaîne de Markov # théorème limite # opérateur quasi-compact # quasi-compacité # noyau de Fourier # noyau de Lipschitz # grande déviation

60F05 ; 60F10 ; 60J10 ; 37A25 ; 37A99 ; 37A50

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