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259
On mixing properties of compact group extensions of hyperbolic systems
 Israel J. Math
"... Abstract. We study compact group extensions of hyperbolic diffeomorphisms. We relate mixing properties of such extensions with accessibility properties of their stable and unstable laminations. We show that generically the correlations decay faster than any power of time. In particular, this is alw ..."
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Cited by 46 (7 self)
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Abstract. We study compact group extensions of hyperbolic diffeomorphisms. We relate mixing properties of such extensions with accessibility properties of their stable and unstable laminations. We show that generically the correlations decay faster than any power of time. In particular, this is always the case for ergodic semisimple extensions as well as for stably ergodic extensions of Anosov diffeomorphisms of infranilmanifolds. 1.
A vectorvalued almost sure invariance principle for hyperbolic dynamical systems
, 2006
"... Abstract We prove an almost sure invariance principle (approximation by ddimensional Brownian motion) for vectorvalued H"older observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom A diffeomorphisms and flows as well as systems modelled by ..."
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Cited by 44 (10 self)
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Abstract We prove an almost sure invariance principle (approximation by ddimensional Brownian motion) for vectorvalued H&quot;older observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom A diffeomorphisms and flows as well as systems modelled by Young towers with moderate tail decay rates. In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a 2dimensional Brownian motion. 1 Introduction The scalar almost sure invariance principle (ASIP), or approximation by onedimensional Brownian motion, is a strong statistical property of sequences of random variables introduced by Strassen [40, 41]. It implies numerous other statistical limit laws including the central limit theorem, the functional central limit theorem, and the law of the iterated logarithm. See [23, 38] and references therein for a survey of consequences of the ASIP.
Escape rates and conditionally invariant measures
, 2005
"... We consider dynamical systems on domains that are not invariant under the dynamics – for example, a system with a hole in the phase space – and raise issues regarding the meaning of escape rates and conditionally invariant measures. Equating observable events with sets of positive Lebesgue measure, ..."
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Cited by 41 (6 self)
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We consider dynamical systems on domains that are not invariant under the dynamics – for example, a system with a hole in the phase space – and raise issues regarding the meaning of escape rates and conditionally invariant measures. Equating observable events with sets of positive Lebesgue measure, we are led quickly to conditionally invariant measures that are absolutely continuous with respect to Lebesgue (a.c.c.i.m.). Comparisons with SRB measures are inevitable, yet there are important differences. Via informal discussions and examples, this paper seeks to clarify the ideas involved. It includes also a brief review of known results and possible directions of further work in this developing subject.
Central Limit Theorems and Invariance Principles for Lorenz Attractors
, 2006
"... We prove statistical limit laws for Hölder observations of the Lorenz attractor, and more generally for geometric Lorenz attractors. In particular, we prove the almost sure invariance principle (approximation by Brownian motion). Standard consequences of this result include the central limit theo ..."
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Cited by 40 (17 self)
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We prove statistical limit laws for Hölder observations of the Lorenz attractor, and more generally for geometric Lorenz attractors. In particular, we prove the almost sure invariance principle (approximation by Brownian motion). Standard consequences of this result include the central limit theorem, the law of the iterated logarithm, and the functional versions of these results.
Statistical Limit Theorems for Suspension Flows
, 2004
"... In dynamical systems theory, a standard method for passing from discrete time to continuous time is to construct the suspension flow under a roof function. ..."
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Cited by 39 (17 self)
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In dynamical systems theory, a standard method for passing from discrete time to continuous time is to construct the suspension flow under a roof function.
On dynamics of mostly contracting diffeomorphisms
 Comm. Math. Phys
"... Abstract. Mostly contracting dieomorphisms are the simplest examples of robustly nonuniformly hyperbolic systems. This paper studies the mixing properties of mostly contracting dieomorphisms. 1. Introduction. This paper treats a class of partially hyperbolic systems with nonzero Lyapunov exponent ..."
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Cited by 35 (6 self)
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Abstract. Mostly contracting dieomorphisms are the simplest examples of robustly nonuniformly hyperbolic systems. This paper studies the mixing properties of mostly contracting dieomorphisms. 1. Introduction. This paper treats a class of partially hyperbolic systems with nonzero Lyapunov exponents. Before stating our result let us recall some recent work motivating our research. In recent years there were several advances in understanding
HITTING TIME STATISTICS AND EXTREME VALUE THEORY
, 2008
"... We consider discrete time dynamical system and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory (distribution properties of the partial maximum of stochastic processes). This relation allows to study ..."
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Cited by 35 (12 self)
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We consider discrete time dynamical system and show the link between Hitting Time Statistics (the distribution of the first time points land in asymptotically small sets) and Extreme Value Theory (distribution properties of the partial maximum of stochastic processes). This relation allows to study Hitting Time Statistics with tools from Extreme Value Theory, and vice versa. We apply these results to nonuniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws (with no extra condition on the speed of mixing, for example). We extend these ideas to the subsequent returns to the asymptotically small sets, linking the Poisson statistics of both processes.
Stability of statistical properties in twodimensional piecewise hyperbolic maps Trans.
 Am. Math. Soc.
, 2008
"... Abstract. We investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions. We accomplish such a program in the case of twodimensional maps with uniformly bounded second derivative. For ..."
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Cited by 34 (17 self)
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Abstract. We investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions. We accomplish such a program in the case of twodimensional maps with uniformly bounded second derivative. For the class of systems at hand, we obtain a complete description of the SRB measures, their statistical properties and their stability with respect to many types of perturbations, including deterministic and random perturbations and holes.
Decay of Correlations for Lorentz Gases and Hard Balls
, 2000
"... We discuss rigorous results and open problems on the decay of correlations for dynamical systems characterized by elastic collisions: hard balls, Lorentz gases, Sinai billiards and related models. Recently developed techniques for general dynamical systems with some hyperbolic behavior are discuss ..."
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Cited by 31 (2 self)
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We discuss rigorous results and open problems on the decay of correlations for dynamical systems characterized by elastic collisions: hard balls, Lorentz gases, Sinai billiards and related models. Recently developed techniques for general dynamical systems with some hyperbolic behavior are discussed. These techniques give exponential decay of correlations for many classes of billiards and open the door to further investigations.
Random perturbations of chaotic dynamical systems: stability of the spectrum
 Nonlinearity
, 1998
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