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259
Recurrence Times And Rates Of Mixing
, 1997
"... The setting of this paper consists of a map making "nice" returns to a reference set. Criteria for the existence of equilibria, speed of convergence to equilibria and for the central limit theorem are given in terms of the tail of the return time function. The abstract setting considered a ..."
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Cited by 238 (10 self)
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The setting of this paper consists of a map making "nice" returns to a reference set. Criteria for the existence of equilibria, speed of convergence to equilibria and for the central limit theorem are given in terms of the tail of the return time function. The abstract setting considered arises naturally in differentiable dynamical systems with some expanding or hyperbolic properties. This paper is part of an attempt to understand the speed of mixing and related statistical properties for chaotic dynamical systems. More precisely, we are interested in systems that are expanding or hyperbolic on large parts (though not necessarily all) of their phase spaces. A natural approach to this problem is to pick a suitable reference set, and to regard a part of the system as having "renewed" itself when it makes a "full" return to this set. We obtain in this way a representation of the dynamical system in question, described in terms of a reference set and return times. We propose to study thi...
SRB measures for partially hyperbolic systems whose central direction is mostly expanding
, 2000
"... We construct SinaiRuelleBowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms  the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting  under the assumption that the complementary subbundle is nonuniformly expanding. If the r ..."
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Cited by 197 (44 self)
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We construct SinaiRuelleBowen (SRB) measures supported on partially hyperbolic sets of diffeomorphisms  the tangent bundle splits into two invariant subbundles, one of which is uniformly contracting  under the assumption that the complementary subbundle is nonuniformly expanding. If the rate of expansion (Lyapunov exponents) is bounded away from zero, then there are only finitely many SRB measures. Our techniques extend to other situations, including certain maps with singularities or critical points, as well as diffeomorphisms having only a dominated splitting (and no uniformly hyperbolic subbundle). 1 Introduction The following approach has been most effective in studying the dynamics of complicated systems: one tries to describe the average time spent by typical orbits in different regions of the phase space. According to the ergodic theorem of Birkhoff, such times are well defined for almost all point, with respect to any invariant probability measure. However, the...
What are SRB measures, and which dynamical systems have them?
"... This is a slightly expanded version of the text of a lecture I gave in a conference at Rutgers University in honor of David Ruelle and Yasha Sinai. In this lecture I reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle and Bowen in the 1970s. SRB measures ..."
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Cited by 124 (12 self)
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This is a slightly expanded version of the text of a lecture I gave in a conference at Rutgers University in honor of David Ruelle and Yasha Sinai. In this lecture I reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle and Bowen in the 1970s. SRB measures, as these objects are called, play an important role in the ergodic theory of dissipative dynamical systems with chaotic behavior. Roughly speaking, • SRB measures are the invariant measures most compatible with volume when volume is not preserved; • they provide a mechanism for explaining how local instability on attractors can produce coherent statistics for orbits starting from large sets in the basin. An outline of this paper is as follows. The original work of Sinai, Ruelle and Bowen was carried out in the context of Anosov and Axiom A systems. For these dynamical systems they identified and constructed an invariant measure which is uniquely important from several different points of view. These pioneering works are reviewed in Section 1. Subsequently, a nonuniform, almosteverywhere notion of hyperbolicity expressed in terms of Lyapunov exponents was developed. This notion provided a new framework for the ideas in the last paragraph. While not all of the previous characterizations are equivalent in this broader setting, the central ideas have remained intact, leading to a more general notion of SRB measures. This is discussed in Section 2. The extension above opened the door to the possibility that the dynamics on many attractors are described by SRB measures. Determining if this is (or is not) the case, however, let alone proving it, has turned out to be very challenging. No genuinely nonuniformly hyperbolic examples were known until the early 1990s, when SRB measures were constructed for certain Hénon maps. Today we still do not have a good understanding of which dynamical systems admit SRB measures, but some progress has been made; a sample of it is reported in Section 3.
Limit theorems for partially hyperbolic systems
 Trans. Amer. Math. Soc
"... Abstract. We consider a large class of partially hyperbolic systems containing, among others, ane maps, frame
ows on negatively curved manifolds and mostly contracting dieomorphisms. If the rate of mixing is suciently high the system satises many classical limit theorems of probability theory. 1. ..."
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Cited by 82 (14 self)
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Abstract. We consider a large class of partially hyperbolic systems containing, among others, ane maps, frame
ows on negatively curved manifolds and mostly contracting dieomorphisms. If the rate of mixing is suciently high the system satises many classical limit theorems of probability theory. 1. Introduction. The study of the statistical properties of deterministic systems constitutes an important branch of smooth ergodic theory. According to a modern view, a chaotic behavior of deterministic systems is caused by the exponential instability of nearby trajectories. The best illustra
Large Deviations for Nonuniformly Hyperbolic Systems
, 2006
"... We obtain large deviation estimates for a large class of nonuniformly hyperbolic systems: namely those modelled by Young towers with summable decay of correlations. In the case of exponential decay of correlations, we obtain exponential large deviation estimates given by a rate function. In the case ..."
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Cited by 70 (11 self)
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We obtain large deviation estimates for a large class of nonuniformly hyperbolic systems: namely those modelled by Young towers with summable decay of correlations. In the case of exponential decay of correlations, we obtain exponential large deviation estimates given by a rate function. In the case of polynomial decay of correlations, we obtain polynomial large deviation estimates, and exhibit examples where these estimates are essentially optimal. In contrast with many treatments of large deviations, our methods do not rely on thermodynamic formalism. Hence, for Hölder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton, as well as polynomial estimates in situations where there is not a unique equilibrium measure. 1
Chaotic billiards
 Mathematical Surveys and Monographs, 127. American Mathematical Society
, 2006
"... To Yakov Sinai on the occasion of his 70th birthday The authors are grateful to many colleagues who have read the manuscript and made numerous useful remarks, in particular P. Balint, D. Dolgopyat, C. Liverani, G. Del Magno, and H.K. Zhang. It is a pleasure to acknowledge the warm hospitality of IM ..."
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Cited by 69 (11 self)
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To Yakov Sinai on the occasion of his 70th birthday The authors are grateful to many colleagues who have read the manuscript and made numerous useful remarks, in particular P. Balint, D. Dolgopyat, C. Liverani, G. Del Magno, and H.K. Zhang. It is a pleasure to acknowledge the warm hospitality of IMPA (Rio de Janeiro), where the final version of the book was prepared. We also thank the anonymous referees for helpful comments. Last but not the least, the book was written at the suggestion of Sergei Gelfand and thanks to his constant encouragement. The first author was partially supported by NSF grant DMS0354775 (USA). The second author was partially supported by a Proyecto PDTConicyt (Uruguay). Contents Preface vii Symbols and notation ix
Markov extensions and decay of correlations for certain Hénon maps, Astérisque
, 2000
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Almost sure invariance principle for nonuniformly hyperbolic systems.
 Comm. Math. Phys.
, 2005
"... Abstract We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with fin ..."
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Cited by 63 (14 self)
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Abstract We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizon. Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences.
Decay of Correlations in OneDimensional Dynamics
, 2002
"... We consider multimodal C³ interval maps f satisfying a summability condition on the derivatives Dn along the critical orbits which implies the existence of an absolutely continuous finvariant probability measure µ. If f is nonrenormalizable, µ is mixing and we show that the speed of mixing (decay ..."
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Cited by 48 (17 self)
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We consider multimodal C³ interval maps f satisfying a summability condition on the derivatives Dn along the critical orbits which implies the existence of an absolutely continuous finvariant probability measure µ. If f is nonrenormalizable, µ is mixing and we show that the speed of mixing (decay of correlations) is strongly related to the rate of growth of the sequence (Dn) as n → ∞. We also give sufficient conditions for µ to satisfy the Central Limit Theorem. This applies for example to the quadratic Fibonacci map which is shown to have subexponential decay of correlations.