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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, almost-everywhere notion of hyperbolicity expressed in terms of Lyapunov exponents was developed. This notion provided a new frame-work for the ideas in the last paragraph. While not all of the previous characteriza-tions 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 sys-tems containing, among others, ane maps, frame
ows on neg-atively 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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Abstract. We consider a large class of partially hyperbolic sys-tems containing, among others, ane maps, frame
ows on neg-atively 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 con-stitutes 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-
On mixing properties of compact group extensions of hyperbolic systems
- Israel J. Math
"... Abstract. We study compact group extensions of hyperbolic dif-feomorphisms. 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 dif-feomorphisms. 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.
Statistical stability for robust classes of maps with non-uniform expansion, Ergd
- Th. & Dynam. Sys
"... We consider open sets of maps in a manifold M exhibiting non-uniform expanding behaviour in some domain S ⊂ M. Assuming that there is a forward invariant region containing S where each map has a unique SRB measure, we prove that under general uniformity conditions, the SRB measure varies continuousl ..."
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Cited by 46 (20 self)
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We consider open sets of maps in a manifold M exhibiting non-uniform expanding behaviour in some domain S ⊂ M. Assuming that there is a forward invariant region containing S where each map has a unique SRB measure, we prove that under general uniformity conditions, the SRB measure varies continuously in the L 1-norm with the map. As a main application we show that the open class of maps introduced in [V] fits to this situation, thus proving that the SRB measures constructed in [A] vary continuously with the map. 1
Random perturbations of nonuniformly expanding maps
, 2000
"... We give both sufficient conditions and necessary conditions for the stochastic stability of non-uniformly expanding maps either with or without critical sets. We also show that the number of probability measures describing the statistical asymptotic behaviour of random orbits is bounded by the numbe ..."
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Cited by 43 (24 self)
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We give both sufficient conditions and necessary conditions for the stochastic stability of non-uniformly expanding maps either with or without critical sets. We also show that the number of probability measures describing the statistical asymptotic behaviour of random orbits is bounded by the number of SRB measures if the noise level is small enough. As an application of these results we prove the stochastic stability of certain classes of non-uniformly expanding maps introduced in [Vi1] and [ABV].
On dynamics of mostly contracting diffeomorphisms
- Comm. Math. Phys
"... Abstract. Mostly contracting dieomorphisms are the simplest examples of robustly nonuniformly hyperbolic systems. This pa-per studies the mixing properties of mostly contracting dieomor-phisms. 1. Introduction. This paper treats a class of partially hyperbolic systems with non-zero 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 pa-per studies the mixing properties of mostly contracting dieomor-phisms. 1. Introduction. This paper treats a class of partially hyperbolic systems with non-zero 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 non-uniformly 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.
On differentiability of SRB states for partially hyperbolic systems
- Invent. Math
"... Abstract. Consider a one parameter family of diffeomorphisms fε such that f0 is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let νε be any u-Gibbs state for fε. We prove (Theorem 1) that if A is a C function then the map A → νε(A) is differentia ..."
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Cited by 34 (6 self)
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Abstract. Consider a one parameter family of diffeomorphisms fε such that f0 is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let νε be any u-Gibbs state for fε. We prove (Theorem 1) that if A is a C function then the map A → νε(A) is differentiable at ε = 0. This implies (Corollary 1) that the difference of Birkhoff averages of the perturbed and unperturbed systems is proportional to ε. We apply this result (Corollary 3) to show that if f0 is a time one map of geodesic flow on a unit tangent bundle over a surface of negative curvature then a generic perturbation has a unique SRB measure with good statistical properties. 1.
Lyapunov exponents: How frequently are dynamical systems hyperbolic?
- IN ADVANCES IN DYNAMICAL SYSTEMS. CAMBRIDGE UNIV
, 2004
"... Lyapunov exponents measure the asymptotic behavior of tangent vectors under iteration, positive exponents corresponding to exponential growth and negative exponents corresponding to ex-ponential decay of the norm. Assuming hyperbolicity, that is, that no Lyapunov exponents are zero, Pesin theory p ..."
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Cited by 32 (2 self)
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Lyapunov exponents measure the asymptotic behavior of tangent vectors under iteration, positive exponents corresponding to exponential growth and negative exponents corresponding to ex-ponential decay of the norm. Assuming hyperbolicity, that is, that no Lyapunov exponents are zero, Pesin theory provides detailed geometric information about the system, that is at the basis of several deep results on the dynamics of hyperbolic systems. Thus, the question in the title is central to the whole theory. Here we survey and sketch the proofs of several recent results on genericity of vanishing and non-vanishing Lyapunov exponents. Genericity is meant in both topological and measure-theoretical sense. The results are for dynamical systems (diffeomorphisms) and for linear cocycles, a natural generalization of the tangent map which has an important role in Dynamics as well as in several other areas of Mathematics and its applications. The first section contains statements and a detailed discussion of main results. Outlines of proofs follow. In the last section and the appendices we prove a few useful related results.
Recent Results About Stable Ergodicity
- In Smooth ergodic theory and its applications
, 2000
"... this paper, has been directed toward extending their results beyond Axiom A. ..."
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Cited by 31 (8 self)
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this paper, has been directed toward extending their results beyond Axiom A.
