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19
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...
Stochasticlike behaviour in nonuniformly expanding maps Handbook of Dynamical Systems
, 2006
"... 1.1. Determinism versus randomness 2 1.2. Nonuniform expansivity 2 1.3. General overview of the notes 3 ..."
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Cited by 14 (2 self)
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1.1. Determinism versus randomness 2 1.2. Nonuniform expansivity 2 1.3. General overview of the notes 3
Equilibrium states for nonuniformly expanding maps. Ergodic Theory & Dynamical Systems
, 2003
"... We construct equilibrium states, including measures of maximal entropy, for a large (open) class of nonuniformly expanding maps on compact manifolds. Moreover, we study uniqueness of these equilibrium states, as well as some of their ergodic properties. 1 ..."
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Cited by 13 (1 self)
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We construct equilibrium states, including measures of maximal entropy, for a large (open) class of nonuniformly expanding maps on compact manifolds. Moreover, we study uniqueness of these equilibrium states, as well as some of their ergodic properties. 1
A minimum principle for Lyapunov exponents and a higherdimensional version of a Theorem of Mané
, 2003
"... Abstract. We consider compact invariant sets Λ for C 1 maps in arbitrary dimension. We prove that if Λ contains no critical points then there exists an invariant probability measure with a Lyapunov exponent λ which is the minimum of all Lyapunov exponents for all invariant measures supported on Λ. W ..."
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Cited by 9 (4 self)
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Abstract. We consider compact invariant sets Λ for C 1 maps in arbitrary dimension. We prove that if Λ contains no critical points then there exists an invariant probability measure with a Lyapunov exponent λ which is the minimum of all Lyapunov exponents for all invariant measures supported on Λ. We apply this result to prove that Λ is uniformly expanding if every invariant probability measure supported on Λ is hyperbolic repelling. This generalizes a well known theorem of Mañé to the higherdimensional setting. 1. Introduction and
Nonhyperbolic ergodic measures for nonhyperbolic homoclinic classes
, 2008
"... We prove that there is a residual subset S in Diff 1 (M) such that, for every f ∈ S, any homoclinic class of f containing saddles of different indices (dimension of the unstable bundle) contains also an uncountable support of an invariant ergodic nonhyperbolic (one of the Lyapunov exponents is equa ..."
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Cited by 8 (0 self)
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We prove that there is a residual subset S in Diff 1 (M) such that, for every f ∈ S, any homoclinic class of f containing saddles of different indices (dimension of the unstable bundle) contains also an uncountable support of an invariant ergodic nonhyperbolic (one of the Lyapunov exponents is equal to zero) measure of f. 1
Topological structure of (partially) hyperbolic sets with positive volume
, 2006
"... Abstract. We consider both hyperbolic sets and partially hyperbolic sets attracting a set of points with positive volume in a Riemannian manifold. We obtain several results on the topological structure of such sets for diffeomorphisms whose differentiability is bigger than one. We show in particular ..."
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Cited by 6 (3 self)
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Abstract. We consider both hyperbolic sets and partially hyperbolic sets attracting a set of points with positive volume in a Riemannian manifold. We obtain several results on the topological structure of such sets for diffeomorphisms whose differentiability is bigger than one. We show in particular that there are no partially hyperbolic horseshoes with positive volume for such diffeomorphisms. We also give a description of the limit set of almost every point belonging to a hyperbolic set or a partially hyperbolic set with positive volume. Contents
Ergodic Optimization
"... Let f be a realvalued function defined on the phase space of a dynamical system. Ergodic optimization is the study of those orbits, or invariant probability measures, whose ergodic faverage is as large as possible. In these notes we establish some basic aspects of the theory: equivalent definition ..."
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Cited by 4 (0 self)
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Let f be a realvalued function defined on the phase space of a dynamical system. Ergodic optimization is the study of those orbits, or invariant probability measures, whose ergodic faverage is as large as possible. In these notes we establish some basic aspects of the theory: equivalent definitions of the maximum ergodic average, existence and generic uniqueness of maximizing measures, and the fact that every ergodic measure is the unique maximizing measure for some continuous function. Generic properties of the support of maximizing measures are described in the case where the dynamics is hyperbolic. A number of problems are formulated.
SHADOWING BY NON UNIFORMLY HYPERBOLIC PERIODIC POINTS AND UNIFORM HYPERBOLICITY
, 2006
"... Abstract. We prove that, under a mild condition on the hyperbolicity of its periodic points, a map g which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive a ..."
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Cited by 4 (2 self)
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Abstract. We prove that, under a mild condition on the hyperbolicity of its periodic points, a map g which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question done by A. Katok, in a related context. 1.
MIXING AND DECAY OF CORRELATIONS IN NONUNIFORMLY EXPANDING MAPS
, 2004
"... I discuss recent results on decay of correlations for nonuniformly expanding maps. Throughout the discussion, I address the question of why different dynamical systems have different rates of decay of correlations and how this may reflect underlying geometrical characteristics of the system. ..."
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Cited by 3 (1 self)
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I discuss recent results on decay of correlations for nonuniformly expanding maps. Throughout the discussion, I address the question of why different dynamical systems have different rates of decay of correlations and how this may reflect underlying geometrical characteristics of the system.