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SRB measures for partially hyperbolic systems whose central direction is mostly expanding
, 2000
"... We construct Sinai-Ruelle-Bowen (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 non-uniformly expanding. If the r ..."
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Cited by 197 (44 self)
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We construct Sinai-Ruelle-Bowen (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 non-uniformly 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...
ON THE VOLUME OF SINGULAR-HYPERBOLIC SETS
, 2005
"... Abstract. An attractor Λ for a 3-vector field X is singular-hyperbolic if all its singularities are hyperbolic and it is partially hyperbolic with volume expanding central direction. We prove that C 1+α singularhyperbolic attractors, for some α> 0, always have zero volume, thus extending an analo ..."
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Cited by 8 (4 self)
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Abstract. An attractor Λ for a 3-vector field X is singular-hyperbolic if all its singularities are hyperbolic and it is partially hyperbolic with volume expanding central direction. We prove that C 1+α singularhyperbolic attractors, for some α> 0, always have zero volume, thus extending an analogous result for uniformly hyperbolic attractors. The same result holds for a class of higher dimensional singular attractors. Moreover, we prove that if Λ is a singular-hyperbolic attractor for X then either it has zero volume or X is an Anosov flow. We also present examples of C 1 singular-hyperbolic attractors with positive volume. In addition, we show that C 1 generically we have volume zero for C 1 robust classes of singular-hyperbolic attractors. Contents
Dominated splitting and zero volume for incompressible three-flows
- Nonlinearity
"... ABSTRACT. We prove that there exists an open and dense subset of the incompressible 3-flows of class C 2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincaré flow, then it must be an Anosov flow. With this result we are able ..."
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Cited by 3 (1 self)
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ABSTRACT. We prove that there exists an open and dense subset of the incompressible 3-flows of class C 2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincaré flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of Bochi-Mañé (see [26, 13, 9]) and of Newhouse (see [30, 10]) for flows with singularities. That is we obtain for a residual subset of the C 1 incompressible flows on 3-manifolds that: (i) either all Lyapunov exponents are zero or the flow is Anosov, and (ii) either the flow is Anosov or else the elliptic periodic points are dense in the manifold.
GIBBS-MARKOV STRUCTURES AND LIMIT LAWS FOR PARTIALLY HYPERBOLIC ATTRACTORS WITH MOSTLY EXPANDING CENTRAL DIRECTION
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0 PARTIALLY HYPERBOLIC SETS WITH POSITIVE MEASURE AND ACIP FOR PARTIALLY HYPERBOLIC SYSTEMS
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VOLUME OF PARTIALLY HYPERBOLIC HORSESHOE-LIKE ATTRACTORS
"... Abstract. We prove that any partially hyperbolic horseshoe-like attractor of a C1-generic diffeomorphism has zero volume. By modification of Poincare ́ cross section of the geo-metric model of Lorenz attractor, we build a C1-diffeomorphism with a partially hyperbolic horseshoe-like attractor of posi ..."
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Abstract. We prove that any partially hyperbolic horseshoe-like attractor of a C1-generic diffeomorphism has zero volume. By modification of Poincare ́ cross section of the geo-metric model of Lorenz attractor, we build a C1-diffeomorphism with a partially hyperbolic horseshoe-like attractor of positive volume. 1. introduction A hyperbolic set is a compact invariant set over which the tangent bundle splits into two invariant sub-bundles, one is contracting and the other one is expanding. The Lebesgue measure (volume) of hyperbolic sets is an interesting subject considered in many articles. The scenario was begun by the seminal works of Bowen in 70’s. Bowen has proved in [5] that a hyperbolic attractor of positive volume of a C2-diffeomorphism does contains some stable and unstable manifolds. On the other hand, he showed in [4] the existence of a C1-diffeomorphism admitting a totally disconnected hyperbolic set of positive volume. The issue of the volume and the interior of a hyperbolic sets followed by many authors. For instance, it is shown in [2] that a transitive hyperbolic set which attracts a set with positive volume necessarily attracts a neighborhood of itself. It is also proven in [2] that there are no
