Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes (original) (raw)

Non-hyperbolic ergodic measures with large support

Nonlinearity, 2010

We prove that there is a residual subset S in Diff 1 (M ) such that, for every f ∈ S, any homoclinic class of f with invariant one dimensional central bundle containing saddles of different indices (i.e. with different dimensions of the stable invariant manifold) coincides with the support of some invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of f .

New criteria for ergodicity and non-uniform hyperbolicity

2009

In this work we obtain a new criterion to establish ergodicity and non-uniform hyperbolicity of smooth measures of diffeomorphisms. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of non-zero Lyapunov exponents in some contexts. In the partial hyperbolicity context, we obtain that stably ergodic diffeomorphisms are C^1-dense among volume preserving partially hyperbolic diffeomorphisms with two-dimensional center bundle. This is motivated by a well known conjecture of C. Pugh and M. Shub.

Robust criterion for the existence of nonhyperbolic ergodic measures

We give explicit C1C^1C1-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with one-dimensional center and positive topological entropy on which the center Lyapunov exponent vanishes uniformly. The conditions of the criterion are met on a C1C^1C1-dense and open subset of the set of a diffeomorphisms having a robust cycle. As a corollary, there exists a C1C^1C1-open and dense subset of the set of non-Anosov robustly transitive diffeomorphisms consisting of systems with nonhyperbolic ergodic measures with positive entropy. The criterion is based on a notion of a blender defined dynamically in terms of strict invariance of a family of discs.

A criterion for ergodicity of non-uniformly hyperbolic diffeomorphisms

arXiv preprint arXiv:0710.2353, 2007

In this work we exhibit a new criteria for ergodicity of diffeomorphisms involving conditions on Lyapunov exponents and general position of some invariant manifolds. On one hand we derive uniqueness of SRB-measures for transitive surface diffeomorphisms. On the other hand, using recent results on the existence of blenders we give a positive answer, in the C 1 topology, to a conjecture of Pugh-Shub in the context of partially hyperbolic conservative diffeomorphisms with two dimensional center bundle.

Notes on Ergodicity of Partially Hyperbolic Systems

2008

These lecture notes were written to accompany a minicourse given at the School and Workshop on Dynamical Systems at IMPA (Rio de Janeiro), January 2006. Warning: These notes are unedited lecture notes and do contain errors! Proceed with caution... Let f : M → M be a (pointwise) partially hyperbolic diffeomor-phism. Recall what partially hyperbolic means. At every point in M , there exist tangent vectors that are uniformly contracted by the derivative T f and tangent vectors that are uniformly expanded by T f. The contracted vectors lie in an invariant subbundle E s of T M , called the stable subbundle, and the expanded vectors lie in an invariant subbun-dle E u , called the unstable subbundle. The sum of these subbundles E u ⊕ E s is the hyperbolic part of T M. The rest of T M is also under control: there is a T f-invariant complement E c to E u ⊕ E s in T M , called the center bundle. Tangent vectors in E c may be expanded or contracted under the action of T f , but they are neithe...

$C^r$-prevalence of stable ergodicity for a class of partially hyperbolic systems

Journal of the European Mathematical Society

We prove that for r ∈ N ≥2 ∪ {∞}, for any dynamically coherent, center bunched and strongly pinched volume preserving C r partially hyperbolic diffeomorphism f : X → X, if either (1) its center foliation is uniformly compact, or (2) its center-stable and center-unstable foliations are of class C 1 , then there exists a C 1-open neighbourhood of f in Diff r (X, Vol), in which stable ergodicity is C r-prevalent in Kolmogorov's sense. In particular, we verify Pugh-Shub's stable ergodicity conjecture in this region. This also provides the first result that verifies the prevalence of stable ergodicity in the measuretheoretical sense. Our theorem applies to a large class of algebraic systems. As applications, we give affirmative answers in the strongly pinched region to: 1. an open question of Pugh-Shub in [32]; 2. a generic version of an open question of Hirsch-Pugh-Shub in [25]; and 3. a generic version of an open question of Pugh-Shub in [25]. Contents * M.

Stably ergodic diffeomorphisms which are not partially hyperbolic

Israel Journal of Mathematics, 2004

We show stable ergodicity of a class of conservative diffeomorphisms of T n which do not have any hyperbolic invariant subbundle. Moreover, the uniqueness of SRB (Sinai-Ruelle-Bowen) measure for non-conservative C 1 perturbations of such diffeomorphisms is verified. This class strictly contains non-partially hyperbolic robustly transitive diffeomorphisms constructed by Bonatti-Viana [BV00] and so we answer the question posed there on the stable ergodicity of such systems.

On hyperbolic measures and periodic orbits

Discrete and Continuous Dynamical Systems, 2006

We prove that if a diffeomorphism on a compact manifold preserves a nonatomic ergodic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measure does not locally maximize entropy in the space of invariant ergodic hyperbolic measures, then there exist hyperbolic periodic points that satisfy a multiplicative asymptotic growth and are uniformly distributed with respect to this measure.

On the ergodicity of partially hyperbolic systems

Annals of Mathematics, 2010

Pugh and Shub have conjectured that essential accessibility implies ergodicity for a C 2 , partially hyperbolic, volume-preserving diffeomorphism. We prove this conjecture under a mild center bunching assumption, which is satisfied in particular by all partially hyperbolic systems with 1-dimensional center bundle. We also obtain ergodicity results for C 1Cı partially hyperbolic systems.

Stably Ergodic Dynamical Systems and Partial Hyperbolicity

Journal of Complexity, 1997

In this paper we show that a little hyperbolicity goes a long way toward guaranteeing stable ergodicity, and in fact may be necessary for it. Our main theorem may be interpreted as saying that the same phenomenon producing chaotic behavior (i.e., some hyperbolicity) also leads to robust statistical behavior. Examples to which our theory applies include translations on certain homogeneous spaces and the time-one map of the geodesic flow for a manifold of constant negative curvature.