Path connectedness and entropy density of the space of ergodic hyperbolic measures (original) (raw)

Path connectedness and entropy density of the space of hyperbolic ergodic measures

Contemporary mathematics, 2017

We show that the space of hyperbolic ergodic measures of a given index supported on an isolated homoclinic class is path connected and entropy dense provided that any two hyperbolic periodic points in this class are homoclinically related. As a corollary we obtain that the closure of this space is also path connected.

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.

Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes

Ergodic Theory and Dynamical Systems, 2009

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 non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of f .

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 .

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.

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.

UNIFORM HYPERBOLIC APPROXIMATIONS OF MEASURES WITH NON-ZERO LYAPUNOV EXPONENTS

We show that for any C 1+α diffeomorphism of a compact Rie-mannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there exists a sequence Ω n of compact, topologically transitive, locally maximal, uniformly hyperbolic sets such that for any sequence {μ n } of f-invariant ergodic probability measures with supp(μ n) ⊆ Ω n we have μ n → μ in the weak-* topology.

On a theorem of Furstenberg and the structure of topologically ergodic measures

Proceedings of the American Mathematical Society, 1977

An almost everywhere convergence theorem for topologically ergodic measures stated by Furstenberg for homeomorphisms is extended to Markov operators on C ( X ) C(X) with compact Hausdörff state space. A structure theorem for topologically ergodic measures is obtained in the compact metric case again in the more general setting of Markov operators.

The ratio set of the harmonic measure of a random walk on a hyperbolic group

Israel Journal of Mathematics, 2008

We consider the harmonic measure on the Gromov boundary of a nonamenable hyperbolic group defined by a finite range random walk on the group, and study the corresponding orbit equivalence relation on the boundary. It is known to be always amenable and of type III. We determine its ratio set by showing that it is generated by certain values of the Martin kernel. In particular, we show that the equivalence relation is never of type III0.