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On the existence of non-hyperbolic ergodic measures as the limit of periodic measures

Christian Bonatti, Jinhua Zhang

Abstract

[GIKN] and [BBD1] propose two very different ways for building non hyperbolic measures, [GIKN] building such a measure as the limit of periodic measures and [BBD1] as the $ω$-limit set of a single orbit, with a uniformly vanishing Lyapunov exponent. The technique in [GIKN] was essentially used in a generic setting, as the periodic orbits were built by small perturbations. It is not known if the measures obtained by the technique in [BBD1] are accumulated by periodic measures. In this paper we use a shadowing lemma from [G]: $\bullet$for getting the periodic orbits in [GIKN] without perturbing the dynamics, $\bullet$for recovering the compact set in [BBD1] with a uniformly vanishing Lyapunov exponent by considering the limit of periodic orbits. As a consequence, we prove that there exists an open and dense subset $\mathcal{U}$ of the set of robustly transitive non-hyperbolic diffeomorphisms far from homoclinic tangencies, such that for any $f\in\mathcal{U}$, there exists a non-hyperbolic ergodic measure with full support and approximated by hyperbolic periodic measures. We also prove that there exists an open and dense subset $\mathcal{V}$ of the set of diffeomorphisms exhibiting a robust cycle, such that for any $f\in\mathcal{V}$, there exists a non-hyperbolic ergodic measure approximated by hyperbolic periodic measures.

On the existence of non-hyperbolic ergodic measures as the limit of periodic measures

Abstract

[GIKN] and [BBD1] propose two very different ways for building non hyperbolic measures, [GIKN] building such a measure as the limit of periodic measures and [BBD1] as the -limit set of a single orbit, with a uniformly vanishing Lyapunov exponent. The technique in [GIKN] was essentially used in a generic setting, as the periodic orbits were built by small perturbations. It is not known if the measures obtained by the technique in [BBD1] are accumulated by periodic measures. In this paper we use a shadowing lemma from [G]: for getting the periodic orbits in [GIKN] without perturbing the dynamics, for recovering the compact set in [BBD1] with a uniformly vanishing Lyapunov exponent by considering the limit of periodic orbits. As a consequence, we prove that there exists an open and dense subset of the set of robustly transitive non-hyperbolic diffeomorphisms far from homoclinic tangencies, such that for any , there exists a non-hyperbolic ergodic measure with full support and approximated by hyperbolic periodic measures. We also prove that there exists an open and dense subset of the set of diffeomorphisms exhibiting a robust cycle, such that for any , there exists a non-hyperbolic ergodic measure approximated by hyperbolic periodic measures.

Paper Structure

This paper contains 20 sections, 24 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Results in the global setting
  3. Results in the semi-local setting of robust cycles
  4. Preliminaries
  5. Dominated splitting, partial hyperbolicity and hyperbolicity
  6. Center Lyapunov exponent of ergodic measures supported on a partially hyperbolic set
  7. Homoclinic class
  8. A criterion for ergodicity of the limit measure of periodic measures
  9. A shadowing lemma
  10. Plaque family, hyperbolic time and estimate on the size of the invariant manifold
  11. Control of averages at any scale
  12. Flip flop family and the control at any scale
  13. Blender
  14. Flip-flop configuration
  15. Flip flop configuration and flip flop family
  16. ...and 5 more sections

Key Result

Theorem \oldthetheorem

There exists an open and dense subset $\tilde{\mathcal{T}}(M)$ of $\mathcal{T}(M)$, such that for any $f\in\tilde{\mathcal{T}}(M)$, there exist $k$ non-hyperbolic ergodic measures $\mu_1,\cdots,\mu_k$ such that for any $i=1,\cdots,k$, we have that

Theorems & Definitions (51)

  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem A
  • Theorem B
  • Remark \oldthetheorem
  • Theorem C
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • Remark \oldthetheorem
  • ...and 41 more