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Homoclinic classes for flows: ergodicity and SRB measures

Ygor de Jesus, Marcielis Espitia, Gabriel Ponce

TL;DR

This work extends Hopf- and Pesin-theoretic techniques to nonuniformly hyperbolic flows to analyze ergodicity and SRB measures on homoclinic classes. It defines stable/unstable homoclinic classes for a hyperbolic periodic orbit γ and prove that positive Lebesgue weight on both classes yields a hyperbolic ergodic component, with SRB-measure analogues. A flow version of Katok’s periodic-orbit result links regular hyperbolic SRB measures to periodic orbits and shows ergodic components correspond to single homoclinic classes; moreover, SRB measures with full measure on a class are equal. Collectively, the results provide a robust Hopf-Pesin framework for decomposing nonuniformly hyperbolic flows into ergodic SRB components via homoclinic structure.

Abstract

In this work we intend to study homoclinic classes for some classes of flows. To this end we obtain analogous results those obtained by Hertz-Hertz-Tahzibi-Ures in the flow setting. Namely we prove that if the Lesbegue measure gives positive measure to both stable and unstable homoclinic classes of a periodic hyperbolic orbit, then their intersection constitute an ergodic component. Futhermore, with similar techiniques we state several results concerning regular SRB measures.

Homoclinic classes for flows: ergodicity and SRB measures

TL;DR

This work extends Hopf- and Pesin-theoretic techniques to nonuniformly hyperbolic flows to analyze ergodicity and SRB measures on homoclinic classes. It defines stable/unstable homoclinic classes for a hyperbolic periodic orbit γ and prove that positive Lebesgue weight on both classes yields a hyperbolic ergodic component, with SRB-measure analogues. A flow version of Katok’s periodic-orbit result links regular hyperbolic SRB measures to periodic orbits and shows ergodic components correspond to single homoclinic classes; moreover, SRB measures with full measure on a class are equal. Collectively, the results provide a robust Hopf-Pesin framework for decomposing nonuniformly hyperbolic flows into ergodic SRB components via homoclinic structure.

Abstract

In this work we intend to study homoclinic classes for some classes of flows. To this end we obtain analogous results those obtained by Hertz-Hertz-Tahzibi-Ures in the flow setting. Namely we prove that if the Lesbegue measure gives positive measure to both stable and unstable homoclinic classes of a periodic hyperbolic orbit, then their intersection constitute an ergodic component. Futhermore, with similar techiniques we state several results concerning regular SRB measures.

Paper Structure

This paper contains 11 sections, 16 theorems, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonuniform hyperbolicity
  4. Sinai-Ruelle-Bowen measures
  5. Typical points for smooth and SRB measures
  6. Proof of Theorem \ref{['criteria for ergodicity']}
  7. Proof of the second statement
  8. Proof of the first Statement
  9. Proof of Theorem \ref{['homoclinic class for SRB']}
  10. Proof of Theorem \ref{['SRB ergodic components has full measure homoclinic class']}
  11. Proof of Theorem \ref{['SRB measures with full measure homoclinic class are equal']}

Key Result

Theorem A

Let $\varphi_t:M \to M$ be a $C^{2}$-flow and let $m$ be a smooth $\varphi_t$-invariant measure. If, for a certain hyperbolic non-singular periodic orbit $\gamma$, we have $m(\Lambda^s(\gamma))\cdot m(\Lambda^u(\gamma))>0$, then:

Figures (4)

  • Figure 1: Illustration of the local dynamics around a point $z\in \Lambda(\gamma)$.
  • Figure 2: Representation of the stable holonomy map
  • Figure 3: Construction of the set $B_{\varphi_T(p)}$
  • Figure 4: Case with $\mathrm{dim}W^s(y_0)+\mathrm{dim}W^u(\gamma)=n$ and $\mathrm{dim}W^u(x)+\mathrm{dim}W^s(\gamma)=n$.

Theorems & Definitions (31)

  • Theorem A
  • Remark 1.1
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Theorem 2.2: Oseledets' Decomposition
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • ...and 21 more