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Ergodic measures with multi-zero Lyapunov exponents inside homoclinic classes

Xiaodong Wang, Jinhua Zhang

Abstract

We prove that for $C^1$ generic diffeomorphisms, if a homoclinic class $H(P)$ contains two hyperbolic periodic orbits of indices $i$ and $i+k$ respectively and $H(P)$ has no domination of index $j$ for any $j\in\{i+1,\cdots,i+k-1\}$, then there exists a non-hyperbolic ergodic measure whose $(i+l)^{th}$ Lyapunov exponent vanishes for any $l\in\{1,\cdots, k\}$, and whose support is the whole homoclinic class. We also prove that for $C^1$ generic diffeomorphisms, if a homoclinic class $H(P)$ has a dominated splitting of the form $E\oplus F\oplus G$, such that the center bundle $F$ has no finer dominated splitting, and $H(p)$ contains a hyperbolic periodic orbit $Q_1$ of index $\dim(E)$ and a hyperbolic periodic orbit $Q_2$ whose absolute Jacobian along the bundle $F$ is strictly less than $1$, then there exists a non-hyperbolic ergodic measure whose Lyapunov exponents along the center bundle $F$ all vanish and whose support is the whole homoclinic class.

Ergodic measures with multi-zero Lyapunov exponents inside homoclinic classes

Abstract

We prove that for generic diffeomorphisms, if a homoclinic class contains two hyperbolic periodic orbits of indices and respectively and has no domination of index for any , then there exists a non-hyperbolic ergodic measure whose Lyapunov exponent vanishes for any , and whose support is the whole homoclinic class. We also prove that for generic diffeomorphisms, if a homoclinic class has a dominated splitting of the form , such that the center bundle has no finer dominated splitting, and contains a hyperbolic periodic orbit of index and a hyperbolic periodic orbit whose absolute Jacobian along the bundle is strictly less than , then there exists a non-hyperbolic ergodic measure whose Lyapunov exponents along the center bundle all vanish and whose support is the whole homoclinic class.

Paper Structure

This paper contains 32 sections, 17 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Lyapunov exponents
  4. Chain recurrence
  5. A criterion to the ergodicity of convergence
  6. Perturbation technics
  7. Generic diffeomorphisms
  8. Ergodic measure with multi-zero Lyapunov exponents for the case controlled by norm: Proof of Theorem \ref{['Thm: A']}
  9. Proof of Theorem \ref{['Thm: A']}
  10. Choice of $P_0$, $N_0$ and $\gamma_0$
  11. Construct $P_n$, $N_n$ and $\gamma_n$ inductively
  12. End of proof of Theorem \ref{['Thm: A']}
  13. Good approximation with weaker center Lyapunov exponents: Proof of Proposition \ref{['proposition']}
  14. Construction of the periodic orbit $P_1$
  15. Choice of the integers $m$ and $n$
  16. ...and 17 more sections

Key Result

Theorem 1

For generic $f\in\operatorname{Diff}^1(M)$, consider a hyperbolic periodic orbit $P$ of index $i$. Assume that the homoclinic class $H(P,f)$ contains a hyperbolic periodic orbit $Q$ of index $i-1$, then $H(P,f)$ supports a non-hyperbolic ergodic measure, whose $i^{th}$ Lyapunov exponent vanishes. If

Theorems & Definitions (35)

  • Theorem 1: DGBDG
  • Theorem A
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem B
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem: DGBDG
  • ...and 25 more