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Homoclinic points of symplectic partially hyperbolic systems with 2D centre

Pengfei Zhang

Abstract

We consider a generic symplectic partially hyperbolic diffeomorphism close to direct/skew products of symplectic Anosov diffeomorphisms with area-preserving diffeomorphisms and prove that every hyperbolic periodic point has transverse homoclinic points.

Homoclinic points of symplectic partially hyperbolic systems with 2D centre

Abstract

We consider a generic symplectic partially hyperbolic diffeomorphism close to direct/skew products of symplectic Anosov diffeomorphisms with area-preserving diffeomorphisms and prove that every hyperbolic periodic point has transverse homoclinic points.

Paper Structure

This paper contains 12 sections, 16 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partial hyperbolicity
  4. Dynamical coherence and plaque expansiveness
  5. Symplectic diffeomorphisms
  6. Symplectic partially hyperbolic systems
  7. Birkhoff normal form and nonlinear stability
  8. Homoclinic intersections for surface diffeomorphisms
  9. Symplectic partially hyperbolic systems with 2D center
  10. Kupka--Smale property
  11. Perturbations of the center-leaf twist coefficients
  12. Proof of the main theorem

Key Result

Theorem 1.1

Suppose $r\ge 1$, $f\colon N\to N$ be a $C^r$ symplectic Anosov diffeomorphism, $g \colon S \to S$ an area-preserving diffeomorphism such that $f\times g$ is partially hyperbolic and $4$-normally hyperbolic. Then there is a $C^1$-open neighborhood $\mathcal{U}\subset\mathrm{Diff}^r_{\Omega}(M)$ of

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3: HPSPes04
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7: SX06
  • ...and 17 more