Homoclinic points of symplectic partially hyperbolic systems with 2D centre

TL;DR

This paper proves that in generic symplectic partially hyperbolic systems with 2D center, every hyperbolic periodic point has transverse homoclinic points, highlighting complex dynamical behaviors.

Contribution

It establishes the existence of transverse homoclinic points for all hyperbolic periodic points in a broad class of symplectic partially hyperbolic systems.

Findings

Every hyperbolic periodic point has transverse homoclinic points.

Results apply to systems close to direct/skew products of symplectic Anosov and area-preserving diffeomorphisms.

Highlights generic dynamical complexity in these systems.

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.