Partial Hyperbolicity and Homoclinic Tangencies

TL;DR

This paper demonstrates that any diffeomorphism on a compact manifold can be closely approximated by either those with homoclinic tangencies or those with partial hyperbolic structures, highlighting the ubiquity of complex dynamical behaviors.

Contribution

It establishes a dichotomy showing that any diffeomorphism can be approximated by systems with either homoclinic tangencies or partial hyperbolic structures, advancing understanding of dynamical complexity.

Findings

Any diffeomorphism can be C1 approximated by one with a homoclinic tangency.

Any diffeomorphism can be C1 approximated by one with a partial hyperbolic structure.

The result links hyperbolic and non-hyperbolic dynamics in a unifying framework.

Abstract

We show that any diffeomorphism of a compact manifold can be C1 approximated by diffeomorphisms exhibiting a homoclinic tangency or by diffeomorphisms having a partial hyperbolic structure.