Structurally Stable Homoclinic Classes

TL;DR

This paper investigates the properties of structurally stable homoclinic classes, establishing conditions under which they admit dominated splittings and are hyperbolic, advancing understanding of their stability and structure.

Contribution

It proves that structurally stable homoclinic classes admit dominated splittings and are hyperbolic under certain conditions, addressing a key open question.

Findings

Structurally stable homoclinic classes admit dominated splittings.

Codimension one structurally stable classes are hyperbolic.

Far from homoclinic tangencies, such classes are hyperbolic.

Abstract

In this paper we study structurally stable homoclinic classes. In a natural way, the structural stability for an individual homoclinic class is defined through the continuation of periodic points. Since the homoclinic classes is not innately locally maximal, it is hard to answer whether structurally stable homoclinic classes are hyperbolic. In this article, we make some progress on this question. We prove that if a homoclinic class is structurally stable, then it admits a dominated splitting. Moreover we prove that codimension one structurally stable classes are hyperbolic. Also, if the diffeomorphism is far away from homoclinic tangencies, then structurally stable homoclinic classes are hyperbolic.