Generic bi-Lyapunov stable homoclinic classes

TL;DR

This paper investigates Lyapunov stable homoclinic classes in generic diffeomorphisms, showing they admit dominated splittings and often coincide with the entire manifold, especially in low dimensions.

Contribution

It establishes that Lyapunov stable homoclinic classes have dominated splittings and characterizes their structure in low dimensions, extending understanding of stability in dynamical systems.

Findings

Lyapunov stable classes admit dominated splittings.

In dimension 2, such classes are the whole manifold.

In dimension 3, these classes have nonempty interior.

Abstract

We study, for $C^1$ generic diffeomorphisms, homoclinic classes which are Lyapunov stable both for backward and forward iterations. We prove they must admit a dominated splitting and show that under some hypothesis they must be the whole manifold. As a consequence of our results we also prove that in dimension 2 the class must be the whole manifold and in dimension 3, these classes must have nonempty interior. Many results on Lyapunov stable homoclinic classes for $C^1$-generic diffeomorphisms are also deduced.