Homoclinic classes with shadowing property

TL;DR

This paper proves that for generic $C^1$ diffeomorphisms, isolated homoclinic classes are shadowable precisely when they are hyperbolic basic sets, linking shadowing property to hyperbolicity.

Contribution

It establishes a necessary and sufficient condition for shadowing in isolated homoclinic classes in the context of generic $C^1$ diffeomorphisms.

Findings

Shadowing property characterizes hyperbolic basic sets in isolated homoclinic classes.

For $C^1$ generic diffeomorphisms, shadowable isolated homoclinic classes are exactly hyperbolic.

The result provides a clear criterion connecting shadowing and hyperbolicity in dynamical systems.

Abstract

We show that for $C^1$ generic diffeomorphisms, an isolated homoclinic class is shadowable if and only if homoclinic class is hyperbolic basic set.