Newhouse phenomenon and homoclinic classes

TL;DR

This paper investigates the structure of chain recurrent classes in certain diffeomorphisms, showing they are homoclinic classes with specific periodic points and limits, advancing understanding of dynamical systems.

Contribution

It establishes that for a generic set of diffeomorphisms, non-trivial chain recurrent classes are homoclinic classes with particular periodic point properties.

Findings

Non-trivial chain recurrent classes are homoclinic classes.

Such classes contain periodic points with index 1.

They are Hausdorff limits of sources.

Abstract

We show that for a $C^1$ residual subset of diffeomorphisms far away from tangency, every non-trivial chain recurrent class that is accumulated by sources ia a homoclinic class contains periodic points with index 1 and it's the Hausdorff limit of a family of sources.