Codimension one generic homoclinic classes with interior

TL;DR

This paper investigates generic diffeomorphisms with homoclinic classes having interior, showing that certain dominated splittings imply partial hyperbolicity and transitivity, advancing understanding of complex dynamical behaviors.

Contribution

It proves that for generic diffeomorphisms with a homoclinic class with interior and a codimension one dominated splitting, the system is partially hyperbolic and transitive.

Findings

Homoclinic class with interior implies partial hyperbolicity.

Extreme subbundles being one-dimensional leads to transitivity.

Results apply to generic diffeomorphisms with codimension one dominated splitting.

Abstract

We study generic diffeomorphisms with a homoclinc class with non empty interior and in particular those admitting a codimension one dominated splitting. We prove that if in the finest dominated splitting the extreme subbundles are one dimensional then the diffeomorphism is partially hyperbolic and from this we deduce that the diffeomorphism is transitive.