Transitivity and topological mixing for C1 diffeomorphisms

TL;DR

This paper proves that for generic conservative and transitive C1-diffeomorphisms on connected compact manifolds, topological mixing is guaranteed, using homoclinic class period analysis and the closing lemma.

Contribution

It establishes topological mixing for generic C1-diffeomorphisms by analyzing homoclinic class periods and applying the closing lemma.

Findings

C1-generic conservative diffeomorphisms are topologically mixing

C1-generic transitive diffeomorphisms are topologically mixing

Period control of periodic points is key to the proof

Abstract

We prove that, on connected compact manifolds, both C1-generic conservative diffeomorphisms and C1-generic transitive diffeomorphisms are topologically mixing. This is obtained through a description of the periods of a homoclinic class and by a control of the period of the periodic points given by the closing lemma.