Homoclinic orbits and entropy for three-dimensional flows

TL;DR

This paper proves that any three-dimensional flow with positive topological entropy can be approximated by flows with homoclinic orbits, extending previous results from surface diffeomorphisms to three-dimensional flows.

Contribution

It establishes the approximation of three-dimensional flows with homoclinic orbits for flows with positive entropy, extending prior work on surface diffeomorphisms.

Findings

Flows with positive topological entropy can be approximated by flows with homoclinic orbits.

Extension of previous surface diffeomorphism results to three-dimensional flows.

Abstract

We prove that every $C^1$ three-dimensional flow with positive topological entropy can be $C^1$ approximated by flows with homoclinic orbits. This extends a previous result for $C^1$ surface diffeomorphisms \cite{g}.