On measure expansive diffeomorphisms

TL;DR

This paper investigates measure expansiveness in diffeomorphisms, showing that most are measure expansive away from homoclinic tangencies, while those with tangencies can be approximated by non-measure expansive maps.

Contribution

It demonstrates that measure expansive diffeomorphisms are generic away from homoclinic tangencies and that tangency cases can be approximated by non-measure expansive diffeomorphisms.

Findings

Residual subset diffeomorphisms are measure expansive

Surface diffeomorphisms with homoclinic tangencies can be approximated by non-measure expansive maps

Measure expansiveness is prevalent away from homoclinic tangencies

Abstract

Let $f: M \to M$ be a diffeomorphism defined on a compact boundaryless $d$-dimensional manifold $M$, $d\geq 2$. C. Morales has proposed the notion of measure expansiveness. In this note we show that diffeomorphisms in a residual subset far from homoclinic tangencies are measure expansive. We also show that surface diffeomorphisms presenting homoclinic tangencies can be $C^1$-approximated by non-measure expansive diffeomorphisms.