A Note on Measure-Expansive Diffeomorphisms

TL;DR

This paper establishes the equivalence between countably-expansive and measure-expansive homeomorphisms and explores the implications for the structure of expansive diffeomorphisms in the $C^1$ topology.

Contribution

It proves the equivalence of countably-expansive and measure-expansive homeomorphisms and characterizes the $C^1$-interior of various expansive diffeomorphism sets.

Findings

Countably-expansive and measure-expansive homeomorphisms are equivalent.

The $C^1$-interior of expansive, measure-expansive, and continuum-wise expansive diffeomorphisms coincide.

Abstract

In this note we prove that a homeomorphism is countably-expansive if and only if it is measure-expansive. This result is applied for showing that the $C^1$-interior of the sets of expansive, measure-expansive and continuum-wise expansive $C^1$-diffeomorphisms coincide.