Separating Homeomorphisms

TL;DR

This paper investigates the properties of separating homeomorphisms on totally disconnected compact metric spaces, establishing conditions for expansivity, recurrence, and asymptotic points, and characterizing spaces admitting such homeomorphisms.

Contribution

It provides new results linking separating homeomorphisms with expansivity, recurrence, and the structure of the underlying space, including a characterization of expansivity via cyclic groups.

Findings

Separating homeomorphisms are expansive except at periodic points on totally disconnected spaces.

Minimal separating homeomorphisms are necessarily expansive.

Only finite spaces admit separating or finite expansive recurrent homeomorphisms.

Abstract

We show that on a totally disconnected compact metric space every separating homeomorphisms is expansive except at periodic points. We conclude that minimal separating homeomorphisms are expansive and that every separating homeomorphism has asymptotic points. We show that the only spaces admitting separating (or finite expansive) and recurrent homeomorphisms are finite sets. We apply our results to give a characterization of expansivity in terms of the expansivity of the cyclic group of powers of the homeomorphism.