The structure of pointwise recurrent expansive homeomorphisms

TL;DR

This paper characterizes pointwise recurrent expansive homeomorphisms on compact metric spaces, showing they are conjugate to subshifts, and discusses conditions under which these subshifts are semisimple.

Contribution

It establishes a topological conjugacy between pointwise recurrent expansive homeomorphisms and subshifts, and explores the role of positive recurrence in semisimplicity.

Findings

Recurrent expansive homeomorphisms are conjugate to subshifts.

Positive recurrence implies the subshift is semisimple.

Counterexample shows positive recurrence is necessary for semisimplicity.

Abstract

Let $X$ be a compact metric space and let $f:X\rightarrow X$ be a homeomorphism on $X$. We show that if $f$ is both pointwise recurrent and expansive, then the dynamical system $(X, f)$ is topologically conjugate to a subshift of some symbolic system. Moreover, if $f$ is pointwise positively recurrent, then the subshift is semisimple; a counterexample is given to show the necessity of positive recurrence to ensure the semisimilicity.