Expansive Minimal Flows

TL;DR

This paper characterizes minimal expansive flows without fixed points on compact metric spaces, showing they are suspensions of minimal subshifts and revealing that higher-dimensional regular expansive flows contain infinitely many minimal subsets.

Contribution

It extends Mañé's result to flows, providing a complete characterization of minimal expansive flows without fixed points on compact metric spaces.

Findings

Such flows are on one-dimensional sets

They are equivalent to suspensions of minimal subshifts

Higher-dimensional regular expansive flows have infinitely many minimal subsets

Abstract

In this paper, we extend a Ma\~n\'e's famous result on expansive homeomorphisms, originally presented in [17], to the setting of flows. Specifically, we provide a complete characterization of minimal expansive flows without fixed points on compact metric spaces. We prove that such flows must be defined on one-dimensional sets and are equivalent to the suspension of a minimal subshift. This result significantly improves upon [16] by eliminating the need for their additional hypothesis. Furthermore, we apply our findings to show that any regular expansive flow on a compact metric space of dimension two or higher must contain infinitely many minimal subsets.