Comparability of clopen sets in a zero-dimensional dynamical system

TL;DR

This paper establishes a connection between the comparability of clopen sets in a zero-dimensional dynamical system and the system's unique ergodicity, providing a new criterion for ergodic behavior.

Contribution

It introduces a binary relation on clopen sets and proves its comparability characterizes unique ergodicity in zero-dimensional systems.

Findings

Comparability of clopen sets is equivalent to unique ergodicity.

Provides a new criterion for identifying unique ergodicity.

Links topological structure to ergodic properties.

Abstract

We consider a homeomorphism on a totally disconnected, compact metric space and define a binary relation on the family of clopen subsets. We will show that the comparability of any clopen sets with respect to the relation is equivalent to the unique ergodicity of the homeomorphism.