Unique ergodicity for zero-entropy dynamical systems with the approximate product property

TL;DR

This paper proves that in topological dynamical systems with the approximate product property, zero entropy is equivalent to unique ergodicity, and explores related properties like minimality and periodic points.

Contribution

It establishes the equivalence between zero entropy and unique ergodicity for systems with the approximate product property, extending understanding of their dynamical behavior.

Findings

Zero entropy is equivalent to unique ergodicity under the approximate product property.

Minimality is also equivalent under a stronger condition.

Unique ergodicity implies the approximate product property if periodic points exist.

Abstract

We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronger condition. Moreover, we show that unique ergodicity implies the approximate product property if the system has periodic points.