Topologically Mixing Properties of Multiplicative Integer System

TL;DR

This paper explores the relationship between topologically mixing properties of multiplicative shift spaces and traditional shift spaces, establishing conditions under which these properties are equivalent or related.

Contribution

It provides a detailed analysis linking multiplicative shift spaces' mixing properties to those of classical shift spaces, including new results on directional mixing.

Findings

Multiplicative shift space is transitive if and only if the original shift space is extensible.

Multiplicative shift space is mixing if and only if the original shift space is mixing.

Introduces $l$-directional mixing property and relates it to weak mixing of the original space.

Abstract

Motivated from the study of multiple ergodic average, the investigation of multiplicative shift spaces has drawn much of interest among researchers. This paper focuses on the relation of topologically mixing properties between multiplicative shift spaces and traditional shift spaces. Suppose that $\mathsf{X}_{\Omega}^{(l)}$ is the multiplicative subshift derived from the shift space $\Omega$ with given $l > 1$. We show that $\mathsf{X}_{\Omega}^{(l)}$ is (topologically) transitive/mixing if and only if $\Omega$ is extensible/mixing. After introducing $l$-directional mixing property, we derive the equivalence between $l$-directional mixing property of $\mathsf{X}_{\Omega}^{(l)}$ and weakly mixing property of $\Omega$.