Topological Mixing Properties of Rank-One Subshifts

TL;DR

This paper investigates the topological mixing properties of rank-one subshifts, characterizing their factors and mixing behaviors based on spacer parameters, and providing examples to illustrate the distinctions.

Contribution

It offers a complete characterization of mixing for bounded spacer parameters and introduces new conditions and examples for unbounded spacer parameters.

Findings

Maximal equicontinuous factor is finite for rank-one subshifts.

Complete characterization of weak and strong mixing for bounded spacer parameters.

Examples showing the limitations of characterizations in the unbounded spacer case.

Abstract

We study topological mixing properties and the maximal equicontinuous factor of rank-one subshifts as topological dynamical systems. We show that the maximal equicontinuous factor of a rank-one subshift is finite. We also determine all the finite factors of a rank-one shift with a condition involving the cutting and spacer parameters. For rank-one subshifts with bounded spacer parameter we completely characterize weak mixing and mixing. For rank-one subshifts with unbounded spacer parameter we prove some sufficient conditions for weak mixing and mixing. We also construct some examples showing that the characterizations for the bounded spacer parameter case do not generalize to the unbounded spacer parameter case.