Topo-isomorphisms of irregular Toeplitz subshifts for residually finite groups

TL;DR

This paper constructs irregular Toeplitz subshifts over residually finite groups that are topo-isomorphic extensions of their maximal equicontinuous factors, expanding understanding of their measure and dynamical properties.

Contribution

It introduces conditions for Toeplitz subshifts to have invariant measures as limits of periodic measures and demonstrates the existence of such subshifts for residually finite groups.

Findings

Existence of irregular Toeplitz subshifts with invariant measures as limits of periodic measures

Construction of non-regular extensions of totally disconnected compactifications for amenable groups

Identification of conditions ensuring Toeplitz subshifts are topo-isomorphic to their maximal equicontinuous factors

Abstract

For each countable residually finite group $G$, we present examples of irregular Toeplitz subshifts in $\{0,1\}^G$ that are topo-isomorphic extensions of its maximal equicontinuous factor. To achieve this, we first establish sufficient conditions for Toeplitz subshifts to have invariant probability measures as limit points of periodic invariant measures of $\{0,1\}^G$. Next, we demonstrate that the set of Toeplitz subshifts satisfying these conditions is non-empty. When the acting group $G$ is amenable, this construction provides non-regular extensions of totally disconnected metric compactifications of $G$ that are (Weyl) mean-quicontinuous dynamical systems.