On the complexity of topological conjugacy of Toeplitz subshifts

TL;DR

This paper investigates the complexity of classifying Toeplitz subshifts up to conjugacy, revealing that the problem's difficulty varies significantly between amenable and non-amenable groups, with implications for descriptive set theory.

Contribution

It demonstrates that conjugacy of Toeplitz subshifts is amenable for al, but non-amenable for non-amenable groups, advancing understanding of classification complexity in symbolic dynamics.

Findings

Conjugacy is amenable for al groups.

Conjugacy is non-amenable for non-amenable groups.

Results address a question by Gao, Jackson, and Seward.

Abstract

In this paper, we study the descriptive set theoretic complexity of the equivalence relation of conjugacy of Toeplitz subshifts of a residually finite group $G$. On the one hand, we show that if $G = \mathbb{Z}$, then topological conjugacy on Toeplitz subshifts with separated holes is amenable. In contrast, if $G$ is non-amenable, then conjugacy of Toeplitz $G$-subshifts is a non-amenable equivalence relation. The results were motivated by a general question, asked by Gao, Jackson and Seward, about the complexity of conjugacy for minimal, free subshifts of countable groups.