The complexity of the topological conjugacy problem for Toeplitz subshifts

TL;DR

This paper investigates the Borel complexity of topological conjugacy among Toeplitz subshifts, establishing hyperfiniteness for certain classes and addressing a question posed by Sabok and Tsankov.

Contribution

It proves that topological conjugacy is hyperfinite for Toeplitz subshifts with separated holes and those with growing blocks, advancing understanding of their classification complexity.

Findings

Topological conjugacy is hyperfinite for Toeplitz subshifts with separated holes.

The relation is also hyperfinite for Toeplitz subshifts with growing blocks.

Provides partial answers to a question by Sabok and Tsankov.

Abstract

In this paper, we analyze the Borel complexity of the topological conjugacy relation on Toeplitz subshifts. More specifically, we prove that topological conjugacy of Toeplitz subshifts with separated holes is hyperfinite. Indeed, we show that the topological conjugacy relation is hyperfinite on a larger class of Toeplitz subshifts which we call Toeplitz subshifts with growing blocks. This result provides a partial answer to a question asked by Sabok and Tsankov.