Flow equivalence of sofic beta-shifts

TL;DR

This paper investigates flow equivalence of sofic beta-shifts, constructing covers to classify their invariants, and explores how these shifts relate to each other through flow and group actions, advancing the classification of complex shift spaces.

Contribution

The paper introduces new constructions of covers for sofic beta-shifts, shows their flow invariants depend on a single integer, and connects these to group actions for classification purposes.

Findings

Every strictly sofic beta-shift is 2-sofic.

Flow invariants depend only on a single integer from the beta-expansion of 1.

Beta-shifts are flow equivalent to those with 1<β<2.

Abstract

The Fischer, Krieger, and fiber product covers of sofic beta-shifts are constructed and used to show that every strictly sofic beta-shift is $2$-sofic. Flow invariants based on the covers are computed, and shown to only depend on an single integer easily determined from the $\beta$-expansion of 1. It is shown that any beta-shift is flow equivalent to a beta-shift given by some $1< \beta < 2$, and concrete constructions lead to further reductions of the flow classification problem. For each sofic beta-shift, there is an action of $\mathbb Z/2\mathbb Z$ on the edge shift given by the fiber product, and it is shown precisely when there exists a flow equivalence respecting these $\mathbb Z/2\mathbb Z$-actions. This opens a connection to ongoing efforts to classify general irreducible 2-sofic shifts via flow equivalences of reducible SFTs equipped with $\mathbb Z/2\mathbb Z$-actions.