Flow equivalence and isotopy for subshifts

TL;DR

This paper investigates flow equivalence and isotopy in one-dimensional dynamical systems, especially shifts of finite type, providing new insights and a generalized discretization result relevant for classifying sofic shifts.

Contribution

It demonstrates that orbit-preserving flow maps are not always isotopic, except in certain cases, and extends the fundamental discretization theorem without injectivity or surjectivity constraints.

Findings

Orbit-preserving flow maps are not always isotopic.

In suspension flows of irreducible shifts of finite type, orbit-preserving maps are isotopic.

A generalized discretization theorem applicable without injectivity or surjectivity.

Abstract

We study basic properties of flow equivalence on one-dimensional compact metric spaces with a particular emphasis on isotopy in the group of (self-) flow equivalences on such a space. In particular, we show that an orbit-preserving such map is not always an isotopy, but that this always is the case for suspension flows of irreducible shifts of finite type. We also provide a version of the fundamental discretization result of Parry and Sullivan which does not require that the flow maps are either injective or surjective. Our work is motivated by applications in the classification theory of sofic shift spaces, but has been formulated to supply a solid and accessible foundation for other purposes.