Decidability of the isomorphism and the factorization between minimal substitution subshifts

TL;DR

This paper proves the decidability of isomorphism and factorization problems for minimal substitution subshifts by establishing computable bounds and algorithms, advancing classification methods in dynamical systems.

Contribution

It introduces a computable constant for factor maps and provides an algorithm to decide factor and isomorphism relations between minimal substitution subshifts.

Findings

Existence of a computable radius R for factor maps

Decidability of checking factor maps via sliding block codes

Algorithm to determine factor and isomorphism relations

Abstract

Classification is a central problem for dynamical systems, in particular for families that arise in a wide range of topics, like substitution subshifts. It is important to be able to distinguish whether two such subshifts are isomorphic, but the existing invariants are not sufficient for this purpose. We first show that given two minimal substitution subshifts, there exists a computable constant $R$ such that any factor map between these subshifts (if any) is the composition of a factor map with a radius smaller than $R$ and some power of the shift map. Then we prove that it is decidable to check whether a given sliding block code is a factor map between two prescribed minimal substitution subshifts. As a consequence of these two results, we provide an algorithm that, given two minimal substitution subshifts, decides whether one is a factor of the other and, as a straightforward corollary, whether they are isomorphic.