Flow Equivalence of G-SFTs

TL;DR

This paper classifies G-shifts of finite type up to flow equivalence using algebraic invariants, providing explicit classifications for certain cases and connecting to cellular automata involutions.

Contribution

It introduces an algebraic framework for classifying G-SFTs up to equivariant flow equivalence, including explicit invariants for specific cases.

Findings

Classification reduces to algebraic invariants of poset-blocked matrices over group rings.

Explicit invariants computed for two irreducible components with G=Z_2.

Connections established between G-SFTs and cellular automata involutions.

Abstract

In this paper, a G-shift of finite type (G-SFT) is a shift of finite type together with a free continuous shift-commuting action by a finite group G. We reduce the classification of G-SFTs up to equivariant flow equivalence to an algebraic classification of a class of poset-blocked matrices over the integral group ring of G. For a special case of two irreducible components with G$=\mathbb Z_2$, we compute explicit complete invariants. We relate our matrix structures to the Adler-Kitchens-Marcus group actions approach. We give examples of G-SFT applications, including a new connection to involutions of cellular automata.