A certain synchronizing property of subshifts and flow equivalence

TL;DR

This paper introduces a new class of subshifts called $bb$-synchronizing, proves their invariance under flow equivalence, and establishes new invariants for classifying such subshifts.

Contribution

It defines $bb$-synchronization in subshifts, proves its invariance under flow equivalence, and introduces new flow invariants based on K-groups and Bowen-Franks groups.

Findings

$bb$-synchronization is invariant under flow equivalence.

$bb$-synchronizing K-groups are flow invariants.

$bb$-synchronizing Bowen-Franks groups are flow invariants.

Abstract

We will study a certain synchronizing property of subshifts called $\lambda$-synchronization. The $\lambda$-synchronizing subshifts form a large class of irreducible subshifts containing irreducible sofic shifts. We prove that the $\lambda$-synchronization is invariant under flow equivalence of subshifts. The $\lambda$-synchronizing K-groups and the $\lambda$-synchronizing Bowen-Franks groups are studied and proved to be invariant under flow equivalence of $\lambda$-synchronizing subshifts. They are new flow equivalence invariants for $\lambda$-synchronizing subshifts.