Cuntz-Krieger algebras and one-sided conjugacy of shifts of finite type and their groupoids

TL;DR

This paper links algebraic, groupoid, and dynamical structures of shifts of finite type to characterize their conjugacy classes, providing new insights and counterexamples in the theory of symbolic dynamics.

Contribution

It establishes a complete characterization of one-sided conjugacy of shifts of finite type via associated Cuntz-Krieger algebras and groupoids, extending previous results.

Findings

Characterization of conjugacy via algebraic and groupoid data

Counterexample showing eventual conjugacy does not imply conjugacy

Strengthening of Cuntz and Krieger's results

Abstract

A one-sided shift of finite type $(X_A,\sigma_A)$ determines on the one hand a Cuntz-Krieger algebra $\mathcal{O}_A$ with a distinguished abelian subalgebra $\mathcal{D}_A$ and a certain completely positive map $\tau_A$ on $\mathcal{O}_A$. On the other hand, $(X_A,\sigma_A)$ determines a groupoid $\mathcal{G}_A$ together with a certain homomorphism $\epsilon_A$ on $\mathcal{G}_A$. We show that this data completely characterizes the one-sided conjugacy class of $X_A$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This answers a question of Matsumoto of whether eventual conjugacy implies conjugacy in the negative.