C*-algebras, groupoids and covers of shift spaces

TL;DR

This paper links the dynamics of shift spaces with C*-algebras and groupoids, providing algebraic characterizations of various conjugacy and equivalence relations, especially for sofic shifts with effective groupoids.

Contribution

It introduces a framework connecting shift space relations to isomorphisms of associated groupoids and C*-algebras, extending the classification of shift spaces via algebraic invariants.

Findings

Characterizes conjugacy and orbit equivalence via groupoid and C*-algebra isomorphisms.

Shows how to recover flow and orbit equivalence classes from algebraic data for sofic shifts.

Establishes that continuous orbit equivalence implies flow equivalence in certain shift classes.

Abstract

To every one-sided shift space $\mathsf{X}$ we associate a cover $\tilde{\mathsf{X}}$, a groupoid $\mathcal{G}_{\mathsf{X}}$ and a $\mathrm{C^*}$-algebra $\mathcal{O}_{\mathsf{X}}$. We characterize one-sided conjugacy, eventual conjugacy and (stabilizer preserving) continuous orbit equivalence between $\mathsf{X}$ and $\mathsf{Y}$ in terms of isomorphism of $\mathcal{G}_{\mathsf{X}}$ and $\mathcal{G}_{\mathsf{Y}}$, and diagonal preserving $^*$-isomorphism of $\mathcal{O}_{\mathsf{X}}$ and $\mathcal{O}_{\mathsf{Y}}$. We also characterize two-sided conjugacy and flow equivalence of the associated two-sided shift spaces $\Lambda_{\mathsf{X}}$ and $\Lambda_{\mathsf{Y}}$ in terms of isomorphism of the stabilized groupoids $\mathcal{G}_{\mathsf{X}}\times \mathcal{R}$ and $\mathcal{G}_{\mathsf{Y}}\times \mathcal{R}$, and diagonal preserving $^*$-isomorphism of the stabilized $\mathrm{C^*}$-algebras $\mathcal{O}_{\mathsf{X}}\otimes \mathbb{K}$ and $\mathcal{O}_{\mathsf{Y}}\otimes \mathbb{K}$. Our strategy is to lift relations on the shift spaces to similar relations on the covers. Restricting to the class of sofic shifts whose groupoids are effective, we show that it is possible to recover the continuous orbit equivalence class of $\mathsf{X}$ from the pair $(\mathcal{O}_{\mathsf{X}}, C(\mathsf{X}))$, and the flow equivalence class of $\Lambda_{\mathsf{X}}$ from the pair $(\mathcal{O}_{\mathsf{X}}\otimes \mathbb{K}, C(\mathsf{X})\otimes c_0)$. In particular, continuous orbit equivalence implies flow equivalence for this class of shift spaces.