A groupoid approach to $C^*$-algebras associated with $\lambda$-graph systems and continuous orbit equivalence of subshifts

TL;DR

This paper explores the relationship between $mbda$-graph systems, groupoids, and $C^*$-algebras to understand continuous orbit equivalence and topological conjugacy of subshifts.

Contribution

It introduces a groupoid approach to analyze $C^*$-algebras associated with $mbda$-graph systems and their role in classifying subshifts.

Findings

Characterization of continuous orbit equivalence via groupoids

Conditions for topological conjugacy of subshifts

New connections between $C^*$-algebras and symbolic dynamics

Abstract

A $\lambda$-graph system $\frak L$ is a labeled Bratteli diagram with shift operation. It is a generalized notion of finite labeled graph and presents a subshifts. We will study continuous orbit equivalence of one-sided subshifts and topological conjugacy of two-sided subshifts from the view points of groupoids and $C^*$-algebras constructed from $\lambda$-graph systems.