Orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras

TL;DR

This paper establishes a deep connection between topological orbit equivalence of one-sided topological Markov shifts and isomorphisms of their associated Cuntz-Krieger algebras, revealing new structural insights.

Contribution

It proves that orbit equivalence corresponds to isomorphisms of Cuntz-Krieger algebras preserving certain subalgebras, linking dynamical systems and operator algebras.

Findings

Orbit equivalence characterized by algebra isomorphisms

Homeomorphisms intertwining topological full groups

Structure of automorphisms preserving commutative subalgebras

Abstract

We will prove that one-sided topological Markov shifts $(X_A,\sigma_A)$ and $(X_B,\sigma_B)$ for matrices $A$ and $B$ with entries in $\{0,1\}$ are topologically orbit equivalent if and only if there exists an isomorphism between the Cuntz-Krieger algebras ${\Cal O}_A$ and ${\Cal O}_B$ keeping their commutative $C^*$-subalgerbas $C(X_A)$ and $C(X_B)$. It is also equivalent to the condition that there exists a homeomorphism from $X_A$ to $X_B$ intertwining their topological full groups. We will also study structure of the automorphisms of ${\Cal O}_A$ keeping the commutative $C^*$-algebra $C(X_A)$.