# Journal of Operator Theory

Volume 82, Issue 2, Fall 2019 pp. 253-284.

Topological conjugacy of topological Markov shifts and Ruelle
algebras

**Authors**:
Kengo Matsumoto

**Author institution:**Department of Mathematics,
Joetsu University of Education,
Joetsu, 943-8512, Japan

**Summary: **We will characterize topological conjugation for two-sided topological Markov shifts $(\overline{X}_A,\overline{\sigma}_A)$ in terms of the associated asymptotic Ruelle $C^*$-algebra ${\mathcal{R}}_A$
and its commutative $C^*$-subalgebra $C(\overline{X}_A)$
and the canonical circle action.
We will also show that the extended Ruelle algebra $\widetilde{\mathcal{R}}_A$,
which is a unital and purely infinite version of ${\mathcal{R}}_A$,
together with its commutative $C^*$-subalgebra $C(\overline{X}_A)$
and the canonical torus action $\gamma^A$ is a complete invariant for topological conjugacy of $(\overline{X}_A,\overline{\sigma}_A)$.
The diagonal action of $\gamma^A$ has a unique KMS-state on
$\widetilde{\mathcal{R}}_A$, which is an extension of the Parry measure on $\overline{X}_A$.

**DOI: **http://dx.doi.org/10.7900/jot.2018apr08.2235

**Keywords: **topological Markov shift, topological conjugacy, \'etale groupoid,
Ruelle algebra, Cuntz--Krieger algebra, K-group, KMS-state

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