Topological conjugacy of topological Markov shifts and Ruelle algebras

Abstract

We will characterize topologically conjugate two-sided topological Markov shifts $({\bar{X}}_A,{\bar{\sigma }}_A)$ in terms of the associated asymptotic Ruelle $C^{\ast }$-algebras ${\mathcal{R}}_A$ with its commutative $C^{\ast }$-subalgebras $C({\bar{X}}_A)$ and the canonical circle actions. We will also show that extended Ruelle algebras ${\tilde{\mathcal{R}}}_A$, which are purely infinite version of the asymptotic Ruelle algebras, with its commutative $C^{\ast }$-subalgebras $C({\bar{X}}_A)$ and the canonical torus actions ${\gamma }^A$ are complete invariants for topological conjugacy of two-sided topological Markov shifts. We then have a computable topological conjugacy invariant, written in terms of the underlying matrix, of a two-sided topological Markov shift by using K-theory of the extended Ruelle algebra. The diagonal action of ${\gamma }^A$ has a unique KMS-state on ${\tilde{\mathcal{R}}}_A$, which is an extension of the Parry measure on ${\bar{X}}_A$.

31 pages, Typo in Definition 3.1 and Section 5 were corrected

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• $C^{\ast }$-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms