Topological conjugacy of topological Markov shifts and Ruelle algebras

2017/06/22 by Kengo Matsumoto, Matsumoto, Kengo · 2 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1706.07155

openalex publication_date 2017/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We will characterize topologically conjugate two-sided topological Markov shifts (X_A,σ_A) in terms of the associated asymptotic Ruelle C^*-algebras R_A with its commutative C^*-subalgebras C(X_A) and the canonical circle actions. We will also show that extended Ruelle algebras \widetildeR_A, which are purely infinite version of the asymptotic Ruelle algebras, with its commutative C^*-subalgebras C(X_A) and the canonical torus actions γ^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 γ^A has a unique KMS-state on \widetildeR_A, which is an extension of the Parry measure on X_A.

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