Topological conjugacy of topological Markov shifts and Ruelle algebras

We will characterize topological conjugation for two-sided topological Markov shifts (¯¯¯¯¯XA,¯¯¯σA) in terms of the associated asymptotic Ruelle C∗-algebra RA and its commutative C∗-subalgebra C(¯¯¯¯¯XA) and the canonical circle action. We will also show that the extended Ruelle algebra ˜RA, which is a unital and purely infinite version of RA, together with its commutative C∗-subalgebra C(¯¯¯¯¯XA) and the canonical torus action γA is a complete invariant for topological conjugacy of (¯¯¯¯¯XA,¯¯¯σA). The diagonal action of γA has a unique KMS-state on ˜RA, which is an extension of the Parry measure on ¯¯¯¯¯XA.