  We will characterize topologically conjugate two-sided topological Markov
shifts $(\bar{X}_A,\bar{\sigma}_A)$ in terms of the associated asymptotic
Ruelle $C^*$-algebras ${\mathcal{R}}_A$ with its commutative $C^*$-subalgebras
$C(\bar{X}_A)$ and the canonical circle actions. We will also show that
extended Ruelle algebras ${\widetilde{\mathcal{R}}}_A$, which are purely
infinite version of the asymptotic Ruelle algebras, with its commutative
$C^*$-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 ${\widetilde{\mathcal{R}}}_A$, which is an
extension of the Parry measure on $\bar{X}_A$.
