Hyperbolic structure preserving isomorphisms of Markov shifts. II

This paper is a continuation of 14] and deals with metric isomorphisms of Markov shifts which are finitary and hyperbolic structure preserving. We prove that theβ-function introduced by S. Tuncel in 15] is an invariant of such isomorphisms. Following 5] this result is extended to Gibbs measures arising from functions with summable variation. Finally we prove that, for anyC ² Axiom A diffeomorphism on a basic set Ω, and for any equilibrium state associated with a Hölder continuous function on Ω, the Markov shifts arising from different Markov partitions of Ω are isomorphic via a finitary, hyperbolic structure preserving isomorphism. This fact leads to a rich class of examples of such isomorphisms (other examples are provided by finitary isomorphisms of Markov shifts with finite expected code lengths — cf. 14]).