Hyperbolic structure preserving isomorphisms of Markov shifts

In this paper we consider metric isomorphisms of Markov shifts which are also isomorphisms of the hyperbolic structures of the shift spaces. We prove that such isomorphisms need not be finitary, and that finitary isomorphisms need not preserve the hyperbolic structures unless they have finite expected code lengths. In particular we show that certain explicity computable invariants previously associated with finitary isomorphisms with finite expected code lengths are, in fact, invariants of the hyperbolic structure of the Markov shifts.