Mixing sets and relative entropies for higher-dimensional Markov shifts

We consider certain measurable isomorphism invariants for measure-preserving d-actions on probability spaces, compute them for a class of d-dimensional Markov shifts, and use them to prove that some of these examples are non-isomorphic. The invariants under discussion are of three kinds: the first is associated with the higher-order mixing behaviour of the d-action, and is related—in this class of examples—to an an arithmetical result by David Masser, the second arises from certain relative entropies associated with the d-action, and the third is a collection of canonical invariant σ-algebras. The results of this paper are generalizations of earlier results by Kitchens and Schmidt, and we include a proof of David Masser's unpublished theorem. © 1993, Cambridge University Press. All rights reserved.