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Mixing sets and relative entropies for higher-dimensional Markov shifts

Published online by Cambridge University Press:  19 September 2008

Bruce Kitchens  and
Klaus Schmidt
Bruce Kitchens
Affiliation:
Mathematical Sciences Department, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA
Klaus Schmidt
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK
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Abstract

We consider certain measurable isomorphism invariants for measure-preserving ℤd-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 ℤd-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 ℤd-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of this paper are generalizations of earlier results by Kitchens and Schmidt, and we include a proof of David Masser's unpublished theorem.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 13 , Issue 4 , December 1993 , pp. 705 - 735
Copyright
Copyright © Cambridge University Press 1993

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