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The Bernoulli property for expansive \(\mathbb Z^ 2\) actions on compact groups. (English) Zbl 0789.28014

Summary: We show that an expansive \(\mathbb Z^ 2\)-action on a compact Abelian group is measurably isomorphic to a two-dimensional Bernoulli shift if and only if it has completely positive entropy. The proof uses the algebraic structure of such actions described by Kitchens and Schmidt and an algebraic characterization of the \(K\) property due to Lind, Schmidt and the author. As a corollary, we note that an expansive \(\mathbb Z^ 2\) action on a compact Abelian group is measurably isomorphic to a Bernoulli shift relative to the Pinsker algebra. A further corollary applies an argument of Lind to show that an expansive \(K\) action of \(\mathbb Z^ 2\) on a compact Abelian group is exponentially recurrent. Finally, an example is given of measurable isomorphism without topological conjugacy for \(\mathbb Z^ 2\) actions.

MSC:

28D15 General groups of measure-preserving transformations
22D40 Ergodic theory on groups
28D20 Entropy and other invariants
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