The bernoulli property for expansive ℤ2 actions on compact groups

Abstract

We show that an expansive ℤ2 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 characterisation of theK property due to Lind, Schmidt and the author. As a corollary, we note that an expansive ℤ2 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 expansiveK action of ℤ2 on a compact abelian group is exponentially recurrent. Finally an example is given of measurable isomorphism without topological conjugacy for ℤ2 actions.