Ward, Thomas B. The Bernoulli property for expansive \(\mathbb Z^ 2\) actions on compact groups. (English) Zbl 0789.28014 Isr. J. Math. 79, No. 2-3, 225-249 (1992). 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. Cited in 2 Documents MSC: 28D15 General groups of measure-preserving transformations 22D40 Ergodic theory on groups 28D20 Entropy and other invariants Keywords:expansive \(\mathbb Z^ 2\)-action; compact abelian group; two-dimensional Bernoulli shift; positive entropy; Pinsker algebra; measurable isomorphism PDFBibTeX XMLCite \textit{T. B. Ward}, Isr. J. Math. 79, No. 2--3, 225--249 (1992; Zbl 0789.28014) Full Text: DOI References: [1] Boyd, D., Kronecker’s theorem and Lehmer’s problem for polynomials in several variables, Number Theory, 13, 116-121 (1981) · Zbl 0447.12003 [2] Chu, H., Some results on affine transformations of compact groups, Invent. Math., 28, 161-183 (1975) · Zbl 0299.22009 [3] Conze, J. P., Entropie d’un groupe abélien de transformations, Z. Wahrsch. verw. Geb., 25, 11-30 (1972) · Zbl 0261.28015 [4] Feldman, J.; Rudolph, D. J.; Moore, C. C., Affine extensions of a Bernoulli shift, Trans. Amer. Math. 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