Abstract
An ergodic action a of the direct product of ℤ and\(G = \begin{array}{*{20}c} \infty \\ \oplus \\ {n = 1} \\ \end{array} \mathbb{Z}_2 \), not isomorphic to a product of actions of ℤ and G, is constructed, such that the actions of ℤ and G separately are not ergodic. The actions of ℤ on its ergodic components are metrically isomorphic if and only if these components are taken into one another by the action of G. Finally, the centralizerC α(ℤ×G) is such thatC α(ℤ×G)/α(ℤ×G)≈ℤ2.
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Literature cited
S. I. Bezuglyi and V. Ya. Golodets, “Weak equivalence and the structure of cocycles of an ergodic automorphism,” Preprint, Physical-Technological Institute for Low Temperatures, Academy of Sciences of Ukr. SSR, Parts I and II, 15, 16–88, Khar'kov (1988).
O. N. Ageev, “Dynamical systems with a Lebesgue component of countable multiplicity in the spectrum,” Mat. Sb.,136, No. 3, 307–319 (1988).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 684–688, May, 1991.
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Golodets, V.Y., Sokhet, A.M. Properties of a jointly ergodic action of the direct product of two groups. Ukr Math J 43, 635–639 (1991). https://doi.org/10.1007/BF01058552
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DOI: https://doi.org/10.1007/BF01058552