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Spectrum of positive entropy multidimensional dynamical systems with a mixed time

Author: B. Kaminski
Journal: Proc. Amer. Math. Soc. 124 (1996), 1533-1537
MSC (1991): Primary 28D15; Secondary 60G15
MathSciNet review: 1307534
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Abstract: It is shown that if an abelian countable group $G = G_{1}\oplus G_{2}$ is such that $G_{2}$ is a finite group and every aperiodic positive entropy action $\Phi$ of $G_{1}$ on a Lebesgue probability space $(X,\cal B,\mu)$ has a countable Haar spectrum in the subspace $L^{2}_{0}(X,\mu)\ominus L^{2}_{0}(X,\Pi(\Phi),\mu)$, where $\Pi(\Phi)$ denotes the Pinsker $\sigma$-
algebra of $\Phi$, then every aperiodic positive entropy action of $G$ on $(X,\cal B,\mu)$ has the same property. A positive answer to the question of J.P. Thouvenot is obtained as a corollary.

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  • [FK] S. Ferenci, B. Kaminski, Zero entropy and directional Bernoullicity of a Gaussian $ \mathbb Z^{2}$- action, Proc. Amer. Math. Soc.123 (1995), 3079--3083. CMP 95:15
  • [Ka] Brunon Kamiński, The theory of invariant partitions for 𝑍^{𝑑}-actions, Bull. Acad. Polon. Sci. Sér. Sci. Math. 29 (1981), no. 7-8, 349–362 (English, with Russian summary). MR 640327
  • [Kif1] J. C. Kieffer, A generalized Shannon-McMillan theorem for the action of an amenable group on a probability space, Ann. Probability 3 (1975), no. 6, 1031–1037. MR 0393422
  • [Kif2] J. C. Kieffer, The isomorphism theorem for generalized Bernoulli schemes, Studies in probability and ergodic theory, Adv. in Math. Suppl. Stud., vol. 2, Academic Press, New York-London, 1978, pp. 251–267. MR 517265
  • [Kir] A. A. Kirillov, Dynamical systems, factors and group representations, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 67–80 (Russian). MR 0217256
  • [KL] B. Kamiński and P. Liardet, Spectrum of multidimensional dynamical systems with positive entropy, Studia Math. 108 (1994), no. 1, 77–85. MR 1259025
  • [Pa] William Parry, Topics in ergodic theory, Cambridge Tracts in Mathematics, vol. 75, Cambridge University Press, Cambridge-New York, 1981. MR 614142
  • [RS] V.A. Rokhlin and Y.G. Sinai, Construction and properties of invariant measurable partitions, Dokl. Akad. Nauk SSSR 223 (1975), 1067--1070 (Russian).
  • [Ru] T. de la Rue, Entropie d'un système dynamique gaussien; Cas d'une action de $\mathbb Z^{d}$, C. R. Acad. Sci. Paris, Série I 317 (1993), 191--194.
  • [T] Jean-Paul Thouvenot, Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli, Israel J. Math. 21 (1975), no. 2-3, 177–207 (French, with English summary). Conference on Ergodic Theory and Topological Dynamics (Kibbutz, Lavi, 1974). MR 0399419,

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Additional Information

B. Kaminski

Keywords: Countable Haar spectrum, entropy, Gaussian actions, spectral measure, spectrally natural
Received by editor(s): November 3, 1994
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society