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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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Additional Information

B. Kaminski
Email: bkam@mat.uni.torun.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03186-3
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