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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Ergodic properties that lift to compact group extensions


Author: E. Arthur Robinson
Journal: Proc. Amer. Math. Soc. 102 (1988), 61-67
MSC: Primary 28D05
DOI: https://doi.org/10.1090/S0002-9939-1988-0915717-4
MathSciNet review: 915717
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Abstract: Let $ T$ and $ R$ be measure preserving, $ T$ weakly mixing, $ R$ ergodic, and let $ S$ be conservative ergodic and nonsingular. Let $ \tilde T$ be a weakly mixing compact abelian group extension of $ T$. If $ T \times S$ is ergodic then $ \tilde T \times S$ is ergodic. A corollary is a new proof that if $ T$ is mildly mixing then so is $ \tilde T$. A similar statement holds for other ergodic multiplier properties. Now let $ \tilde T$ be a weakly mixing type $ \alpha $ compact affine $ G$ extension of $ T$ where $ \alpha $ is an automorphism of $ G$. If $ T$ and $ R$ are disjoint and $ \alpha $ or $ R$ has entropy zero, then $ \tilde T$ and $ R$ are disjoint. $ \tilde T$ is uniquely ergodic if and only if $ T$ is uniquely ergodic and $ \alpha $ has entropy zero. If $ T$ is mildly mixing and $ \tilde T$ is weakly mixing then $ \tilde T$ is mildly mixing. We also provide a new proof that if $ \tilde T$ is weakly mixing then $ \tilde T$ has the $ K$-property if $ T$ does.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0915717-4
Article copyright: © Copyright 1988 American Mathematical Society