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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Weak mixing for nonsingular Bernoulli actions of countable amenable groups
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by Alexandre I. Danilenko PDF
Proc. Amer. Math. Soc. 147 (2019), 4439-4450 Request permission

Abstract:

Let $G$ be an amenable discrete countable infinite group, let $A$ be a finite set, and let $(\mu _g)_{g\in G}$ be a family of probability measures on $A$ such that $\inf _{g\in G}\min _{a\in A}\mu _g(a)>0$. It is shown (among other results) that if the Bernoulli shiftwise action of $G$ on the infinite product space $\bigotimes _{g\in G}(A,\mu _g)$ is nonsingular and conservative, then it is weakly mixing. This answers in the positive a question by Z. Kosloff, who proved recently that the conservative Bernoulli $\mathbb {Z}^d$-actions are ergodic. As a byproduct, we prove a weak version of the pointwise ratio ergodic theorem for nonsingular actions of $G$.
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Additional Information
  • Alexandre I. Danilenko
  • Affiliation: Institute for Low Temperature Physics & Engineering of National Academy of Sciences of Ukraine, 47 Nauky Avenue, Kharkiv, 61103, Ukraine
  • MR Author ID: 265198
  • Email: alexandre.danilenko@gmail.com
  • Received by editor(s): July 16, 2018
  • Received by editor(s) in revised form: August 4, 2018, January 20, 2019, and January 31, 2019
  • Published electronically: June 10, 2019
  • Communicated by: Nimish Shah
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4439-4450
  • MSC (2010): Primary 37A20, 37A40
  • DOI: https://doi.org/10.1090/proc/14572
  • MathSciNet review: 4002554