Ergodicity and type of nonsingular Bernoulli actions

@article{Bjorklund2019ErgodicityAT,
  title={Ergodicity and type of nonsingular Bernoulli actions},
  author={Michael Bjorklund and Zemer Kosloff and Stefaan Vaes},
  journal={arXiv: Dynamical Systems},
  year={2019}
}
We determine the Krieger type of nonsingular Bernoulli actions $G \curvearrowright \prod_{g \in G} (\{0,1\},\mu_g)$. When $G$ is abelian, we do this for arbitrary marginal measures $\mu_g$. We prove in particular that the action is never of type II$_\infty$ if $G$ is abelian and not locally finite, answering Krengel's question for $G = \mathbb{Z}$. When $G$ is locally finite, we prove that type II$_\infty$ does arise. For arbitrary countable groups, we assume that the marginal measures stay… 
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References

SHOWING 1-10 OF 35 REFERENCES
Weak mixing for nonsingular Bernoulli actions of countable amenable groups
  • A. I. Danilenko
  • Mathematics
    Proceedings of the American Mathematical Society
  • 2019
Let $G$ be an amenable discrete countable infinite group, $A$ a finite set, and $(\mu_g)_{g\in G}$ a family of probability measures on $A$ such that $\inf_{g\in G}\min_{a\in A}\mu_g(a)>0$. It is
Bernoulli actions of type III1 and L2-cohomology
We conjecture that a countable group G admits a nonsingular Bernoulli action of type III1 if and only if the first L2-cohomology of G is nonzero. We prove this conjecture for all groups that admit at
K-property for Maharam extensions of non-singular Bernoulli and Markov shifts
It is shown that each conservative non-singular Bernoulli shift is either of type $\mathit{II}_{1}$ or $\mathit{III}_{1}$ . Moreover, in the latter case the corresponding Maharam extension of the
Proving ergodicity via divergence of ergodic sums
A classical fact in ergodic theory is that ergodicity is equivalent to almost everywhere divergence of ergodic sums of all nonnegative integrable functions which are not identically zero. We show two
C*-Algebras and Finite-Dimensional Approximations
Fundamental facts Basic theory: Nuclear and exact $\textrm{C}^*$-algebras: Definitions, basic facts and examples Tensor products Constructions Exact groups and related topics Amenable traces and
On the K property for Maharam extensions of Bernoulli shifts and a question of Krengel
We show that the Maharam extension of a type III, conservative and nonsingular K Bernoulli is a K-transformation. This together with the fact that the Maharam extension of a conservative
Transformations without finite invariant measure have finite strong generators
The following theorem is proved: If T is a nonsingular invertible transformation in a separable probability space (Ω, F, μ) and there exists no T-invariant probability measure μo << μ, then the
Bernoulli actions of amenable groups with weakly mixing Maharam extensions
We provide a simple criterion for a non-singular and conservative Bernouilli action to have a weakly mixing Maharam extension. As an application, we show that every countable amenable group admits a
AN INTRODUCTION TO INFINITE ERGODIC THEORY (Mathematical Surveys and Monographs 50)
By Jon Aaronson: 284 pp., US$79.00, isbn 0 8218 0494 4 (American Mathematical Society, 1997).
An introduction to infinite ergodic theory
Non-singular transformations General ergodic and spectral theorems Transformations with infinite invariant measures Markov maps Recurrent events and similarity of Markov shifts Inner functions
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2
3
4
...