# 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…

## 12 Citations

Sinai factors of nonsingular systems: Bernoulli shifts and Anosov flows

- Mathematics
- 2021

We show that a totally dissipative system has all nonsingular systems as factors, but that this is no longer true when the factor maps are required to be finitary. In particular, if a nonsingular…

KMS states on $C_c^{*}(\mathbb{N}^2)$

- Mathematics
- 2022

Let C∗ c (N ) be the universal C∗-algebra generated by a semigroup of isometries {v(m,n) : m,n ∈ N} whose range projections commute. We analyse the structure of KMS states on C∗ c (N ) for the time…

Classification results for nonsingular Bernoulli crossed products

- Mathematics
- 2022

We prove rigidity and classification results for type III factors given by nonsingular Bernoulli actions of the free groups and more general free product groups. This includes a large family of…

On Absolutely Continuous Invariant Measures and Krieger-Types of Markov Subshifts

- Mathematics
- 2020

It is shown that for a non-singular conservative shift on a topologically mixing Markov subshift with Doeblin Condition the only possible absolutely continuous shift-invariant measure is a Markov…

The orbital equivalence of Bernoulli actions and their Sinai factors

- Mathematics
- 2020

Given a countable amenable group G and 0 < L < 1, we give an elementary construction of a type-III:L Bernoulli group action. In the case where G is the integers, we show that our nonsingular…

Bernoulli actions of type III$_0$ with prescribed associated flow

- Mathematics
- 2021

We prove that many, but not all injective factors arise as crossed products by nonsingular Bernoulli actions of the group Z . We obtain this result by proving a completely general result on the…

Ergodic cocycles of IDPFT systems and non-singular Gaussian actions

- MathematicsErgodic Theory and Dynamical Systems
- 2021

Abstract It is proved that each Gaussian cocycle over a mildly mixing Gaussian transformation is either a Gaussian coboundary or sharply weak mixing. The class of non-singular infinite direct…

Ergodic theory of affine isometric actions on Hilbert spaces

- MathematicsGeometric and Functional Analysis
- 2021

The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we…

Boundary and Rigidity of Nonsingular Bernoulli Actions

- Mathematics
- 2020

Let $ G $ be a countable discrete group and consider a nonsingular Bernoulli shift action $ G \curvearrowright \prod_{g\in G }(\{0,1\},\mu_g)$ with two base points. When $ G $ is exact, under a…

## References

SHOWING 1-10 OF 28 REFERENCES

Weak mixing for nonsingular Bernoulli actions of countable amenable groups

- MathematicsProceedings 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

- Mathematics
- 2017

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

- MathematicsErgodic Theory and Dynamical Systems
- 2019

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

- Mathematics
- 2018

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

- Mathematics
- 2008

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

- Mathematics
- 2012

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

- Mathematics
- 1970

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

- Mathematics
- 2018

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)

- Mathematics
- 1999

By Jon Aaronson: 284 pp., US$79.00, isbn 0 8218 0494 4 (American Mathematical Society, 1997).

An introduction to infinite ergodic theory

- Mathematics
- 1997

Non-singular transformations General ergodic and spectral theorems Transformations with infinite invariant measures Markov maps Recurrent events and similarity of Markov shifts Inner functions…