# Nonsingular Poisson Suspensions

@article{Danilenko2022NonsingularPS, title={Nonsingular Poisson Suspensions}, author={Alexandre I. Danilenko and Zemer Kosloff and Emmanuel Roy}, journal={Journal d'Analyse Math{\'e}matique}, year={2022}, volume={146}, pages={741-790} }

The classical Poisson functor associates to every infinite measure preserving dynamical system ( X, μ, T ) a probability preserving dynamical system ( X *, μ*, T * ) called the Poisson suspension of T . In this paper we generalize this construction: a subgroup Aut 2 ( X, μ ) of μ -nonsingular transformations T of X is specified as the largest subgroup for which T * is μ *-nonsingular. The topological structure of this subgroup is studied. We show that a generic element in Aut 2 ( X, μ ) is…

## 6 Citations

### 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…

### Generic nonsingular Poisson suspension is of type $III_1$

- Mathematics
- 2020

It is shown that for a dense $G_\delta$-subset of the subgroup of nonsingular transformations (of a standard infinite $\sigma$-finite measure space) whose Poisson suspensions are nonsingular, the…

### BERNOULLI ACTIONS OF TYPE III WITH PRESCRIBED ASSOCIATED FLOW

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2022

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

### Krieger’s type for ergodic non-singular Poisson actions of non-(T) locally compact groups

- MathematicsErgodic Theory and Dynamical Systems
- 2022

It is shown that each locally compact second countable non-(T) group G admits non-strongly ergodic weakly mixing IDPFT Poisson actions of any possible Krieger type. These actions are amenable if…

### Generic non-singular Poisson suspension is of type III1

- MathematicsErgodic Theory and Dynamical Systems
- 2021

Abstract It is shown that for a dense
$G_\delta $
-subset of the subgroup of non-singular transformations (of a standard infinite
$\sigma $
-finite measure space) whose Poisson suspensions are…

### Krieger’s type of nonsingular Poisson suspensions and IDPFT systems

- Mathematics
- 2020

Given an infinite countable discrete amenable group $\Gamma$, we construct explicitly sharply weak mixing nonsingular Poisson $\Gamma$-actions of each Krieger's type: $III_\lambda$, for…

## References

SHOWING 1-10 OF 34 REFERENCES

### A survey on spectral multiplicities of ergodic actions

- MathematicsErgodic Theory and Dynamical Systems
- 2011

Abstract Given a transformation T of a standard measure space (X,μ), let ℳ(T) denote the set of spectral multiplicities of the Koopman operator UT defined in $L^2(X,\mu )\ominus \Bbb C$ by UTf:=f∘T.…

### 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…

### Isometric Group Actions on Banach Spaces and Representations Vanishing at Infinity

- Mathematics
- 2006

Our main result is that the simple Lie group G = Sp(n, 1) acts metrically properly isometrically on Lp(G) if p > 4n + 2. To prove this, we introduce Property $ \left( {\text{BP}_\text{0}^\text{V} }…

### Furstenberg entropy values for nonsingular actions of groups without property (T)

- Mathematics
- 2015

Let $G$ be a discrete countable infinite group that does not have Kazhdan's property ~(T) and let $\kappa$ be a generating probability measure on $G$. Then for each $t>0$, there is a type $III_1$…

### Generic nonsingular Poisson suspension is of type $III_1$

- Mathematics
- 2020

It is shown that for a dense $G_\delta$-subset of the subgroup of nonsingular transformations (of a standard infinite $\sigma$-finite measure space) whose Poisson suspensions are nonsingular, the…

### On the category of certain classes of transformations in ergodic theory

- Mathematics
- 1965

The main purpose of this paper is to establish some category theorems for certain classes of "invertible measurable and nonsingular transformations" on the unit interval. We chose our setting to be…

### Isometric Group Actions on Hilbert Spaces: Growth of Cocycles

- Mathematics
- 2007

Abstract.We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following…

### Irreducible affine isometric actions on Hilbert spaces

- Mathematics
- 2014

We undertake a systematic study of irreducible affine isometric actions of locally compact groups on Hilbert spaces. It turns out that, while that are a few parallels of this study to the by now…

### The Poisson model of the Fock space and representations of current groups

- Mathematics
- 2012

The quasi-Poisson measures are considered, i.e., the σ-finite measures given by a density with respect to a Poisson measure. Representations of current groups are constructed in Hilbert spaces of…

### Abelian cocycles for nonsingular ergodic transformations and the genericity of type III1 transformations

- Mathematics
- 1987

The authors prove that in the space of nonsingular transformations of a Lebesgue probability space the type III1 ergodic transformations form a denseGδ set with respect to the coarse topology. They…