Nonsingular Bernoulli actions of arbitrary Krieger type

Ergodicity and type of nonsingular Bernoulli actions

We determine the Krieger type of nonsingular Bernoulli actions $G \curvearrowright \prod_{g \in G} (\{0,1\},μ_g)$. When $G$ is abelian, we do this for arbitrary marginal measures $μ_g$. 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 away from $0$ and $1$. When $G$ has only one end, we prove that the Krieger type is always I, II$_1$ or III$…

Generic nonsingular Poisson suspension is of type $III_1$

It is shown that for a dense $G_δ$-subset of the subgroup of nonsingular transformations (of a standard infinite $σ$-finite measure space) whose Poisson suspensions are nonsingular, the corresponding Poisson suspensions are ergodic and of Krieger's type $III_1$.

K-property for Maharam extensions of nonsingular Bernoulli and Markov shifts

It is shown that each conservative nonsingular Bernoulli shift is either of type $II_1$ or $III_1$. Moreover, in the latter case the corresponding Maharam extension of the shift is a $K$-automorphism. This extends earlier results obtained by Z.~Kosloff for the equilibrial shifts. Nonequilibrial shifts of type $III_1$ are constructed. We further generalize (partly) the main results to nonsingular Markov shifts.

Nonsingular Gaussian actions: beyond the mixing case

Every affine isometric action $α$ of a group $G$ on a real Hilbert space gives rise to a nonsingular action $\hatα$ of $G$ on the associated Gaussian probability space. In the recent paper [AIM19], several results on the ergodicity and Krieger type of these actions were established when the underlying orthogonal representation $π$ of $G$ is mixing. We develop new methods to prove ergodicity when $π$ is only weakly mixing. We determine the type of $\hatα$ in full generality. Using Cantor measures, we give examples of type III$_1$ ergodic Gaussian actions of $\mathbb{Z}$ whose underlying represe…

Some factors of nonsingular Bernoulli shifts

We give elementary constructions of factors of nonsingular Bernoulli shifts. In particular, we show that all nonsingular Bernoulli shifts on a finite number of symbols which satisfy the Doeblin condition have a factor that is equivalent to an independent and identically distributed system. We also prove that there are type-III:1 Bernoulli shifts of every possible ergodic index, answering a question of Danilenko and Lemanczyk (Ergodic Theory Dynam. Systems, 39(12):3292-3321, 2019).

Weak mixing for nonsingular Bernoulli actions of countable amenable groups

Let $G$ be an amenable discrete countable infinite group, $A$ a finite set, and $(μ_g)_{g\in G}$ a family of probability measures on $A$ such that $\inf_{g\in G}\min_{a\in A}μ_g(a)>0$. It is shown (among other results) that if the Bernoulli shiftwise action of $G$ on the infinite product space $\bigotimes_{g\in G}(A,μ_g)$ is nonsingular and conservative then it is weakly mixing. This answers in positive a question by Z.~Kosloff who proved recently that the conservative Bernoulli $\Bbb Z^d$-actions are ergodic. As a byproduct, we prove a weak version of the pointwise ratio ergodic theorem for n…

Bernoulli actions of type III_1 and L^2-cohomology

We conjecture that a countable group $G$ admits a nonsingular Bernoulli action of type III$_1$ if and only if the first $L^2$-cohomology of $G$ is nonzero. We prove this conjecture for all groups that admit at least one element of infinite order. We also give numerous explicit examples of type III$_1$ Bernoulli actions of the group of integers and the free groups, with different degrees of ergodicity.

非奇异系统的 Sinai 因子：Bernoulli 平移与 Anosov 流

Sinai factors of nonsingular systems: Bernoulli shifts and Anosov flows

本文研究了非奇异系统在有限态因子映射下的限制，证明满足 Doeblin 条件的 Bernoulli 平移的有限态 iid 因子熵受 H(p) 约束，并将此结论推广至 Anosov 微分同胚，确定了其 iid 因子的 Kolmogorov-Sinai 熵上界。

Nonsingular Poisson Suspensions

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. Topological structure of this subgroup is studied. We show that a generic element in Aut$_2(X,μ)$ is ergodic and of Krieger type III$_1$. Let $G$ be a locally compact Polish group and let $A:G\to\text{Aut}_2…

The orbital equivalence of Bernoulli actions and their Sinai factors

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 shifts have independent and identically distributed factors.

具有指定关联流的III$_0$型伯努利作用

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

该研究针对群作用与算子代数交叉问题，证明了关联流的无穷可分结构性质，构造了指定关联流的伯努利作用与Poisson悬浮作用，还得到单射因子的张量积结构结论。

通过倾斜的Følner几何研究$\ell^{2}$-上同调的不对称性

Asymmetry of $\ell^{2}$-cohomology via skewed Følner geometry

本文研究有限生成群的两个标准$\ell^{2}$-Dirichlet结构，证明有限生成 nilpotent 群的$\ell^{2}$-不对称性由虚拟交换性决定，引入了左方案机制，并展示了非虚拟交换的 nilpotent 群的 Bernoulli 方案构造。

不可数超有限性与随机比率遍历定理

Uncountable Hyperfiniteness and The Random Ratio Ergodic Theorem

本文刻画局部紧群自由作用轨道关系的超有限性等价于超紧性，给出 Ornstein-Weiss 与 Danilenko 随机比率遍历定理的不可数推广，并导出非奇异 Bernoulli 作用的 Hopf 二分法。