具有指定关联流的III$_0$型伯努利作用

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.

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.

Nonsingular Bernoulli actions of arbitrary Krieger type

We prove that every infinite amenable group admits Bernoulli actions of any possible Krieger type, including type $II_\infty$ and type $III_0$. We obtain this result as a consequence of general results on the ergodicity and Krieger type of nonsingular Bernoulli actions $G \curvearrowright \prod_{g \in G} (X_0,μ_g)$ with arbitrary base space $X_0$, both for amenable and for nonamenable groups. Earlier work focused on two point base spaces $X_0 = \{0,1\}$, where type $II_\infty$ was proven not to occur.

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$…

关于Bernoulli移位上的KMS态

On the KMS states for the Bernoulli shift

本文研究Bernoulli移位交叉积上由单坐标势确定的流，证明对任意非零β存在II_∞型极值β-KMS态，并在势有理相关时确定III_λ型极值态存在的λ范围。

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 stable type III_1 Bernoulli action, answering a recent question by Vaes and Wahl.

Krieger's finite generator theorem for actions of countable groups II

We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving actions of countable groups introduced in the previous paper. We prove that every free ergodic action with finite Rokhlin entropy admits generating partitions which are almost Bernoulli, strengthening the theorem of Abért--Weiss that all free actions weakly contain Bernoulli shifts. We then use this result to study the Rokhlin entropy of Bernoulli shifts. Under the assumption that every countable group admits a free ergodic action of positive Rokhlin entropy, we prove that: (i) the Rokhlin ent…

III$_0$型作用的轨道等价超刚性

Orbit equivalence superrigidity for type III$_0$ actions

该研究首次证明λ≠1时III_λ型作用的轨道等价超刚性，这类作用由SL(n,R)稠密子群在球面S^{n-1}上作用的斜积生成，可具备任意指定伴随流，推动了相关算子代数与动力系统领域的研究。

Krieger's type of nonsingular Poisson suspensions and IDPFT systems

Given an infinite countable discrete amenable group $Γ$, we construct explicitly sharply weak mixing nonsingular Poisson $Γ$-actions of each Krieger's type: $III_λ$, for $λ\in[0,1]$, and $II_\infty$. The result is new even for $Γ=\Bbb Z$. As these Poisson suspension actions are over very special dissipative base, we obtain also new examples of sharply weak mixing nonsingular Bernoulli $Γ$-actions and IDPFT systems of each possible Krieger's type.

内作用在$C^*$-代数上的平稳态

Stationary states on a $C^*$-algebra for an inner action

本文研究可数离散群在$C^*$-代数上内作用的平稳态，证明其状态空间为Choquet单纯形，给出迹态与纯非迹态的分解及判别，并应用于自由群和性质(T)群，最后给出Bauer单纯形的充分条件。