论文标题
非泊松悬浮液
Nonsingular Poisson Suspensions
论文作者
论文摘要
经典的泊松函子与每个无限度量保持动力学系统$（x，μ，t）$ a概率保留动力系统$（x^*，μ^*，t _*）$称为$ t $的泊松悬架。在本文中，我们概括了这种结构：$ $ $ $ $ $ $ t $ $ x $的亚组$ _2（x，μ）$ $ t $ t $ t $ t $ t $ t $ t $ t $ t _*$ t _*$是$μ^*$ - 非单词。研究了该亚组的拓扑结构。我们表明，自动$ _2（x，μ）$中的一个通用元素是ergodic，而Krieger型III $ _1 $。令$ g $为本地紧凑的抛光群，让$ a：g \ to \ text {aut} _2（x，μ）$为$ g $ - action。我们调查了泊松悬架的动态属性$ a _*$ a $ a $的活性属性，以$ g $与$ a $相关的$ g $。结果表明，$ g $具有属性（t），并且仅当每个非语言泊松$ g $ action都承认绝对不变的概率。如果$ g $没有属性$（t）$，那么对于$ g $和$ t> 0 $上的每个生成概率$κ$，构建了非词汇poisson $ g $ - 的furstenberg $κ$ -entropy为$ t $。
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 $A:G\to\text{Aut}_2(X,μ)$ be a $G$-action. We investigate dynamical properties of the Poisson suspension $A_*$ of $A$ in terms of an affine representation of $G$ associated naturally with $A$. It is shown that $G$ has property (T) if and only if each nonsingular Poisson $G$-action admits an absolutely continuous invariant probability. If $G$ does not have property $(T)$ then for each generating probability $κ$ on $G$ and $t>0$, a nonsingular Poisson $G$-action is constructed whose Furstenberg $κ$-entropy is $t$.
