标准Borel空间的Borel自同构群上的拓扑
Topologies on the group of Borel automorphisms of a standard Borel space
摘要：
本文研究标准Borel空间$(X, B)$的Borel自同构群Aut（X，B）上的拓扑.引入了几种拓扑并找出了它们之间所有可能的关系，其中之一$τ$是遍历论中广泛使用的一致拓扑的直接模拟.我们考虑$Aut(X, B)$的最自然的子集，并找到它们的闭包。特别是，我们描述了封闭的里程表，周期性的，非周期性的，不可压缩的，和光滑的自同构相对于定义的拓扑形成的子集。证明了周期Borel自同构集在$Aut(X, B)$（Rokhlin引理）中关于$τ$是稠密的.它表明，$τ$ -封闭的里程表（和秩1 Borel自同构）符合所有非周期自同构的集合。对每一个非周期自同构$T\in Aut(X, B)$，定义并研究了Borel-Bratteli图的概念.证明了每一个非周期Borel自同构$T$都同构于作用在有序Borel-Bratteli图的无限路空间上的Vershik变换.给出了这一结果的几个应用。
英文摘要：
The paper is devoted to the study of topologies on the group Aut(X,B) of all Borel automorphisms of a standard Borel space $(X, B)$. Several topologies are introduced and all possible relations between them are found. One of these topologies, $τ$, is a direct analogue of the uniform topology widely used in ergodic theory. We consider the most natural subsets of $Aut(X, B)$ and find their closures. In particular, we describe closures of subsets formed by odometers, periodic, aperiodic, incompressible, and smooth automorphisms with respect to the defined topologies. It is proved that the set of periodic Borel automorphisms is dense in $Aut(X, B)$ (Rokhlin lemma) with respect to $τ$. It is shown that the $τ$-closure of odometers (and of rank 1 Borel automorphisms) coincides with the set of all aperiodic automorphisms. For every aperiodic automorphism $T\in Aut(X, B)$, the concept of a Borel-Bratteli diagram is defined and studied. It is proved that every aperiodic Borel automorphism $T$ is isomorphic to the Vershik transformation acting on the space of infinite paths of an ordered Borel-Bratteli diagram. Several applications of this result are given.
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