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全不连通的局部紧群的自同构的Nub

The Nub of an Automorphism of a Totally Disconnected, Locally Compact Group

George Willis

发表信息：
Ergod. Th. Dynam. Sys. 34 (2014) 1365-1394
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摘要：

对于一个全不连通的的局部紧群$G$的任何自同构$α$，都有一个$G$的紧的$α$稳定子群，这里称为$α$的\emph{nub}，在这个子群上$α$的作用是拓扑传递的。紧群自同构的拓扑传递作用在拓扑动力学中得到了广泛的研究，所得结果通过nub转移到一般局部紧群自同构的研究中.给出了$α$的收缩群在块中稠密的一个新的证明，但同时也证明了双侧收缩群不一定是稠密的.它还表明，每对$(G,α)$，与$G$紧和$α$拓扑传递，是一个逆极限的对有“有限深度”和类似的Schreier精化和约旦-Hölder定理持有有限深度的对。

英文摘要：

To any automorphism, $α$, of a totally disconnected, locally compact group, $G$, there is associated a compact, $α$-stable subgroup of $G$, here called the \emph{nub} of $α$, on which the action of $α$ is topologically transitive. Topologically transitive actions of automorphisms of compact groups have been studied extensively in topological dynamics and results obtained transfer, via the nub, to the study of automorphisms of general locally compact groups. A new proof that the contraction group of $α$ is dense in the nub is given, but it is seen that the two-sided contraction group need not be dense. It is also shown that each pair $(G,α)$, with $G$ compact and $α$ topologically transitive, is an inverse limit of pairs that have `finite depth' and that analogues of the Schreier Refinement and Jordan-Hölder Theorems hold for pairs with finite depth.

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