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

TL;DR

This paper introduces the concept of the nub of an automorphism in totally disconnected, locally compact groups, linking topological dynamics with group automorphism analysis, and explores properties like density of contraction groups and finite depth structures.

Contribution

It defines the nub of an automorphism, provides a new proof regarding contraction groups, and establishes finite depth decompositions and analogues of classical theorems for these structures.

Findings

The contraction group of an automorphism is dense in the nub.

Two-sided contraction groups need not be dense.

Pairs with finite depth satisfy Schreier and Jordan-Hölder type theorems.

Abstract

To any automorphism, $\alpha$, of a totally disconnected, locally compact group, $G$, there is associated a compact, $\alpha$-stable subgroup of $G$, here called the \emph{nub} of $\alpha$, on which the action of $\alpha$ 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 $\alpha$ 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 $(G,\alpha)$, with $G$ compact and $\alpha$ topologically transitive, is an inverse limit of pairs that have `finite depth' and that analogues of the Schreier Refinement and Jordan-H\"older Theorems hold for pairs with finite depth.