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

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.