The nub of an automorphism of a totally disconnected, locally compact group (Q2925269)

Let \(\alpha\) be an automorphism of a totally disconnected, locally compact topological group \(G\). Then the index \([\alpha(V):\alpha(V)\cap V]\) is finite, for each compact open subgroup \(V\) of \(G\). If the minimum index is attained at \(V\), then \(V\) is called minimizing. The intersection of all minimizing subgroups is called the nub of \(\alpha\). Define \(V_+:=\bigcap_{n=0}^\infty \alpha^n(V)\) and \(V_-:=\bigcap_{n=0}^\infty \alpha^{-n}(V)\). If \(V=V_+V_-\), then \(V\) is called tidy above. If \(\bigcup_{n=0}^\infty \alpha^n(V_+)\) and \(\bigcup_{n=0}^\infty \alpha^{-n}(V_-)\) are closed, then \(V\) is called tidy below. It is known from the author's previous work [J. Algebra 237, No. 1, 142--164 (2001; Zbl 0982.22001)] that a compact open subgroup \(V\) is minimizing if and only if \(V\) is both tidy above and tidy below. The author shows that \(V\) is tidy below if and only if \(V\) contains the nub of \(\alpha\) (Corollary 4.2). The nub of \(\alpha\) is the largest compact, \(\alpha\)-stable subgroup of \(G\) having no relatively open, proper, \(\alpha\)-stable subgroups (Corollary 4.4). This implies that the nub is the largest closed \(\alpha\)-stable subgroup on which \(\alpha\) acts ergodically (Proposition 4.4). Now assume that \(G\) is a totally disconnected, compact group (i.e., a profinite group). The author says that an automorphism \(\alpha\) of \(G\) has finite depth if \(\bigcap_{n=-\infty}^\infty\alpha^n(V)=\{e\}\) for some identity neighbourhood \(V\) in \(G\) (such automorphisms are also called expansive). For \(\alpha\) of finite depth, the nub is shown to be open in \(G\) (Lemma 5.1). The author studies automorphisms of finite depth (including variants of the Jordan-Hölder and Schreier Refinement theorems) and their relations to general automorphisms. Notably, for any automorphism \(\alpha\) of a profinite group, \((G,\alpha)\) is a projective limit of suitable pairs \((G_j,\alpha_j)\) with \(\alpha_j\) of finite depth (Proposition 5.1). If \(\alpha\) is an automorphism of a profinite group \(G\) and \(N\) an \(\alpha\)-stable closed normal subgroup of \(G\), then \(\alpha\) has finite depth if and only if both \(\alpha|_N\) and the automorphism of \(G/N\) induced by \(\alpha\) have finite depth (Proposition 6.1). As shown by \textit{C. R. E. Raja} and the reviewer, this result extends to expansive automorphisms of non-compact groups [``Expansive automorphisms of totally disconnected, locally compact groups'', Preprint, \url{arXiv:1312.5875}].