Expansive automorphisms of totally disconnected, locally compact groups

Abstract

We study automorphisms $\alpha$ of a totally disconnected, locally compact group $G$ which are expansive in the sense that, for some identity neighbourhood $U$, the sets $\alpha^n(U)$ (for integers $n$) intersect in the trivial group. Notably, we prove that the automorphism induced by $\alpha$ on $G/N$ for an $\alpha$-stable closed normal subgroup $N$ of $G$ is always expansive. Further results involve the associated contraction groups $U_\alpha$ consisting of all $x$ in $G$ such that $\alpha^n(x) \to e$ as $n$ tends to infinity. If $\alpha$ is expansive, then $W := U_\alpha U_{\alpha^{-1}}$ is an open identity neighbourhood in $G$. We give examples where $W$ fails to be a subgroup. However, $W$ is a nilpotent open subgroup whenever $G$ is a closed subgroup of a general linear group over the $p$-adic numbers. Further results are devoted to the divisible and torsion parts of $U_\alpha$, and to the so-called "nub" $U_0$ of an expansive automorphism $\alpha$ (the intersection of the closures of $U_\alpha$ and $U_{\alpha^{-1}}$).