Using results from topological groups and topological dynamics for locally compact spaces, Aoki has shown that when a group automorphism of a locally compact totally disconnected group is ergodic under the Haar measure, the group is compact. We provide some remarks on Aoki's proof. Also we present a new proof of his result using the structure of locally compact totally disconnected groups established by Willis.

Willis has established a structure theory of locally compact totally disconnected groups. An important feature of this theory is the notion of a tidy subgroup. In this paper we provide several results regarding these subgroups.

In this paper we study the centre of a simply connected analytic group. We present two structure theorems on the centres of such groups. We also study maximal tori and maximal compact subgroups of an analytic group, and as a result, we extend a theorem by Goto.

Let G be an analytic group and let Aut G denote the group of all topological group automorphisms of G. We investigate when Aut G is almost algebraic. We provide various conditions, some of which are known, under which Aut G is almost algebraic. We also provide examples showing that there does not seem to be a clear and concise way to characterise G so that Aut G is almost algebraic in terms of the maximal central torus in G.