Thursday 13 June 2024, 16:00 - 17:00 in HG03.085
George Willis (Newcastle)
Dynamics of automorphisms of locally compact groups
Each automorphism, \(\alpha\), of a locally compact group, \(G\), determines a dynamical system \((G,\mathbf{Z})\) with \(\mathbf{Z}\) acting by iterates of \(\alpha\). If \(G\) is a Lie group, then the action of the linear transformation \(\mathrm{ad}(\alpha)\) on the Lie algebra \(\mathfrak{g}\) models the dynamics. This suffices to treat connected locally compact groups via the solution to Hilbert's \(5^{\rm{th}}\) problem and approximation by Lie groups.
In the case of totally disconnected locally compact (t.d.l.c.) groups, Lie algebra methods apply to Lie groups over t.d.l.c. fields but these do not go close to approximating general t.d.l.c. groups. Much can be said about the action of \(\alpha\) on t.d.l.c. \(G\) all the same. There are strong parallels with the linear case, abstract models of important features exist and the techniques may be used to answer questions from the literature. The talk will survey these ideas.