Automorphisms of Locally Compact Groups, Symbolic Dynamics and the Scale Function

Bruce P. Kitchens

Published 2023-03-02, updated 2024-03-22Version 3

It is shown how to model any automorphism of a totally disconnected, locally compact group by a symbolic dynamical system. The model is an inverse limit of a product of a full-shift, on a finite number of symbols, with one of two types of systems. One is a countable discrete space with a permutation having every point periodic and the other is an essentially wandering, countable state Markov shift. Some of the ideas used are from dynamics and some from the study of totally disconnected, locally compact groups. The later ideas concern the scale function and tidy subgroups. There is a discussion of the connections between those ideas and the dynamical ideas. It is seen that only the essentially wandering, countable state Markov shift affects the scale function. Finally, it's shown that transitivity or ergodicity with respect to Haar measure implies that the system has no countable discrete or essentially wandering component.