Abstract: Automorphisms of compact groups provide some of the simplest examples of dynamical systems. I will give a brief overview of the following natural question. When do two group automorphisms share a dynamical invariant? More generally, can the fibres and a section be described for a dynamical invariant? For many natural invariants like measurable isomorphism, topological conjugacy, or periodic point data the answer is either surprisingly difficult or indeed not known.
