Density of finitely supported invariant measures for automorphisms of compact abelian groups

Abstract

We study the structure of invariant measures for continuous automorphisms of compact metrizable abelian groups satisfying the descending chain condition. We show that the finitely supported invariant measures are weak-* dense in the space of all invariant probability measures, and that if the system is ergodic with respect to Haar measure, then the finitely supported ergodic invariant measures are also dense. A key ingredient in the proof is a variant of the specification property, which we establish for the ergodic systems in this class. Our results also yield the following two consequences: first, that every finitely generated group that is a semidirect product of $\mathbb{Z}$ with a countable abelian group $G$ is Hilbert-Schmidt stable; and second, a Livshitz-type theorem characterizing the uniform closure of coboundaries arising from continuous functions in terms of vanishing on periodic orbits. We also construct an example showing that, in general dynamical systems, the property of having dense finitely supported invariant measures does not pass to product systems.