On the continuity of F{\o}lner averages

It is known that if each point $x$ of a dynamical system is generic for some invariant measure $\mu_x$, then there is a strong connection between certain ergodic and topological properties of that system. In particular, if the acting group is abelian and the map $x\mapsto \mu_x$ is continuous, then every orbit closure is uniquely ergodic. In this note, we show that if the acting group is not abelian, orbit closures may well support more than one ergodic measure even if $x\mapsto \mu_x$ is continuous. We provide examples of such a situation via actions of the group of all orientation-preserving homeomorphisms on the unit interval as well as the Lamplighter group. To discuss these examples, we need to extend the existing theory of weakly mean equicontinuous group actions to allow for multiple ergodic measures on orbit closures and to allow for actions of general amenable groups. These extensions are achieved by adopting an operator-theoretic approach.