Suppose a discrete group $\Gamma$ acts on a connected compact metrizable space $X$ by homeomorphisms. Denote such a topological dynamical system by $(X,\Gamma)$.

Question: is there any $(X,\Gamma)$ such that the set of ergodic $\Gamma$-invariant Borel probability measures on $X$ is infinite, countable and closed (equipped with weak-$*$ topology)?