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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)?

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up vote 5 down vote accepted

Let $T\colon [0,1] \to [0,1]$ be a homeomorphism such that $T(1/n)=1/n$ for all $n \geq 1$ and $T(x)<x$ for all other $x \in (0,1]$. If $\frac{1}{m+1}<x<\frac{1}{m}$ then $T^n(x)$ is monotone decreasing, hence convergent, and by continuity its limit must be fixed by $T$, so necessarily $\lim_{n\to\infty} T^n(x)=1/(m+1)$. It follows that the only ergodic invariant measures are supported on fixed points, so the set of ergodic measures is precisely the set of Dirac measures supported on either $\frac{1}{n}$ or $0$.

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