3
$\begingroup$

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)?

$\endgroup$
5
$\begingroup$

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 $0$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.