Let $G$ be a discrete abelian group which acts minimally on a compact (for comfort, compact and metrizable) space $X$ by homeomorphisms. If the action is proximal in addition then $X$ must be a singleton. Why?
- By minimality, I mean orbit of every $x\in X$ is dense in $X$.
- By proximally, I mean for every $x, y\in X$ there is a sequence $\{g_n\}_n$ such that $\lim_n g_{n}.x=\lim_n g_{n}.y=z\in X$
OK. It doesn't as hard as I supposed. Indeed it is an elementary problem.
Let $g\in G$ and $x\in X$. $(x, g.x)$ is proximal so there is a net $\{h_i\}$ in $G$ such that $\lim_{i}h_i.x=\lim_{i}h_{i}.(g.x)=z\in X$. Also $G$ is abelian so $\lim_{i}h_{i}.(g.x)=\lim_{i}g.(h_{i}.x)=g.\lim_{i}h_{i}.x$. Thus we must $z=gz$ for all $g\in G$ i.e., the action has a fixed point wherease the action is minimal therefore $X$ is a singleton.