mixing action

Let $X$ be a topological space and let $G$ be a semigroup.An action $\varphi ={\{{\varphi }_g\}}_{g\in G}$ of $G$ on $X$ is (topologically) mixingif, given any two open subsets $U$, $V$ of $X$,the intersection $U\cap {\varphi }_g(V)$ is nonemptyfor all $g\in G$ except at most finitely many.

Example 1.Let $F:X\to X$ be a continuous function.Then $F$ is topologically mixing if and only ifthe action of the monoid $\mathbb{N}$ on $X$defined by ${\varphi }_n(x)=F^n(x)$is mixing according to the definition given above.

Example 2.Suppose $X$ is a discrete nonempty set and $G$ is a group;endow $X^G$ with the product topology.The action of $G$ on $X^G$ defined by

$${\varphi }_g(c)(z)=c(g\cdot z)\forall c:G\to X$$

is mixing.

To prove this fact,we may suppose without loss of generalitythat $U$ and $V$ are two cylindric setsof the form:

$U$$=$$\mathrm{\{}c\in X^G\mid c{|}_E=u{|}_E\}$

$V$$=$$\mathrm{\{}c\in X^G\mid c{|}_F=v{|}_F\}$

for suitable finite subsets $E,F\subseteq G$ and functions $u,v:G\to X$.Then the only chance for $U\cap {\varphi }_g(V)$ to be empty,is that $e=gf$ for some $e\in E$, $f\in F$such that $u(e)\ne v(f)$:but then, $g\in E{F}^{-1}$, which is finite.