43.3 Topological Recurrence

Definition 43.3.1 (Periodic).label Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $x$ is periodic with respect to $G$ if $Gx$ is finite.

Definition 43.3.2 (Return Set).label Let $G \curvearrowright X$ be a $G$-flow, $x \in X$, and $\emptyset \ne U \subset X$, then the return set of $x$ to $U$ is

Definition 43.3.3 (Recurrence).label Let $G$ be an infinite group, $G \curvearrowright X$ be a $G$-flow, and $x \in X$, then $x$ is recurrent if for every $U \in \cn_{X}(x)$, $N(x, U)$ is infinite.

Definition 43.3.4 (Syndetic).label Let $G$ be a group and $S \subset G$, then $S$ is (left) syndetic if there exists $F \subset G$ finite $FS = G$.

Lemma 43.3.5.label Let $G$ be a group and $H \subset G$ be a subgroup, then $H$ is syndetic if and only if the index of $H$ in $G$ is finite.

Definition 43.3.6 (Uniformly Recurrent).label Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $x$ is uniformly recurrent if for every $U \in \cn_{X}(x)$, $N(x, U)$ is syndetic.