\section{Topological Recurrence}
\label{section:tds-recurrence}

\begin{definition}[Periodic]
\label{definition:tds-periodic}
    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $x$ is \textbf{periodic} with respect to $G$ if $Gx$ is finite. 
\end{definition}

\begin{definition}[Return Set]
\label{definition:tds-return-set}
    Let $G \curvearrowright X$ be a $G$-flow, $x \in X$, and $\emptyset \ne U \subset X$, then the \textbf{return set} of $x$ to $U$ is
    \[
    N(x, U) = \bracsn{g \in G|gx \in U}
    \]
\end{definition}


\begin{definition}[Recurrence]
\label{definition:tds-recurrence}
    Let $G$ be an infinite group, $G \curvearrowright X$ be a $G$-flow, and $x \in X$, then $x$ is \textbf{recurrent} if for every $U \in \cn_X(x)$, $N(x, U)$ is infinite.
\end{definition}

\begin{definition}[Syndetic]
\label{definition:group-syndetic}
    Let $G$ be a group and $S \subset G$, then $S$ is \textbf{(left) syndetic} if there exists $F \subset G$ finite $FS = G$. 
\end{definition}

\begin{lemma}
\label{lemma:subgroup-syndetic}
    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. 
\end{lemma}
% Proof omitted. 

\begin{definition}[Uniformly Recurrent]
\label{definition:tds-uniformly-recurrent}
    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $x$ is \textbf{uniformly recurrent} if for every $U \in \cn_X(x)$, $N(x, U)$ is syndetic. 
\end{definition}