Housekeeping after class.

src/dyn/index.tex

@@ -2,3 +2,4 @@
 \label{part:dynamics}
 
 \input{./top/index.tex}
+\input{./notation.tex}

src/dyn/notation.tex

@@ -0,0 +1,12 @@
+\chapter{Notations}
+\label{chap:dyn-notations}
+
+\begin{tabular}{lll}
+    \textbf{Notation} & \textbf{Description} & \textbf{Source} \\
+    \hline
+
+    $G \curvearrowright X$ & Topological dynamical system. & \autoref{definition:topological-dynamical-system}\\
+    $N(x, U)$ & Return set of $x$ to $U$. & \autoref{definition:tds-return-set}
+
+\end{tabular}
+

src/dyn/top/def.tex

@@ -3,7 +3,7 @@
 
 \begin{definition}[Topological Dynamical System]
 \label{definition:topological-dynamical-system}
-    Let $G$ be a group and $X$ be a compact Hausdorff space, then a \textbf{$G$-flow/topological dynamical system} is a homomorphism $\alpha: G \to \text{Aut}(X)$. In which case, the system is denoted by $G \curvearrowright X$. 
+    Let $G$ be a group and $X$ be a (non-empty) compact Hausdorff space, then a \textbf{$G$-flow/topological dynamical system} is a homomorphism $\alpha: G \to \text{Aut}(X)$. In which case, the system is denoted by $G \curvearrowright X$. 
 \end{definition}
 
 \begin{definition}[Invariant]
@@ -53,12 +53,48 @@
     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{example}
+    \label{example:z-syndetic}
+    Let $S \subset \integer$, then $S$ has \textbf{bounded gaps} if
+    \[
+    \sup_{n \in \integer}d(n, S) < \infty
+    \]
+
+    The set $S$ has bounded gaps if and only if it is syndetic. 
+\end{example}
+% Proof omitted. 
+
 \begin{definition}[Minimality]
 \label{definition:tds-minimal}
     Let $G \curvearrowright X$ be a $G$-flow, then the following are equivalent:
     \begin{enumerate}
         \item $G \curvearrowright X$ admits no proper subflows.
         \item Every point in $X$ is topologically transitive. 
+        \item For every $x \in X$ and $\emptyset \ne U \subset X$ open, $N(x, U)$ is syndetic in $G$. 
     \end{enumerate}
     
 
@@ -68,6 +104,34 @@
     (1) $\Rightarrow$ (2): Let $x \in X$, then $\ol{Gx}$ with the restricted action is a $G$-subflow of $X$. As $\emptyset \ne \ol{Gx}$ and $G \curvearrowright X$ admits no proper subflows, $\ol{Gx} = x$. 
 
     (2) $\Rightarrow$ (1): Let $G \curvearrowright Y$ be a $G$-subflow of $G \curvearrowright X$. For any $x \in Y$, $Y \supset \ol{Gx} = X$, so $Y = X$. 
+
+    (2) $\Rightarrow$ (3): Let $\emptyset \ne U \subset X$ be open. Since $G \curvearrowright X$ is minimal, $Gx \cap U \ne \emptyset$ for all $x \in X$. Thus $X \subset G^{-1}U = GU$. By compactness of $X$, there exists $F \subset G$ finite such that $FU = X$. 
+
+    Let $x \in X$ be arbitrary, then for each $g \in G$, $gx \in X = FU$. Thus there exists $h \in F$ with $gx \in hU$ and $h^{-1}gx \in U$. Hence $h^{-1}g \in N(x, U)$, and $g \in hN(x, U) \subset FN(x, U)$.  
+\end{proof}
+
+\begin{lemma}
+\label{lemma:tds-minimal-recurrent}
+    Let $G$ be an infinite group and $G \curvearrowright X$ be minimal $G$-flow, then every point in $X$ is recurrent. 
+\end{lemma}
+\begin{proof}
+    Let $x \in X$ and $U \in \cn_X(x)$, then $N(x, U)$ is syndetic. As $G$ is infinite, so is $N(x, U)$. 
+\end{proof}
+
+
+
+\begin{theorem}[Existence of Minimal Subflows]
+\label{theorem:tds-minimal-existence}
+    Let $G \curvearrowright X$ be a $G$-flow, then there exists $Y \subset X$ closed such that $G \curvearrowright Y$ is minimal. 
+\end{theorem}
+% Proof omitted. 
+
+\begin{corollary}
+\label{corollary:tds-recurrent-existence}
+    Let $G$ be an infinite group and $G \curvearrowright X$ be a $G$-flow, then $X$ admits a recurrent point. 
+\end{corollary}
+\begin{proof}
+    By \autoref{theorem:tds-minimal-existence}, there exists $Y \subset X$ closed such that $G \curvearrowright X$ is minimal. In which case, \autoref{lemma:tds-minimal-recurrent} implies that every point in $Y$ is recurrent. 
 \end{proof}