\section{Topological Dynamical Systems}
\label{section:topological-dynamical-systems}

\begin{definition}[Topological Dynamical System]
\label{definition:topological-dynamical-system}
    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]
\label{definition:tds-invariant}
    Let $G \curvearrowright X$ be a $G$-flow and $A \subset X$, then the following are equivalent:
    \begin{enumerate}
        \item $GA \subset A$.
        \item $gA = A$ for all $g \in G$.
    \end{enumerate}
    
    If the above holds, then $A$ is an \textbf{invariant} subset of $G$. 
\end{definition}
\begin{proof}
    (1) $\Rightarrow$ (2): Let $g \in G$, then $gA \subset A$ and $g^{-1}A \subset A$ by (1). In which case, $A = gg^{-1}A \subset gA$, so $gA = A$. 
\end{proof}


\begin{definition}[Subflow]
\label{definition:tds-sub}
    Let $G \curvearrowright X$ be a $G$-flow and $A \subset X$ be closed\footnote{So $A$ is also compact Hausdorff. } and invariant, then $G \curvearrowright A$ under the restricted action is a \textbf{$G$-subflow}. 
\end{definition}

\begin{definition}[Orbit]
\label{definition:tds-orbit}
    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $Gx$ is the \textbf{orbit} of $x$ under $G$, and $\ol{Gx}$ is the \textbf{orbit closure} of $x$ under $G$. 
\end{definition}

\begin{definition}[Transitive]
\label{definition:tds-transitive}
    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then $x$ is \textbf{topologically transitive} with respect to $G$ if $Gx$ is dense in $X$. 
    
    The flow $G \curvearrowright X$ is \textbf{point transitive} if it admits a topologically transitive point. 

    Finally, $G \curvearrowright X$ is \textbf{topologically transitive} if for any $U, V \subset X$ non-empty and open, there exists $g \in G$ such that $gU \cap V \ne \emptyset$\footnote{Approximations of points may be translated into approximations of other points. }. 
\end{definition}

\begin{proposition}
\label{proposition:tds-point-transitive}
    Let $G \curvearrowright X$ be a point transitive $G$-flow, then $G \curvearrowright X$ is also topologically transitive. 
\end{proposition}
\begin{proof}
    Let $x \in G$ be a topologically transitive point and $U, V \subset X$ be non-empty and open. Since $Gx$ is dense in $G$, $Gx \cap U \ne \emptyset$ and $Gx \cap V \ne \emptyset$. After translating $x$, assume without loss of generality that $x \in U$, then $GU \supset Gx \cap V \ne \emptyset$. Therefore there exists $g \in G$ such that $gU \cap V \ne \emptyset$. 
\end{proof}

\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}[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}


\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 $G$. In particular, $x$ is uniformly recurrent. 
    \end{enumerate}
    

    If the above holds, then $G \curvearrowright X$ is \textbf{minimal}.
\end{definition}
\begin{proof}
    (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{definition}[Minimal Point]
\label{definition:tds-minimal-point}
    Let $G \curvearrowright X$ be a $G$-flow and $x \in X$, then the following are equivalent:
    \begin{enumerate}
        \item $G \curvearrowright \ol{Gx}$ is a minimal $G$-subflow.
        \item $x$ is uniformly recurrent. 
    \end{enumerate}
\end{definition}
\begin{proof}
    (1) $\Rightarrow$ (2): See \autoref{definition:tds-minimal}. 

    (2) $\Rightarrow$ (1): Let $Y = \ol{Gx}$ and $y \in Y$. For any $U \in \cn_X(x)$, since $X$ is a compact Hausdorff space, there exists $V \in \cn_X(x)$ with $\ol{V} \subset U$. As $N(x, V)$ is syndetic in $G$, there exists $F \subset G$ finite such that $G = F^{-1}N(x, V)$. 

    Since $\ol{Gx} = Y$, there exists a filter $\fF \subset 2^G$ such that $\fF x \to y$. As $G = \bigcup_{f \in F}N(x, V)$, there exists $f \in F$ such that $f^{-1}N(x, V) \in \fF$. Thus $y = \lim \fF x \subset \ol{f^{-1}N(x, V)} \subset \ol{f^{-1}V}$. Equivalently, $fy \in \ol{V}$, and $Gy \cap U \supset Gy \cao f^{-1}\ol{V} \ne \emptyset$. As this holds for all $U \in \cn_X(x)$, $x \in \ol{Gy}$, and $\ol{Gy} = \ol{Gx} = Y$. Therefore every point in $Y$ is transitive, and $G \curvearrowright Y$ is a minimal subflow. 
\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. In which case, every point in $Y$ is a 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}



\begin{definition}[Bernoulli Shift]
\label{definition:bernoulli-shift}
    Let $G$ be a group, $X = 2^G$ equipped with the product topology, and
    \[
    \alpha: X \to X \quad \alpha_g(x)(h) = x(g^{-1}h)
    \]

    Viewing $2^G$ as the power set of $G$, $\alpha_g(A) = gA$ for any $A \subset G$. The $G$-flow $G \curvearrowright X$ is the \textbf{Bernoulli shift} of $G$, and $G$-subflows of $X$ are \textbf{subshifts} of $G$. 
\end{definition}

\begin{example}[The Shift]
    \label{example:bernoulli-shift-z}

    Let $\integer \curvearrowright 2^\integer$ be the Bernoulli shift of $\integer$, then:
    \begin{enumerate}
        \item For any $x, y \in 2^\integer$, $y \in \ol{\integer x}$ if and only if for each $A \subset \integer$, there exists $n \in \integer$ such that $y|_A = nx|_A$. In other words, $x$ contains a copy of any segment of $y$\footnote{Topological conditions in Bernoulli shifts translate into combinatorial information. }. 
        \item $\integer \curvearrowright 2^\integer$ is point transitive, and hence topologically transitive\footnote{It is easier to show topological transitivity: move the trees until they are disjoint}. 
        \item 
    \end{enumerate}
    
\end{example}