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\lecture{15}{2023-12-05}{}
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\lecture{15}{2023-12-05}{}
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Recall:
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\begin{definition}+
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Let $X$ be a set.
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A \vocab{group action} of a group $G$ on $X$
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is a function
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$\alpha\colon G \times X \to X$
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such that
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\begin{itemize}
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\item $\forall x \in X.~\alpha(1_G,x) = x$,
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\item $\forall g,h \in G, x \in X.~\alpha(gh,x) = \alpha(g,\alpha(h,x))$.
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\end{itemize}
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Often we will abbreviate $\alpha(g,x)$ as $g\cdot x$.
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\end{definition}
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\begin{remark}+
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Group actions of a group $G$ on a set $X$
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correspond to group-homomorphisms
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$G \to \Sym(X)$.
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Indeed for a group action $\alpha\colon G \times X \to X$
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consider
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\begin{IEEEeqnarray*}{rCl}
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G&\longrightarrow & \Sym(X) \\
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g&\longmapsto & (x \mapsto g \cdot x).
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\end{IEEEeqnarray*}
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\end{remark}
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\begin{theorem}[The Boundedness Theorem]
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\begin{theorem}[The Boundedness Theorem]
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\yalabel{Boundedness Theorem}{Boundedness}{thm:boundedness}
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\yalabel{Boundedness Theorem}{Boundedness}{thm:boundedness}
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@ -64,7 +38,35 @@ Recall:
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\todo{TODO: Copy from official notes}
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\todo{TODO: Copy from official notes}
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\end{proof}
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\end{proof}
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\pagebreak
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\section{Abstract Topological Dynamics}
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\section{Abstract Topological Dynamics}
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Recall:
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\begin{definition}+
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Let $X$ be a set.
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A \vocab{group action} of a group $G$ on $X$
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is a function
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$\alpha\colon G \times X \to X$
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such that
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\begin{itemize}
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\item $\forall x \in X.~\alpha(1_G,x) = x$,
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\item $\forall g,h \in G, x \in X.~\alpha(gh,x) = \alpha(g,\alpha(h,x))$.
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\end{itemize}
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Often we will abbreviate $\alpha(g,x)$ as $g\cdot x$.
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\end{definition}
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\begin{remark}+
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Group actions of a group $G$ on a set $X$
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correspond to group-homomorphisms
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$G \to \Sym(X)$.
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Indeed for a group action $\alpha\colon G \times X \to X$
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consider
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\begin{IEEEeqnarray*}{rCl}
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G&\longrightarrow & \Sym(X) \\
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g&\longmapsto & (x \mapsto g \cdot x).
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\end{IEEEeqnarray*}
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\end{remark}
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\begin{definition}
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\begin{definition}
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Let $T$ be a topological group\footnote{usually $T = \Z$ with the discrete topology}
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Let $T$ be a topological group\footnote{usually $T = \Z$ with the discrete topology}
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