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\subsection{Sheet 6}
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\lecture{07}{2023-11-07}{}
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\lecture{07}{2023-11-07}{}
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\begin{proposition}
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\begin{proposition}
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\lecture{15}{2023-12-05}{}
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\lecture{15}{2023-12-05}{}
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\begin{theorem}[The Boundedness Theorem]
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\begin{theorem}[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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Let $X$ be Polish, $C \subseteq X$ coanalytic,
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Let $X$ be Polish, $C \subseteq X$ coanalytic,
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are Borel subsets of $X$.
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are Borel subsets of $X$.
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\end{theorem}
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\end{theorem}
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\begin{proof}
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\begin{proof}
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% Let
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Let
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% \begin{IEEEeqnarray*}{rCl}
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\begin{IEEEeqnarray*}{rCl}
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% x \prec y&:\iff& x,y \in A \land \phi(x) < \phi(y)\\
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x \prec y&:\iff& x,y \in A \land \phi(x) < \phi(y)\\
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% &\iff& x,y \in A \land y \not\le_\phi^\ast x.
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&\iff& x,y \in A \land y \not\le_\phi^\ast x.
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% \end{IEEEeqnarray*}
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\end{IEEEeqnarray*}
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% Since $A$ is analytic,
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Since $A$ is analytic,
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% this relation is analytic and wellfounded on $X$.
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this relation is analytic and wellfounded on $X$.
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% By \yaref{thm:kunenmartin}
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By \yaref{thm:kunenmartin}
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% we get $\rho(\prec) < \omega_1$.
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we get $\rho(\prec) < \omega_1$.
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% Thus $\sup \{\phi(x) : x \in A\} < \omega_1$.
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Thus $\sup \{\phi(x) : x \in A\} < \omega_1$.
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%
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% Since $D_\xi = \bigcup_{\eta < \xi} E_\xi$,
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% it suffices to check $E_\xi \in \Sigma_1^1(X)$.
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% If $\alpha = \sup \{\phi(x) : x \in C\}$,
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% we have $E_\xi = E_\alpha$ for all $\alpha < \xi < \omega_1$.
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\todo{TODO: Copy from official notes}
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Since $D_\xi = \bigcup_{\eta < \xi} E_\xi$,
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it suffices to check $E_\xi \in \Sigma_1^1(X)$.
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Let $\alpha \coloneqq \sup \{\phi(x) : x \in C\}$.
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Then $E_\xi = E_\alpha$ for all $\alpha < \xi < \omega_1$.
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Consider $\xi \le \alpha$.
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\begin{itemize}
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\item If there exists $x_0 \in C$ with $\phi(x_0) \ge \xi$,
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pick such $x_0$ of minimal rank.
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Then for all $y \in X$ we have
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\begin{IEEEeqnarray*}{rClr}
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y \in E_\xi &\iff& y \in C \land \phi(y) \le \xi\\
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&\iff& y \le^\ast_\phi x_0 & ~ \text{ coanalytic}\\
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&\iff& x_0 \not<^\ast_\phi y & ~ \text{ analytic}\\
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\end{IEEEeqnarray*}
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So $E_\xi$ is Borel.
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% TODO If $\alpha < \omega_1$, this also shows that $E_\alpha$ is Borel?
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\item If there exists no such $x_0$
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then $\xi = \alpha$
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and
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\[
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E_\xi = E_\alpha = \bigcup_{\eta < \alpha}
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\]
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is a countable union of Borel sets by the previous case.
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\end{itemize}
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\end{proof}
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\end{proof}
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\pagebreak
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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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% \subsection*{Basic Definitions}
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% TODO: move to appendix?
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Recall:
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Recall:
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\begin{definition}+
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\begin{definition}+
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Let $X$ be a set.
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Let $X$ be a set.
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\end{itemize}
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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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Often we will abbreviate $\alpha(g,x)$ as $g\cdot x$.
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For $x \in X$,
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the \vocab{orbit} of $x$ is defined as
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\[
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G\cdot x \coloneqq \{g\cdot x : g \in G\}.
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\]
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A group action is called \vocab{transitive}
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iff $g \mapsto g \cdot x$ is surjective for all $x \in X$,
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i.e.~iff the action has exactly one orbit.
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For $x \in X$,
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the \vocab{stabilizer subgroup}
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of $G$ with respect to $x$ is
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\[
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G_x \coloneqq \{g \in G : g\cdot x = x\}.
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\]
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\end{definition}
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\end{definition}
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\begin{remark}+
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\begin{remark}+
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Group actions of a group $G$ on a set $X$
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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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correspond to group homomorphisms
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$G \to \Sym(X)$.
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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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Indeed for a group action $\alpha\colon G \times X \to X$
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consider
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consider
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\end{IEEEeqnarray*}
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\end{IEEEeqnarray*}
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\end{remark}
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\end{remark}
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\begin{definition}+
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A group $G$ with a topology
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is a \vocab{topological group}
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iff
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\begin{IEEEeqnarray*}{rCl}
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G \times G&\longrightarrow & G \\
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(x,y) &\longmapsto & x \cdot y
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\end{IEEEeqnarray*}
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and
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\begin{IEEEeqnarray*}{rCl}
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G&\longrightarrow & G \\
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x&\longmapsto & x^{-1}
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\end{IEEEeqnarray*}
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are continuous.
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\end{definition}
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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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Recall that $S_1 = \{z \in \C : |z| = 1\}$.
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Recall that $S_1 = \{z \in \C : |z| = 1\}$.
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Let $X = S_1$, $T = S_1$
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Let $X = S_1$, $T = S_1$
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$(\alpha,\beta) \mapsto \alpha + \beta$ is isometric.%
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$(\alpha,\beta) \mapsto \alpha + \beta$ is isometric.%
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\footnote{Note that this means $x \cdot \alpha$ in complex numbers, but we consider the abelian group structure of $S^1$}
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\footnote{Note that here we consider the abelian group structure of $S^1$
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and $\alpha + \beta$ denotes the addition of \emph{angles},
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i.e.~$\alpha \cdot \beta$ in complex numbers.}
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\end{example}
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\end{example}
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\begin{definition}
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\begin{definition}
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\label{def:isometricextension}
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Let $X,Y$ be compact metric spaces
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Let $X,Y$ be compact metric spaces
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and $\pi\colon (X,T) \to (Y,T)$ a factor map.
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and $\pi\colon (X,T) \to (Y,T)$ a factor map.
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Then $(X,T)$ is an \vocab{isometric extension}
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Then $(X,T)$ is an \vocab{isometric extension}
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of $(Y,T)$ if there is a real valued $\rho$
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of $(Y,T)$ if there is
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defined on $\{(x_1,x_2) \in X^2 : \pi(x_1) = \pi(x_2)\}$
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$\rho\colon X\times_Y X \to \R$%
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\footnote{Recall that in the category of topological spaces
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the \vocab{fiber product} of
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$A \xrightarrow{f} C$, $B \xrightarrow{g} C$
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is $A \times_C B = \{(a,b) \in A \times B: f(a) = g(b)\}$,
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i.e. $X \times_Y X = \{(x_1,x_2) \in X^2 : \pi(x_1) = \pi(x_2)\}$.}
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such that
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such that
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\begin{enumerate}[(a)]
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\begin{enumerate}[(a)]
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\item $\rho$ is continuous.
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\item $\rho$ is continuous.
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\item For each $y \in Y$, $\rho$ is a metric on the fibre
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\item For each $y \in Y$, $\rho$ is a metric on the fiber
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$X_y \coloneqq \{x \in X: \pi(x) = y\}$.
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$X_y \coloneqq \{x \in X: \pi(x) = y\}$.
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\item $\forall t \in T.~\rho(tx_1,tx_2) = \rho(x_1,x_2)$.
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\item $\forall t \in T.~\rho(tx_1,tx_2) = \rho(x_1,x_2)$.
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\item $\forall y,y' \in Y.~$
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\item $\forall y,y' \in Y.~$
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A flow is isometric iff it is an isometric extension
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A flow is isometric iff it is an isometric extension
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of the trivial flow,
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of the trivial flow,
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i.e.~the flow acting on a singleton.
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i.e.~the flow acting on a singleton.
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Indeed maps $\rho\colon X\times_\star X = X^2 \to 2$
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as in \yaref{def:isometricextension}
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correspond to metrics witnessing that the flow is isometric.
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% TODO THINK ABOUT THIS!
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% TODO THINK ABOUT THIS!
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\end{remark}
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\end{remark}
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\begin{proposition}
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\begin{proposition}
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\[
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\[
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\forall \epsilon > 0.~\exists \delta > 0.~d(x,y) < \delta \implies \forall n.~d(h^n(x), h^n(y)) < \epsilon
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\forall \epsilon > 0.~\exists \delta > 0.~d(x,y) < \delta \implies \forall n.~d(h^n(x), h^n(y)) < \epsilon
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\]
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\]
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we have by the Arzela-Ascoli-Theorem % TODO REF
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we have by the Arzel\`a-Ascoli-Theorem % TODO REF
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that $G$ is compact.
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that $G$ is compact.
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$G$ is a closure of a of a topological group,
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$G$ is a closure of a topological group,
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hence it is a topological group,
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hence it is a topological group.
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i.e.~$g \mapsto g^{-1}$ and $(g,h) \mapsto gh$ are continuous.
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% TODO THINK ABOUT THIS
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% TODO THINK ABOUT THIS
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Moreover since $\Z$ is abelian,
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Moreover since $\Z$ is abelian,
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$\forall n,m \in \Z.~h^n \cdot h^m = h^m \cdot h^n$,
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$\forall n,m \in \Z.~h^n \cdot h^m = h^m \cdot h^n$,
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so $G$ is abelian.
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so $G$ is abelian.
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% TODO THINK ABOUT THIS
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% TODO THINK ABOUT THIS
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Take any $x \in X$ and consider
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Take any $x \in X$ and consider the orbit
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the orbit % TODO DEFINITION
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$G \cdot x = \{f(x) : f \in G\}$.
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$G \cdot x = \{f(x) : f \in G\}$.
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Since $\Z \acts X$ is minimal,
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Since $\Z \acts X$ is minimal,
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i.e.~every orbit is dense,
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i.e.~every orbit is dense,
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\end{subproof}
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\end{subproof}
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Since $G\cdot x$ is compact and dense,
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Since $G\cdot x$ is compact and dense,
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we get $G \cdot x = X$.
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we get $G \cdot x = X$,
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% TODO THINK ABOUT THIS
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since compact subsets of Hausdorff spaces are closed.
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Let $\Gamma = \{f \in G : f(x) = x\} < G$ % TODO \triangleright?
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Let $\Gamma = \{f \in G : f(x) = x\} < G$
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be the stabilizer group. % TODO DEFINITION
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be the stabilizer group.
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Then $\Gamma \subseteq G$ is closed.
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Then $\Gamma \subseteq G$ is closed.
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Take $K \coloneqq \faktor{G}{\Gamma}$ with the quotient topology.
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Take $K \coloneqq \faktor{G}{\Gamma}$ with the quotient topology.
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$\Z \acts K \equiv \Z \acts X$
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$\Z \acts K \equiv \Z \acts X$
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% and $h$ is a claimed.
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% and $h$ is a claimed.
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\todo{Copy from official notes}
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\todo{Copy from official notes}
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% TODO Definition transitive group action.
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\end{proof}
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\end{proof}
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\begin{definition}
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\begin{definition}
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Let $(X,T)$ be a flow
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Let $(X,T)$ be a flow
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