\lecture{16}{2023-12-08}{} % \begin{definition} % % TODO % Isomorphism from $T \acts X$ to $T \acts Y$ : % Bijection $X \xrightarrow{b} Y$ % such that $b(tx) = t b(x)$. % \end{definition} $X$ is always compact metrizable. \begin{theorem} Every minimal isometric flow $(X,\Z)$ for $X$ a compact metrizable space% \footnote{Such a flow is uniquely determined by $h\colon X \to X, x \mapsto 1\cdot x$.} is isomorphic to an abelian group rotation $(K, \Z)$, with $K$ an abelian compact group and $h(x) = x + \alpha$ for all $x \in K$ \end{theorem} \begin{example} Let $\alpha \in S^1$ and $h\colon x \mapsto x + \alpha$.\footnote{Again $x + \alpha$ denotes $x \cdot \alpha$ in $\C$.} \end{example} \begin{proof} The action of $1$ determines $h$ and $n \in \Z \leadsto h^n$. Consider \[ \{h^n : n \in \Z\} \subseteq \cC(X,X) = \{f\colon X \to X : f \text{continuous}\}, \] where the topology is the uniform convergence topology. Let $G = \overline{ \{h^n : n \in \Z\} } \subseteq \cC(X,X)$. Since \[ \forall \epsilon > 0.~\exists \delta > 0.~d(x,y) < \delta \implies \forall n.~d(h^n(x), h^n(y)) < \epsilon \] we have by the Arzela-Ascoli-Theorem % TODO REF that $G$ is compact. $G$ is a closure of a of a topological group, hence it is a topological group, i.e.~$g \mapsto g^{-1}$ and $(g,h) \mapsto gh$ are continuous. % TODO THINK ABOUT THIS Moreover since $\Z$ is abelian, $\forall n,m \in \Z.~h^n \cdot h^m = h^m \cdot h^n$, so $G$ is abelian. % TODO THINK ABOUT THIS Take any $x \in X$ and consider the orbit % TODO DEFINITION $G \cdot x = \{f(x) : f \in G\}$. Since $\Z \acts X$ is minimal, i.e.~every orbit is dense, we have that $G \cdot x$ is dense in $X$. \begin{claim} $G \cdot x$ is compact. \end{claim} \begin{subproof} Since $\Z \acts X$ is continuous, $g \mapsto g x$ is continuous: Let $g_n$ be a sequence in $G$ such that $g_n \to g$. Then $g_n x \to gx$, since the topology on $\cC(X,X)$ is the uniform convergence topology. Therefore the compactness of $G$ implies that the orbit $Gx$ is compact. \end{subproof} Since $G\cdot x$ is compact and dense, we get $G \cdot x = X$. % TODO THINK ABOUT THIS Let $\Gamma = \{f \in G : f(x) = x\} < G$ % TODO \triangleright? be the stabilizer group. % TODO DEFINITION Then $\Gamma \subseteq G$ is closed. Take $K \coloneqq \faktor{G}{\Gamma}$ with the quotient topology. $K$ is an abelian compact group and $G \to Gx$ gives a homeomorphism $K = \faktor{G}{\Gamma} \to Gx = X$. Conclusion: $\Z \acts K \equiv \Z \acts X$ % and $h$ is a claimed. \todo{Copy from official notes} % TODO Definition transitive group action. \end{proof} \begin{definition} Let $(X,T)$ be a flow and $(Y,T)$ a factor of $(X,T)$.% \footnote{i.e. there exists a continuous surjection $\pi\colon X \twoheadrightarrow Y$ commuting with the action, i.e.~$\forall t \in T. x \in X.~\pi(tx) = t \pi(x)$. Warning: Fürstenberg called factors subflows. % TODO: Definition } Suppose there is $\eta \in \Ord$ such that for any $\xi < \eta$ there is a factor $(X_\xi, T)$ of $(X,T)$ \begin{enumerate}[(a)] \item $(X_0, T) = (Y,T)$ and $(X_\eta, T) = (X,T)$. \item If $\xi < \xi'$, then $(X_\xi, T)$ is a factor of $(X_{\xi'}, T)$ ``inside $(X,T)$'', i.e.~$\pi_\xi = \pi_{\xi, \xi'} \circ \pi_{\xi'}$. \item $\forall \xi < \eta.~ (X_{\xi + 1}, T)$ is an isometric extension of $(X_\xi, T)$. \item $\xi \le \eta$ is a limit, then $(X_\xi, T)$ is a limit of $\{(X_\alpha,T), \alpha < \xi\}$. \end{enumerate} % https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzAsNSwiWSJdLFsxLDIsIlhfe1xceGknfSJdLFsxLDMsIlhfXFx4aSJdLFswLDEsIlxccGkiLDAseyJjdXJ2ZSI6NH1dLFswLDIsIlxccGlfe1xceGknfSIsMix7ImN1cnZlIjotMX1dLFswLDMsIlxccGlfe1xceGl9IiwyLHsiY3VydmUiOjJ9XSxbMiwzLCJcXHBpX3tcXHhpLCBcXHhpJ30iXV0= \[\begin{tikzcd} X \\ \\ & {X_{\xi'}} \\ & {X_\xi} \\ \\ Y \arrow["\pi", curve={height=24pt}, from=1-1, to=6-1] \arrow["{\pi_{\xi'}}"', curve={height=-6pt}, from=1-1, to=3-2] \arrow["{\pi_{\xi}}"', curve={height=12pt}, from=1-1, to=4-2] \arrow["{\pi_{\xi, \xi'}}", from=3-2, to=4-2] \end{tikzcd}\] Then we say that $(X,T)$ is a \vocab{quasi-isometric extension} of $(Y,T)$. \end{definition} \begin{definition} If $(Y,T)$ is trivial, i.e.~$|Y| = 1$, then a quasi-isometric extension $(X,T)$ of $(Y,T)$ is called a \vocab{quasi-isometric flow}. \end{definition} \begin{corollary} Every quasi-isometric flow is distal. \end{corollary} \begin{proof} \todo{TODO} % The trivial flow is distal. \end{proof} \begin{theorem}[Fürstenberg] Every minimal distal flow is quasi-isometric. \end{theorem} Therefore one can talk about ranks of distal minimal flows. \begin{definition} Let $(X, \Z)$ be distal minimal. Then $\rank((X,\Z)) \coloneqq \min \{\eta : (X, \Z) \cong (X_\eta, \Z)\}$ where $(X_{\eta}, \Z)$ is as from the definition of quasi-isometric flows, i.e.~$\rank((X,\Z))$ is the minimal height such that a tower as in the definition exists. \end{definition} \begin{theorem}[Beleznay-Foreman] Let $T = \Z$. \begin{itemize} \item For any $\alpha < \omega_1$, there is a distal minimal flow of rank $\alpha$. \item Distal flows form a $\Pi^1_1$-complete set: \todo{Move the explanations to a remark} For a fixed compact metric space $X$, view the flows $(X,\Z)$ as a subset of $\cC(X,X)$. Note that $\cC(X,X)$ is Polish. Then the minimal flows on $X$ are a Borel subset of $\cC(X,X)$. But we want to look all flows at the same time. The Hilbert cube $[0,1]^{\N}$ embeds all compact metric spaces. Thus we consider $K([0,1]^{\N})$, the space of compact subsets of $[0,1]^{\N}$.\todo{move definition} $K([0,1]^{\N})$ is a Polish space. Consider $K(([0,1]^\N)^2)$. A flow $\Z \acts X$ corresponds to the graph of \begin{IEEEeqnarray*}{rCl} X &\longrightarrow & X \\ 1&\longmapsto & 1 \cdot x \end{IEEEeqnarray*} and this graph is an element of $K(([0,1]^{\N})^2)$. \item Moreover, this rank is a $\Pi^1_1$-rank. \end{itemize} \end{theorem} \fi