\lecture{16}{2023-12-08}{} $X$ 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 some fixed $\alpha \in K$ such that $h(x) = x + \alpha$ 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} \gist{% The action of $1$ determines $h$. Consider \[ \{h^n : n \in \Z\} \subseteq \cC(X,X)\gist{ = \{f\colon X \to X : f \text{ continuous}\}}{}, \] where the topology is the uniform convergence topology. Let $G = \overline{ \{h^n : n \in \Z\} } \subseteq \cC(X,X)$. Since the family $\{h^n : n \in \Z\}$ is uniformly equicontinuous, i.e.~ \[ \forall \epsilon > 0.~\exists \delta > 0.~d(x,y) < \delta \implies \forall n.~d(h^n(x), h^n(y)) < \epsilon, % Here we use isometric \] we have by the Arzel\`a-Ascoli-Theorem % TODO REF that $G$ is compact. $G$ is a closure of a topological group, hence it is a topological group, by \yaref{fact:topsubgroupclosure}. Since $h^n$ and $h^m$ commute for all $n, m \in \Z$, we obtain that $G$ is abelian. Take any $x \in X$ and consider the orbit $G \cdot x$. 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 $G \cdot x = X$ (compact subsets of Hausdorff spaces are closed). Let $G_x = \{f \in G : f(x) = x\} < G$ be the stabilizer subgroup. Note that $G_x \subseteq G$ is closed. Take $K \coloneqq \faktor{G}{G_x}$ with the quotient topology. There is a continuous bijection \begin{IEEEeqnarray*}{rCl} K &\longrightarrow & X \\ f G_x &\longmapsto & f(x). \end{IEEEeqnarray*} By compactness this is a homeomorphism, so this is an isomorphism between flows. For $\alpha = h$ we get that a flow $\Z \acts X$ corresponds to $\Z \acts K$ with $(1,x) \mapsto x + \alpha$. }{ \begin{itemize} \item $G \coloneqq \overline{\{h^n : n \in \Z\} } \subseteq \cC(X,X)$. \item $G$ is compact (Arzela-Ascoli), abelian topological group (closure of ab. top. group) \item Take any $x \in G$. \item $Gx$ is compact (since $g \mapsto gx$ is continuous and $G$ is compact) \item Stabilizer $G_x$ is closed. $K \coloneqq \faktor{G}{G_x}$, $K \to X, fG_x \mapsto f(x)$. \end{itemize} } \end{proof} \begin{definition} Let $(X,T)$ be a flow and $(Y,T)$ a factor of $(X,T)$. Suppose there is $\eta \in \Ord$ such that for any $\xi < \eta$ there is a factor $(X_\xi, T)$ of $(X,T)$ with factor map $\pi_\xi\colon X \to X_\xi$ such that \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.~$\pi_\xi = \pi_{\xi, \xi'} \circ \pi_{\xi'}$, where $\pi_{\xi,\xi'}\colon X_{\xi'} \to X_\xi$ is the factor map. \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} The trivial flow is distal. Apply \yaref{prop:isomextdistal} and \yaref{prop:limitdistal}. \end{proof} \begin{theorem}[Furstenberg] \label{thm:furstenberg} Every minimal distal flow is quasi-isometric. \end{theorem} By Zorn's lemma, this will follow from \begin{theorem}[Furstenberg] \label{thm:l16:3} Let $(X, T)$ be a minimal distal flow and let $(Y, T)$ be a proper factor.% \footnote{i.e.~$(X,T)$ and $(Y,T)$ are not isomorphic} Then there is another factor $(Z,T)$ of $(X,T)$ which is a proper isometric extension of $Y$. % https://q.uiver.app/#q=WzAsMyxbMiwwLCIoWCxUKSJdLFswLDAsIihZLFQpIl0sWzEsMSwiKFosVCkiXSxbMCwxXSxbMCwyXSxbMiwxLCJcXHRleHR7aXNvbWV0cmljIGV4dGVuc2lvbn0iLDFdXQ== \[\begin{tikzcd} {(Y,T)} && {(X,T)} \\ & {(Z,T)} \arrow[from=1-3, to=1-1] \arrow[from=1-3, to=2-2] \arrow["{\text{isometric extension}}"{description}, from=2-2, to=1-1] \end{tikzcd}\] \end{theorem} \yaref{thm:furstenberg} allows us to talk about ranks of distal minimal flows: \begin{definition}[{\cite[{}13.1]{Furstenberg}}] \label{def:floworder} Let $(X,T)$ be a quasi-isometric flow, and let $\eta$ be the smallest ordinal such that there exists a quasi-isometric system $\{(X_\xi, T), \xi \le \eta\}$ with $(X,T) = (X_\xi, T)$. Then $\eta$ is called the \vocab{rank} or \vocab{order} of the flow and is denoted by $\rank((X,T))$. \end{definition} \begin{definition}+ Let $X$ be a topological space. Let $K(X)$ denote the set of all compact subspaces of $X$ and $K(X)^\ast \coloneqq K(X)\setminus \{\emptyset\}$. If $d \le 1$ is a metric on $X$, we can equip $K(X)$ with a metric $d_H$ given by \begin{IEEEeqnarray*}{rClr} d_H(\emptyset, \emptyset) &\coloneqq & 0,\\ d_H(K, \emptyset) &\coloneqq & 1 & K \neq \emptyset,\\ d_H(K_0, K_1) &\coloneqq & \max \{\max_{x \in K_0}d(x,K_1), \max_{x \in K_1} d(x,K_0)\} & K_0,K_1 \neq \emptyset. \end{IEEEeqnarray*} The topology induced by the metric is given by basic open subsets\footnote{cf.~\yaref{s9e2}} of the form $[U_0; U_1,\ldots, U_n]$, for $U_0,\ldots, U_n \overset{\text{open}}{\subseteq} X$, where \[ [U_0; U_1,\ldots,U_n] \coloneqq \{K \in K(X) | K \subseteq U_0 \land \forall 1\le i\le n.~K \cap U_i \neq \emptyset\}. \] \end{definition} We want to view flows as a metric space. For a fixed compact metric space $X$, we can view the flows $(X,\Z)$ as a subset of $\cC(X,X)$. Note that $\cC(X,X)$ is Polish.\footnote{cf.~\yaref{s1e4}} Then the minimal flows on $X$ are a Borel subset of $\cC(X,X)$.\footnote{Exercise} % TODO However we do not want to consider only flows on a fixed space $X$, but we want to look all flows at the same time. The Hilbert cube $\bH = [0,1]^{\N}$ embeds all compact metric spaces. Thus we can consider $K(\bH)$, the space of compact subsets of $\bH$. $K(\bH)$ is a Polish space.\footnote{cf.~\yaref{s9e2}, \yaref{s12e4}} % TODO LEARN EXERCISES Consider $K(\bH^2)$. A flow $\Z \acts X$ corresponds to the graph of \begin{IEEEeqnarray*}{rCl} X &\longrightarrow & X \\ x&\longmapsto & 1 \cdot x \end{IEEEeqnarray*} and this graph is an element of $K(\bH^2)$. \begin{theorem}[Beleznay-Foreman] \label{thm:beleznay-foreman} Consider $\Z$-flows. \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, where flows are identified with their graphs as elements of $K(\bH^2)$ as above. \item Moreover, this rank is a $\Pi^1_1$-rank. \end{itemize} \end{theorem}