\lecture{15}{2023-12-05}{} % TODO ANKI-MARKER \begin{theorem}[Boundedness Theorem] \yalabel{Boundedness Theorem}{Boundedness}{thm:boundedness} Let $X$ be Polish, $C \subseteq X$ coanalytic, $\phi\colon C \to \omega_1$ a coanalytic rank on $C$, $A \subseteq C$ analytic, i.e.~$A \in \Sigma^1_1(X)$. Then $\sup \{\phi(x) : x \in A\} < \omega_1$. Moreover for all $\xi < \omega_1$, \[ D_\xi \coloneqq \{x \in C : \phi(x) < \xi\} \] and \[ E_\xi \coloneqq \{x \in C : \phi(x) \le \xi\} \] are Borel subsets of $X$. \end{theorem} \begin{proof} Let \begin{IEEEeqnarray*}{rCl} x \prec y&:\iff& x,y \in A \land \phi(x) < \phi(y)\\ &\iff& x,y \in A \land y \not\le_\phi^\ast x. \end{IEEEeqnarray*} Since $A$ is analytic, this relation is analytic and wellfounded on $X$. By \yaref{thm:kunenmartin} we get $\rho(\prec) < \omega_1$. Thus $\sup \{\phi(x) : x \in A\} < \omega_1$. Since $D_\xi = \bigcup_{\eta < \xi} E_\xi$, it suffices to check $E_\xi \in \Sigma_1^1(X)$. Let $\alpha \coloneqq \sup \{\phi(x) : x \in C\}$. Then $E_\xi = E_\alpha$ for all $\alpha < \xi < \omega_1$. Consider $\xi \le \alpha$. \begin{itemize} \item If there exists $x_0 \in C$ with $\phi(x_0) \ge \xi$, pick such $x_0$ of minimal rank. Then for all $y \in X$ we have \begin{IEEEeqnarray*}{rClr} y \in E_\xi &\iff& y \in C \land \phi(y) \le \xi\\ &\iff& y \le^\ast_\phi x_0 & ~ \text{ coanalytic}\\ &\iff& x_0 \not<^\ast_\phi y & ~ \text{ analytic}\\ \end{IEEEeqnarray*} So $E_\xi$ is Borel. % TODO If $\alpha < \omega_1$, this also shows that $E_\alpha$ is Borel? \item If there exists no such $x_0$ then $\xi = \alpha$ and \[ E_\xi = E_\alpha = \bigcup_{\eta < \alpha} \] is a countable union of Borel sets by the previous case. \end{itemize} \end{proof} \pagebreak \section{Abstract Topological Dynamics} % \subsection*{Basic Definitions} % TODO: move to appendix? Recall: \begin{definition}+ Let $X$ be a set. A \vocab{group action} of a group $G$ on $X$ is a function $\alpha\colon G \times X \to X$ such that \begin{itemize} \item $\forall x \in X.~\alpha(1_G,x) = x$, \item $\forall g,h \in G, x \in X.~\alpha(gh,x) = \alpha(g,\alpha(h,x))$. \end{itemize} Often we will abbreviate $\alpha(g,x)$ as $g\cdot x$. For $x \in X$, the \vocab{orbit} of $x$ is defined as \[ G\cdot x \coloneqq \{g\cdot x : g \in G\}. \] A group action is called \vocab{transitive} iff $g \mapsto g \cdot x$ is surjective for all $x \in X$, i.e.~iff the action has exactly one orbit. For $x \in X$, the \vocab{stabilizer subgroup} of $G$ with respect to $x$ is \[ G_x \coloneqq \{g \in G : g\cdot x = x\}. \] \end{definition} \begin{remark}+ Group actions of a group $G$ on a set $X$ correspond to group homomorphisms $G \to \Sym(X)$. Indeed for a group action $\alpha\colon G \times X \to X$ consider \begin{IEEEeqnarray*}{rCl} G&\longrightarrow & \Sym(X) \\ g&\longmapsto & (x \mapsto g \cdot x). \end{IEEEeqnarray*} \end{remark} \begin{definition}+ A group $G$ with a topology is a \vocab{topological group} iff \begin{IEEEeqnarray*}{rCl} G \times G&\longrightarrow & G \\ (x,y) &\longmapsto & x \cdot y \end{IEEEeqnarray*} and \begin{IEEEeqnarray*}{rCl} G&\longrightarrow & G \\ x&\longmapsto & x^{-1} \end{IEEEeqnarray*} are continuous. \end{definition} \begin{definition} \label{def:flow} Let $T$ be a topological group\footnote{usually $T = \Z$ with the discrete topology} and let $X$ be a compact metrizable space. A \vocab{flow} $(X, T)$, sometimes denoted $T \acts X$ is a continuous action \begin{IEEEeqnarray*}{rCl} T \times X&\longrightarrow & X \\ (t,x) &\longmapsto & tx. \end{IEEEeqnarray*} A flow is \vocab{minimal} iff every orbit is dense. $(Y,T)$ is a \vocab{subflow} of $(X,T)$ if $Y \subseteq X$ and $Y$ is invariant under $T$, i.e.~$\forall t \in T,y \in Y.~ty \in Y$. A flow $(X,T)$ is \vocab{isometric} iff there is a metric $d$ on $X$ such that for all $t \in T$ the map \begin{IEEEeqnarray*}{rCl} a_t\colon X &\longrightarrow & X \\ x &\longmapsto & tx \end{IEEEeqnarray*} is an \vocab{isometry}, i.e.~$\forall t \in T.~\forall x,y \in X.~d(a_t(x),a_t(y)) = d(x,y)$. If $(X,T)$ is a flow, then a pair $(x,y)$, $x \neq y$ is \vocab{proximal} iff \[ \exists z \in X.~\exists (t_n)_{n < } \in T^{\omega}.~t_n x \xrightarrow{n \to \infty} z \land t_n y \xrightarrow{n \to \infty} z. \] A flow is \vocab{distal} iff it has no proximal pair. \end{definition} \begin{definition}+ Let $(T,X)$ and $(T,Y)$ be flows. A \vocab{factor map} $\pi\colon (T,X) \to (T,Y)$ is a continuous surjection $X \twoheadrightarrow Y$ commuting with the group action, i.e.~$\forall t \in T, x \in X.~\pi(t\cdot x) = t\cdot \pi(x)$. If such a factor map exists, we also say that $(T,Y)$ is a \vocab{factor} of $(T,X)$. An \vocab{isomorphism} from $(T,X)$ to $(T,Y)$ is a homeomorphism $X \leftrightarrow Y$ commuting with the group action. \end{definition} \begin{warning}+ What is called ``factor'' here is called ``subflow'' by Furstenberg. \end{warning} \begin{remark} Note that a flow is minimal iff it has no proper subflows. \end{remark} \begin{example} Recall that $S_1 = \{z \in \C : |z| = 1\}$. Let $X = S_1$, $T = S_1$ $(\alpha,\beta) \mapsto \alpha + \beta$ is isometric.% \footnote{Note that here we consider the abelian group structure of $S^1$ and $\alpha + \beta$ denotes the addition of \emph{angles}, i.e.~$\alpha \cdot \beta$ in complex numbers.} \end{example} \begin{definition} \label{def:isometricextension} Let $X,Y$ be compact metric spaces and $\pi\colon (X,T) \to (Y,T)$ a factor map. Then $(X,T)$ is an \vocab{isometric extension} of $(Y,T)$ if there is $\rho\colon X\times_Y X \to \R$% \footnote{Recall that in the category of topological spaces the \vocab{fiber product} of $A \xrightarrow{f} C$, $B \xrightarrow{g} C$ is $A \times_C B = \{(a,b) \in A \times B: f(a) = g(b)\}$, i.e. $X \times_Y X = \{(x_1,x_2) \in X^2 : \pi(x_1) = \pi(x_2)\}$.} such that \begin{enumerate}[(a)] \item $\rho$ is continuous. \item For each $y \in Y$, $\rho$ is a metric on the fiber $X_y \coloneqq \{x \in X: \pi(x) = y\}$. \item $\forall t \in T.~\rho(tx_1,tx_2) = \rho(x_1,x_2)$. \item $\forall y,y' \in Y.~$ the metric spaces $(X_y, \rho)$ and $(X_{y'}, \rho)$ are isometric. \end{enumerate} \end{definition} \begin{remark} A flow is isometric iff it is an isometric extension of the trivial flow, i.e.~the flow acting on a singleton. Indeed maps $\rho\colon X\times_\star X = X^2 \to 2$ as in \yaref{def:isometricextension} correspond to metrics witnessing that the flow is isometric. % TODO THINK ABOUT THIS! \end{remark} \begin{proposition} An isometric extension of a distal flow is distal. \end{proposition} \begin{proof} Let $\pi\colon X\to Y$ be an isometric extension. Towards a contradiction, suppose that $x_1,x_2 \in X$ are proximal. Take $z \in X$ and $(g_n) \in T^{\omega}$ such that $g_n x_1 \to z$ and $g_n x_2 \to z$. Then $g_n \pi(x_1) \to \pi(z)$ and $g_n \pi(x_2) \to \pi(z)$, so by distality of $Y$ we have $\pi(x_1) = \pi(x_{2})$. Then $\rho(g_n x_1, g_n x_2)$ is defined and equal to $\rho(x_1,x_2)$. By the continuity of $\rho$, we get $\rho(g_n x_1, g_n x_2) \to \rho(z,z) = 0$. Therefore $\rho(x_1,x_2) = 0$. Hence $x_1 = x_2$ $\lightning$. \end{proof} \begin{definition} Let $\Sigma = \{(X_i, T) : i \in I\} $ be a collection of factors of $(X,T)$. % TODO State precise definition of a factor Let $\pi_i\colon (X,T) \to (X_i, T)$ denote the factor maps. Then $(X, T)$ is the \vocab{limit} of $\Sigma$ iff \[ \forall x_1,x_2 \in X.~\exists i \in I.~\pi_i(x_1) \neq \pi_i(x_2). \] % TODO think about abstract nonsense \end{definition} \begin{proposition} A limit of distal flows is distal. \end{proposition} \begin{proof} Let $(X,T)$ be a limit of $\Sigma = \{(X_i, T) : i \in I\}$. Suppose that each $(X_i, T)$ is distal. If $(X,T)$ was not distal, then there were $x_1, x_2, z \in X$ and a sequence $(g_n)$ in $T$ with $g_n x_1 \to z$ and $g_n x_2 \to z$. Take $i \in I$ such that $\pi_i(x_1) \neq \pi_i(x_2)$. But then $g_n \pi_i(x_1) \to \pi_i(z)$ and $g_n \pi_i(x_2) \to \pi_i(z)$, which is a contradiction since $(X_i, T)$ is distal. \end{proof}