diff --git a/inputs/lecture_07.tex b/inputs/lecture_07.tex index 82929d9..169fc76 100644 --- a/inputs/lecture_07.tex +++ b/inputs/lecture_07.tex @@ -1,5 +1,3 @@ -\subsection{Sheet 6} - \lecture{07}{2023-11-07}{} \begin{proposition} diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex index e9ebacd..cd49e91 100644 --- a/inputs/lecture_15.tex +++ b/inputs/lecture_15.tex @@ -1,6 +1,6 @@ \lecture{15}{2023-12-05}{} -\begin{theorem}[The Boundedness Theorem] +\begin{theorem}[Boundedness Theorem] \yalabel{Boundedness Theorem}{Boundedness}{thm:boundedness} Let $X$ be Polish, $C \subseteq X$ coanalytic, @@ -14,32 +14,55 @@ \] and \[ - E_\xi \coloneqq \{x \in C : \phi(x) \le \xi\} + E_\xi \coloneqq \{x \in C : \phi(x) \le \xi\} \] 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 $E_\xi \in \Sigma_1^1(X)$. -% If $\alpha = \sup \{\phi(x) : x \in C\}$, -% we have $E_\xi = E_\alpha$ for all $\alpha < \xi < \omega_1$. + 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 $\sup \{\phi(x) : x \in A\} < \omega_1$. - \todo{TODO: Copy from official notes} + 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. @@ -53,10 +76,30 @@ Recall: \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 + correspond to group homomorphisms $G \to \Sym(X)$. Indeed for a group action $\alpha\colon G \times X \to X$ consider @@ -66,7 +109,21 @@ Recall: \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} Let $T$ be a topological group\footnote{usually $T = \Z$ with the discrete topology} @@ -111,19 +168,27 @@ Recall: 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 this means $x \cdot \alpha$ in complex numbers, but we consider the abelian group structure of $S^1$} + \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 a real valued $\rho$ - defined on $\{(x_1,x_2) \in X^2 : \pi(x_1) = \pi(x_2)\}$ + 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 fibre + \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 $\forall y,y' \in Y.~$ @@ -135,6 +200,9 @@ Recall: 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} diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index 30d999b..d029ade 100644 --- a/inputs/lecture_16.tex +++ b/inputs/lecture_16.tex @@ -27,7 +27,7 @@ $X$ is always compact metrizable. and $n \in \Z \leadsto h^n$. Consider \[ - \{h^n : n \in \Z\} \subseteq \cC(X,X) = \{f\colon X \to X : f \text{continuous}\}, + \{h^n : n \in \Z\} \subseteq \cC(X,X) = \{f\colon X \to X : f \text{ continuous}\}, \] where the topology is the uniform convergence topology. Let $G = \overline{ \{h^n : n \in \Z\} } \subseteq \cC(X,X)$. @@ -35,20 +35,18 @@ $X$ is always compact metrizable. \[ \forall \epsilon > 0.~\exists \delta > 0.~d(x,y) < \delta \implies \forall n.~d(h^n(x), h^n(y)) < \epsilon \] - we have by the Arzela-Ascoli-Theorem % TODO REF + we have by the Arzel\`a-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. + $G$ is a closure of a topological group, + hence it is a topological group. % 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 + Take any $x \in X$ and consider the orbit $G \cdot x = \{f(x) : f \in G\}$. Since $\Z \acts X$ is minimal, i.e.~every orbit is dense, @@ -73,11 +71,11 @@ $X$ is always compact metrizable. \end{subproof} Since $G\cdot x$ is compact and dense, - we get $G \cdot x = X$. - % TODO THINK ABOUT THIS + we get $G \cdot x = X$, + since compact subsets of Hausdorff spaces are closed. - Let $\Gamma = \{f \in G : f(x) = x\} < G$ % TODO \triangleright? - be the stabilizer group. % TODO DEFINITION + Let $\Gamma = \{f \in G : f(x) = x\} < G$ + be the stabilizer group. Then $\Gamma \subseteq G$ is closed. Take $K \coloneqq \faktor{G}{\Gamma}$ with the quotient topology. @@ -89,7 +87,6 @@ $X$ is always compact metrizable. $\Z \acts K \equiv \Z \acts X$ % 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