moved definitions

inputs/lecture_07.tex

@@ -1,5 +1,3 @@
-\subsection{Sheet 6}
-
 \lecture{07}{2023-11-07}{}
 
 \begin{proposition}

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,
@@ -19,27 +19,50 @@
     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 $\{(x_1,x_2) \in X^2 : \pi(x_1) = \pi(x_2)\}$
+    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}

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