lecture 16

inputs/facts.tex

@@ -0,0 +1,36 @@
+\subsection{Topological Dynamics}
+
+\begin{fact}[\url{https://math.stackexchange.com/a/801106}]
+    \label{fact:topsubgroupclosure}
+    Let $H$ be a topological group
+    and $G \subseteq H$ a subgroup.
+    Then $\overline{G}$ is a topological
+    group.
+
+    Moreover if $H$ is Hausdorff and $G$ is abelian,
+    then is $\overline{G}$ is abelian.
+\end{fact}
+\begin{proof}
+    Let $g,h \in \overline{G}$. We need to show that $g\cdot h \in \overline{G}$.
+    Take some open neighbourhood $g \cdot h \in U \overset{\text{open}}{\subseteq} H$.
+    Let $V \overset{\text{open}}{\subseteq} H \times H$
+    be the preimage of $U$ under $(a,b) \mapsto a \cdot b$.
+    Let $A \times B \subseteq V$ for some open neighbourhoods of $g$ resp.~$h$.
+    Take $g' \in A \cap G$ and $h' \in B \cap G$.
+    Then $g'h' \in U \cap G$,
+    hence $U \cap G \neq \emptyset$.
+
+    Similarly one shows that $\overline{G}$ is closed under inverse images.
+
+    Now suppose that $H$ is Hausdorff and $G$ is abelian.
+    Consider $f\colon (g,h) \mapsto [g,h]$\footnote{Recall that the \vocab{commutator} is $[g,h] \coloneqq g^{-1}h gh^{-1}$.}.
+    Clearly this is continuous.
+    Since $G$ is abelian, we have  $f(G\times G) = \{1\}$.
+    Since $H$ is Hausdorff, $\{1\}$ is closed, so
+    \[
+    \{1\} = \overline{f(G \times G)} \supseteq f(\overline{G \times G}) = f(\overline{G} \times \overline{G}).
+    \]
+    
+\end{proof}
+
+

inputs/lecture_15.tex

@@ -161,6 +161,27 @@ Recall:
     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 Fürstenberg.
+\end{warning}
+
+
+
 \begin{remark}
     Note that a flow is minimal iff it has no proper subflows.
 \end{remark}

inputs/lecture_16.tex

@@ -1,12 +1,5 @@
 \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}
@@ -16,20 +9,20 @@ $X$ is always compact metrizable.
     is isomorphic to an abelian group rotation
     $(K, \Z)$, with
     $K$ an abelian compact group
-    and $h(x) = x + \alpha$ for all $x \in K$
+    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{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 $n \in  \Z \leadsto h^n$.
+    The action of $1$ determines $h$.
     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.
+    where the topology is the uniform convergence topology. % TODO REF EXERCISE
     Let $G = \overline{ \{h^n : n \in \Z\} } \subseteq \cC(X,X)$.
     Since
     \[
@@ -39,15 +32,12 @@ $X$ is always compact metrizable.
     that $G$ is compact.
 
     $G$ 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
+    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
-    $G \cdot x = \{f(x) : f \in G\}$.
+    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 $X$.
@@ -65,44 +55,45 @@ $X$ is always compact metrizable.
         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 $G \cdot x = X$,
-    since compact subsets of Hausdorff spaces are closed.
+    we get $G \cdot x = X$
+    (compact subsets of Hausdorff spaces are closed).
 
-    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.
+    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.
 
-    $K$ is an abelian compact group
-    and $G \to Gx$ gives
-    a homeomorphism $K = \faktor{G}{\Gamma} \to Gx = X$.
+    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.
 
-    Conclusion:
-    $\Z \acts K \equiv \Z \acts X$
-    % and $h$ is a claimed.
-    \todo{Copy from official notes}
+    For $\alpha = h$ we get that
+    a flow $\Z \acts X$ corresponds to $\Z \acts K$
+    with $(1,x) \mapsto x + \alpha$.
 \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
-    }
+    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.~$\pi_\xi = \pi_{\xi, \xi'} \circ \pi_{\xi'}$.
+            ``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  $\{(X_\alpha,T), \alpha < \xi\}$.
@@ -138,10 +129,28 @@ $X$ is always compact metrizable.
     % The trivial flow is distal.
 \end{proof}
 
-\begin{theorem}[Fürstenberg]
+\begin{theorem}[Furstenberg]
+    \label{thm:furstenberg}
     Every minimal distal flow is quasi-isometric.
 \end{theorem}
-Therefore one can talk about ranks of distal minimal flows.
+By Zorn's lemma, this will follow from
+\begin{theorem}[Furstenberg]
+    Let $(X, T)$ be a minimal distal flow
+    and let $(Y, T)$ be a proper factor,
+    i.e.~$(X,T)$ and $(Y,T)$ are note isomorphic.
+    Then there is another factor $(Z,T)$ of $(X,T)$
+    which is a proper isometric extension of  $Y$.
+
+    % https://q.uiver.app/#q=WzAsMyxbMiwwLCIoWCxUKSJdLFswLDAsIihZLFQpIl0sWzEsMSwiKFosVCkiXSxbMSwwXSxbMiwwXSxbMSwyLCJcXHRleHR7aXNvbWV0cmljIGV4dGVuc2lvbn0iLDFdXQ==
+\[\begin{tikzcd}
+	{(Y,T)} && {(X,T)} \\
+	& {(Z,T)}
+	\arrow[from=1-1, to=1-3]
+	\arrow[from=2-2, to=1-3]
+	\arrow["{\text{isometric extension}}"{description}, from=1-1, to=2-2]
+\end{tikzcd}\]
+\end{theorem}
+\yaref{thm:furstenberg} allows us to talk about ranks of distal minimal flows:
 \begin{definition}
     Let $(X, \Z)$ be distal minimal.
     Then $\rank((X,\Z)) \coloneqq  \min \{\eta : (X, \Z) \cong (X_\eta, \Z)\}$
@@ -150,33 +159,61 @@ Therefore one can talk about ranks of distal minimal flows.
     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}
+\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{see Exercise Sheet 9% TODO REF
+    }
+    $[U_0; U_1,\ldots, U_n]$, $U_0,\ldots, U_n \overset{\text{open}}{\subseteq} X$,
+    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 $X$,
-            view the flows $(X,\Z)$
-            as a subset of $\cC(X,X)$.
+we can 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 $K(([0,1]^\N)^2)$.
+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.\todo{Exercise}
+Consider $K(\bH^2)$.
 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 $K(([0,1]^{\N})^2)$.
+and this graph is an element of $K(\bH^2)$.
+
+\begin{theorem}[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}
-\fi

logic3.tex

@@ -60,6 +60,10 @@
 \input{inputs/tutorial_07}
 \input{inputs/tutorial_08}
 
+\section{Facts}
+\input{inputs/facts}
+
+
 
 \PrintVocabIndex