lecture 16

inputs/lecture_15.tex

@@ -107,13 +107,12 @@ Recall:
 \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.
-%     % TODO: In the official notes it says \alpha + \beta, but this is no group action.
-%     % Maybe \alpha * \beta ?
-% \end{example}
+\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 this means $x \cdot \alpha$ in complex numbers, but we consider the abelian group structure of $S^1$}
+\end{example}
 
 \begin{definition}
     Let $X,Y$ be compact metric spaces

inputs/lecture_16.tex

@@ -0,0 +1,185 @@
+\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}
+    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 $h(x) = x + \alpha$ 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}
+    The action of $1$ determines $h$
+    and $n \in  \Z \leadsto h^n$.
+    Consider
+    \[
+        \{h^n : n \in \Z\}  \subseteq  \cC(X,X) = \{f\colon  X \to X : f \text{continuous}\},
+    \]
+    where the topology is the uniform convergence topology.
+    Let $G = \overline{ \{h^n : n \in \Z\} } \subseteq \cC(X,X)$.
+    Since
+    \[
+    \forall \epsilon > 0.~\exists \delta > 0.~d(x,y) < \delta \implies \forall n.~d(h^n(x), h^n(y)) < \epsilon
+    \]
+    we have by the Arzela-Ascoli-Theorem % TODO REF
+    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.
+    % 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
+    $G \cdot x = \{f(x) : f \in G\}$.
+    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 $G \cdot x = X$.
+    % TODO THINK ABOUT THIS
+
+    Let $\Gamma = \{f \in G : f(x) = x\} <  G$ % TODO \triangleright?
+    be the stabilizer group. % TODO DEFINITION
+    Then $\Gamma \subseteq G$ is closed.
+    Take $K \coloneqq \faktor{G}{\Gamma}$ with the quotient topology.
+
+    $K$ is an abelian compact group
+    and $G \to Gx$ gives
+    a homeomorphism $K = \faktor{G}{\Gamma} \to Gx = X$.
+
+    Conclusion:
+    $\Z \acts K \equiv \Z \acts X$
+    % 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
+    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
+    }
+    Suppose there is $\eta \in \Ord$
+    such that for any $\xi < \eta$
+    there is a factor $(X_\xi, T)$ of  $(X,T)$
+    \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.~$\pi_\xi = \pi_{\xi, \xi'} \circ \pi_{\xi'}$.
+        \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}
+    \todo{TODO}
+    % The trivial flow is distal.
+\end{proof}
+
+\begin{theorem}[Fürstenberg]
+    Every minimal distal flow is quasi-isometric.
+\end{theorem}
+Therefore one can talk about ranks of distal minimal flows.
+\begin{definition}
+    Let $(X, \Z)$ be distal minimal.
+    Then $\rank((X,\Z)) \coloneqq  \min \{\eta : (X, \Z) \cong (X_\eta, \Z)\}$
+    where $(X_{\eta}, \Z)$ is as from the definition of quasi-isometric flows,
+    i.e.~$\rank((X,\Z))$ is the minimal height such 
+    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}
+            For a fixed compact metric space $X$,
+            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 $K(([0,1]^\N)^2)$.
+            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 $K(([0,1]^{\N})^2)$.
+        \item Moreover, this rank is a $\Pi^1_1$-rank.
+    \end{itemize}
+\end{theorem}
+\fi

logic3.tex

@@ -40,6 +40,7 @@
 \input{inputs/lecture_13}
 \input{inputs/lecture_14}
 \input{inputs/lecture_15}
+\input{inputs/lecture_16}