lecture 23

inputs/lecture_21.tex

@@ -84,5 +84,3 @@ Let $X_n \coloneqq  (S^1)^n$ and $X \coloneqq (S^1)^{\N}$.
 \begin{corollary}
     The order of $(X,\tau)$ is $\omega$.
 \end{corollary}
-\todo{I could not attend lecture 21 as I was sick. The official notes on the lecture are very short.
-    Is something missing in the official notes?}

inputs/lecture_22.tex

@@ -127,19 +127,21 @@ For this we define
 % \end{example}
 
 \begin{theorem}[Beleznay Foreman]
+    \label{thm:distalminimalofallranks}
     Whenever $I = \eta$ for some $\eta  < \omega_1$,
     then
     \[
-    \{\overline{f} \in \mathbb{K}_I : E_I(\overline{f}) \text{ is distal, minimal and of rank$\eta$}\} % TODO rank = order
+    \{\overline{f} \in \mathbb{K}_I : E_I(\overline{f}) \text{ is distal, minimal and of rank $\eta$}\} % TODO rank = order
     \]
     is comeager in $\mathbb{K}_I$.
     In particular such flows exist.
 \end{theorem}
 \begin{proof}[sketch]
+    \leavevmode
     \begin{itemize}
         \item Distality:
             For all $\overline{f} \in \mathbb{K}_I$,
-            the flow $E_I \overline{f}$is distal.
+            the flow $E_I \overline{f}$ is distal.
             This is the same as for iterated skew shifts.
             %  TODO since for $\overline{x}, \overline{y} \in \mathbb{K}^I$,
             % $d(x_\alpha, y_\alpha) = d((f(\overline{x})_\alpha, (f(\overline{y})_\alpha))$.
@@ -176,7 +178,7 @@ For this we define
             Fix a countable dense set $(\overline{x_n})$ in $\mathbb{K}^I$.
             For $\epsilon \in \Q$ let
             \begin{IEEEeqnarray*}{rCl}
-                V_{j,m,n,\epsilon} &\coloneqq & \{\overline{f} \in \mathbb{K}_I : \\
+                V_{j,m,n,\epsilon} &\coloneqq  \{\overline{f} \in \mathbb{K}_I :& \\
                 &&\text{if } \Pi_{j+1}(\overline{x}_n) = \Pi_{j+1}(\overline{x_m}),\\
                 &&\text{then there are $k_m$, $k_n$, $\overline{z}$ such that}\\
                 &&\pi_j(\overline{x_n}) = \pi_j(\overline{z}), \forall k> j+1.~z_k = 1,\\

inputs/lecture_23.tex

@@ -0,0 +1,167 @@
+\lecture{23}{2024-01-19}{More sketches of ideas of Beleznay and Foreman}
+
+\begin{notation}
+    Let $X$ be a Polish space and $\cP$ a property of elements of $X$,
+    then we say that $x_0 \in X$ is \vocab{generic} 
+    if
+    \[
+    A_\cP \coloneqq \{x  \in X \colon  \cP(x)\}
+    \]
+    is comeager
+    and $x_0 \in A_\cP$.
+\end{notation}
+For example let $X = \mathbb{K}_I$
+and $\cP$ the property of being a distal minimal flow.
+\begin{abuse}
+    We will usually omit $\cP$.
+\end{abuse}
+
+
+Let $I$ be a linear order
+
+\begin{theorem}[Beleznay and Foreman]
+    The set of distal minimal flows is $\Pi_1^1$-complete.
+\end{theorem}
+
+\begin{proof}[sketch]
+    Consider $\WO(\N) \subset \LO(\N)$.
+    We know that this is $\Pi_1^1$-complete. % TODO ref
+
+    Let
+    \begin{IEEEeqnarray*}{rCll}
+        S & \coloneqq  & \{ x \in \LO(\N) :& x \text{ has a least element},\\
+          &&& \text{for any $t$, there is $t \oplus 1$, the successor of $t$.}\}
+    \end{IEEEeqnarray*}
+    \todo{Exercise sheet 12}
+    $S$ is Borel.
+
+    We will % TODO ? 
+    construct a reduction
+    \begin{IEEEeqnarray*}{rCl}
+        M \colon S &\longrightarrow & C(\mathbb{K}^\N,\mathbb{K})^\N. %\\
+        % \alpha &\longmapsto &  M(\alpha)
+    \end{IEEEeqnarray*}
+    We want that $\alpha \in \WO(\N) \iff M(\alpha)$
+    codes a distal minimal flow of rank $\alpha$.
+
+    \begin{enumerate}[1.]
+        \item For any $\alpha \in S$, $M(\alpha)$ is a code for
+            a flow which is coded by a generic $(f_i)_{i \in I}$.
+            Specifically we will take a flow
+            corresponding to some $(f_i)_{i \in I}$
+            which is in the intersection of all
+            $U_n$, $V_{j,m,n,\frac{p}{q}}$
+            (cf.~proof of \yaref{thm:distalminimalofallranks}).
+
+        \item If $\alpha \in \WO(\N)$,
+            then additionally $(f_i)_{i \in I}$ will code
+            a distal minimal flow of ordertype $\alpha$.
+    \end{enumerate}
+    
+    One can get a Borel map $S \ni \alpha \mapsto \{T_n^{\alpha} : n \in \N\}$,
+    such that $T^{\alpha}_n$ is closed,
+    $T^{\alpha}_n \neq \emptyset$, $\diam(T^\alpha_n) \xrightarrow{n \to  \infty} 0$,
+    $T^\alpha_{n+1} \subseteq  T^\alpha_n$,
+    $T^{\alpha}_n \subseteq  W^{\alpha}_n$,
+    where $W^{\alpha}_n$ is an enumeration of $U_m^\alpha$,$V^\alpha_{j,m,n,\frac{p}{q}}$.
+    Then $(f_i)_{i \in I} \in  \bigcap_{n} T_{n}^\alpha$.
+\end{proof}
+\begin{lemma}
+    Let $\{(X_\xi, T) : \xi  \le \eta\}$ be a normal
+    quasi-isometric system
+    and $\{(Y_i, T) : i \in I\}$
+    such that
+    \begin{enumerate}[(i)]
+        \item $I \in S$ and additionally $I$ has a largest element.
+        \item $Y_0$ is the trivial flow and  $Y_\infty = X_\eta$,
+            where $0$ and $\infty$ denote the minimal
+            resp.~maximal element of $I$.
+        \item $\forall i < j$
+            % https://q.uiver.app/#q=WzAsMyxbMCwwLCIoWF9cXGV0YSwgVCkiXSxbMSwwLCJZX2oiXSxbMSwxLCJZX2kiXSxbMCwxLCJcXHBpX2oiXSxbMCwyLCJcXHBpX2kiLDJdLFsxLDIsIlxccGleal9pIl1d
+\[\begin{tikzcd}
+	{(X_\eta, T)} & {Y_j} \\
+	& {Y_i}
+	\arrow["{\pi_j}", from=1-1, to=1-2]
+	\arrow["{\pi_i}"', from=1-1, to=2-2]
+	\arrow["{\pi^j_i}", from=1-2, to=2-2]
+\end{tikzcd}\]
+        \item If $i \in I$ is a limit (i.e.~there does not exist
+            an immediate predecessor),
+            then $(Y_i,T)$ is the inverse limit
+            of  $\{(Y_j,T) : j < i\}$
+            with respect to the factor maps.
+        \item $(Y_{i\oplus 1}, T)$ is a maximal isometric
+            extension of $(Y_i, T)$
+            in $(X_\eta, T)$.
+    \end{enumerate}
+
+    Then $I$ is well-ordered with $\otp(Y) = \eta + 1$.
+\end{lemma}
+
+\begin{theorem}[Beleznay Foreman]
+    The order %TODO (Furstenberg rank)
+    is a $\Pi^1_1$-rank.
+\end{theorem}
+For the proof one shows that $\le^\ast$ and $<^\ast$
+are  $\Pi^1_1$, where
+\begin{enumerate}[(1)]
+    \item $p_1 \le^\ast p_2$ iff $p_1$ codes
+        a distal minimal flow and if
+        $p_2$ also codes a distal minimal flow,
+        then $\mathop{order}(p_1) \le \mathop{order}(p_2)$.
+    \item $p_1 <^\ast p_2$ iff $p_1$ codes
+        a distal minimal flow and if
+        $p_2$ also codes a distal minimal flow,
+        then $\mathop{order}(p_1) < \mathop{order}(p_2)$.
+\end{enumerate}
+
+One uses that $(Y_{i+1}, T)$ is a maximal
+isometric extension of $(Y_i,T)$
+ind $(X,T)$
+iff for all $x_1,x_2$ from a fixed countable dense set
+in $X$,
+for all $i$ with $\pi_{i\oplus 1}(x_1) = \pi_{i \oplus 1}(x_2)$,
+there is a sequence $(z_k)$ such that $\pi_i(z_k) = \pi_i(x_1)$,
+$F(z_k, x_1) \to  0$, $F(z_k, x_2) \to 0$.
+
+\begin{proposition}
+    The order of a minimal distal flow on a separable,
+    metric space is countable.
+\end{proposition}
+\begin{proof}
+    Let $(X,\Z)$ be such a flow,
+    i.e.~ $X$ is separable, metric and compact.
+
+    Produce a normal quasi-isometric system
+    \[
+    \{(X_\alpha, \Z) : \alpha \le \beta\}
+    \]
+    with $(X_\beta, \Z) = (X,\Z)$.
+    We need to show that $\beta < \omega_1$.
+
+    Let $\pi_\alpha\colon (X,\Z) \to (X_\alpha, \Z)$.
+    Fix $x_0 \in X$.
+    For every $\alpha$
+    consider $\pi_\alpha^{-1}\left( \pi_\alpha(x_0) \right)
+    = F_\alpha \overset{\text{closed}}{\subseteq} X$.
+
+    \begin{itemize}
+        \item For $\alpha_1 < \alpha_2 \le \beta$
+            we have that $F_{\alpha_1} \supseteq F_{\alpha_2}$.
+        \item For limits $\gamma \le \beta$,
+            we have that $F_\gamma = \bigcap_{\alpha < \gamma} F_\alpha$,
+            since $(X_\gamma,\Z)$ is the inverse limit of
+            $\{(X_{\alpha}, \Z):\alpha < \gamma\}$.
+        \item For all $\alpha < \beta$, $F_{\alpha+1} \subsetneq F_\alpha$,
+            because $\pi^{\alpha+1}_\alpha \colon (X_{\alpha+1},\Z) \to (X_\alpha,\Z)$
+            is not a bijection
+            and all the fibers are isomorphic.
+    \end{itemize}
+
+    So $(F_\alpha)_{\alpha \le \beta}$ is a strictly
+    increasing chain of closed subsets.
+    But $X$ is second countable,
+    so $\beta$ is countable.
+\end{proof}
+
+