diff --git a/inputs/lecture_21.tex b/inputs/lecture_21.tex index 662931c..3b5a660 100644 --- a/inputs/lecture_21.tex +++ b/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?} diff --git a/inputs/lecture_22.tex b/inputs/lecture_22.tex index 2cffaf6..44943fb 100644 --- a/inputs/lecture_22.tex +++ b/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,\\ diff --git a/inputs/lecture_23.tex b/inputs/lecture_23.tex new file mode 100644 index 0000000..85d6e55 --- /dev/null +++ b/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} + +