From ca0594ea697e18b68ac79d0e1bdc14e0b51498e4 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Sat, 9 Dec 2023 18:23:59 +0100 Subject: [PATCH] lecture 16 --- inputs/facts.tex | 36 +++++++++ inputs/lecture_15.tex | 21 +++++ inputs/lecture_16.tex | 179 +++++++++++++++++++++++++----------------- logic3.tex | 4 + 4 files changed, 169 insertions(+), 71 deletions(-) create mode 100644 inputs/facts.tex diff --git a/inputs/facts.tex b/inputs/facts.tex new file mode 100644 index 0000000..f749a4c --- /dev/null +++ b/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} + + diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex index cd49e91..4f64adc 100644 --- a/inputs/lecture_15.tex +++ b/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} diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index d029ade..f6f2f00 100644 --- a/inputs/lecture_16.tex +++ b/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 - - Take any $x \in X$ and consider the orbit - $G \cdot x = \{f(x) : f \in G\}$. + 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 $G \cdot x$. 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 $\{(X_\alpha,T), \alpha < \xi\}$. @@ -138,45 +129,91 @@ $X$ 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 $\rank((X,\Z)) \coloneqq \min \{\eta : (X, \Z) \cong (X_\eta, \Z)\}$ where $(X_{\eta}, \Z)$ is as from the definition of quasi-isometric flows, - i.e.~$\rank((X,\Z))$ is the minimal height such + i.e.~$\rank((X,\Z))$ is the minimal height such that a tower as in the definition exists. \end{definition} +\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 $X$, +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 $\cC(X,X)$. + +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 $K(\bH^2)$. + \begin{theorem}[Beleznay-Foreman] - Let $T = \Z$. + 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: - \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 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 diff --git a/logic3.tex b/logic3.tex index 975d19b..5a3f644 100644 --- a/logic3.tex +++ b/logic3.tex @@ -60,6 +60,10 @@ \input{inputs/tutorial_07} \input{inputs/tutorial_08} +\section{Facts} +\input{inputs/facts} + + \PrintVocabIndex