diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex index 87d5d4e..e9ebacd 100644 --- a/inputs/lecture_15.tex +++ b/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 diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex new file mode 100644 index 0000000..30d999b --- /dev/null +++ b/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 diff --git a/logic3.tex b/logic3.tex index b4f6e4b..975d19b 100644 --- a/logic3.tex +++ b/logic3.tex @@ -40,6 +40,7 @@ \input{inputs/lecture_13} \input{inputs/lecture_14} \input{inputs/lecture_15} +\input{inputs/lecture_16}