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@ -107,13 +107,12 @@ Recall:
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\begin{remark}
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\begin{remark}
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Note that a flow is minimal iff it has no proper subflows.
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Note that a flow is minimal iff it has no proper subflows.
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\end{remark}
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\end{remark}
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% \begin{example}
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\begin{example}
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% Recall that $S_1 = \{z \in \C : |z| = 1\}$.
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Recall that $S_1 = \{z \in \C : |z| = 1\}$.
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% Let $X = S_1$, $T = S_1$
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Let $X = S_1$, $T = S_1$
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% $(\alpha,\beta) \mapsto \alpha + \beta$ is isometric.
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$(\alpha,\beta) \mapsto \alpha + \beta$ is isometric.%
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% % TODO: In the official notes it says \alpha + \beta, but this is no group action.
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\footnote{Note that this means $x \cdot \alpha$ in complex numbers, but we consider the abelian group structure of $S^1$}
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% % Maybe \alpha * \beta ?
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\end{example}
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% \end{example}
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\begin{definition}
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\begin{definition}
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Let $X,Y$ be compact metric spaces
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Let $X,Y$ be compact metric spaces
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185
inputs/lecture_16.tex
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185
inputs/lecture_16.tex
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@ -0,0 +1,185 @@
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\lecture{16}{2023-12-08}{}
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% \begin{definition}
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% % TODO
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% Isomorphism from $T \acts X$ to $T \acts Y$ :
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% Bijection $X \xrightarrow{b} Y$
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% such that $b(tx) = t b(x)$.
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% \end{definition}
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$X$ is always compact metrizable.
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\begin{theorem}
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Every minimal isometric flow $(X,\Z)$
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for $X$ a compact metrizable space%
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\footnote{Such a flow is uniquely determined by $h\colon X \to X, x \mapsto 1\cdot x$.}
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is isomorphic to an abelian group rotation
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$(K, \Z)$, with
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$K$ an abelian compact group
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and $h(x) = x + \alpha$ for all $x \in K$
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\end{theorem}
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\begin{example}
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Let $\alpha \in S^1$
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and $h\colon x \mapsto x + \alpha$.\footnote{Again $x + \alpha$ denotes $x \cdot \alpha$ in $\C$.}
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\end{example}
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\begin{proof}
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The action of $1$ determines $h$
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and $n \in \Z \leadsto h^n$.
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Consider
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\[
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\{h^n : n \in \Z\} \subseteq \cC(X,X) = \{f\colon X \to X : f \text{continuous}\},
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\]
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where the topology is the uniform convergence topology.
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Let $G = \overline{ \{h^n : n \in \Z\} } \subseteq \cC(X,X)$.
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Since
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\[
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\forall \epsilon > 0.~\exists \delta > 0.~d(x,y) < \delta \implies \forall n.~d(h^n(x), h^n(y)) < \epsilon
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\]
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we have by the Arzela-Ascoli-Theorem % TODO REF
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that $G$ is compact.
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$G$ is a closure of a of a topological group,
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hence it is a topological group,
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i.e.~$g \mapsto g^{-1}$ and $(g,h) \mapsto gh$ are continuous.
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% TODO THINK ABOUT THIS
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Moreover since $\Z$ is abelian,
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$\forall n,m \in \Z.~h^n \cdot h^m = h^m \cdot h^n$,
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so $G$ is abelian.
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% TODO THINK ABOUT THIS
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Take any $x \in X$ and consider
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the orbit % TODO DEFINITION
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$G \cdot x = \{f(x) : f \in G\}$.
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Since $\Z \acts X$ is minimal,
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i.e.~every orbit is dense,
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we have that $G \cdot x$ is dense in $X$.
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\begin{claim}
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$G \cdot x$ is compact.
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\end{claim}
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\begin{subproof}
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Since $\Z \acts X$ is continuous,
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$g \mapsto g x$ is continuous:
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Let $g_n$ be a sequence in $G$
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such that $g_n \to g$.
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Then $g_n x \to gx$,
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since the topology on $\cC(X,X)$
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is the uniform convergence topology.
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Therefore the compactness of $G$ implies
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that the orbit $Gx$ is compact.
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\end{subproof}
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Since $G\cdot x$ is compact and dense,
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we get $G \cdot x = X$.
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% TODO THINK ABOUT THIS
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Let $\Gamma = \{f \in G : f(x) = x\} < G$ % TODO \triangleright?
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be the stabilizer group. % TODO DEFINITION
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Then $\Gamma \subseteq G$ is closed.
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Take $K \coloneqq \faktor{G}{\Gamma}$ with the quotient topology.
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$K$ is an abelian compact group
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and $G \to Gx$ gives
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a homeomorphism $K = \faktor{G}{\Gamma} \to Gx = X$.
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Conclusion:
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$\Z \acts K \equiv \Z \acts X$
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% and $h$ is a claimed.
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\todo{Copy from official notes}
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% TODO Definition transitive group action.
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\end{proof}
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\begin{definition}
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Let $(X,T)$ be a flow
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and $(Y,T)$ a factor of $(X,T)$.%
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\footnote{i.e. there exists a continuous surjection $\pi\colon X \twoheadrightarrow Y$
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commuting with the action, i.e.~$\forall t \in T. x \in X.~\pi(tx) = t \pi(x)$.
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Warning: Fürstenberg called factors subflows.
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% TODO: Definition
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}
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Suppose there is $\eta \in \Ord$
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such that for any $\xi < \eta$
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there is a factor $(X_\xi, T)$ of $(X,T)$
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\begin{enumerate}[(a)]
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\item $(X_0, T) = (Y,T)$ and $(X_\eta, T) = (X,T)$.
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\item If $\xi < \xi'$, then $(X_\xi, T)$ is a factor of $(X_{\xi'}, T)$
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``inside $(X,T)$'', i.e.~$\pi_\xi = \pi_{\xi, \xi'} \circ \pi_{\xi'}$.
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\item $\forall \xi < \eta.~ (X_{\xi + 1}, T)$ is an isometric extension of $(X_\xi, T)$.
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\item $\xi \le \eta$ is a limit, then $(X_\xi, T)$
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is a limit of $\{(X_\alpha,T), \alpha < \xi\}$.
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\end{enumerate}
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% https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzAsNSwiWSJdLFsxLDIsIlhfe1xceGknfSJdLFsxLDMsIlhfXFx4aSJdLFswLDEsIlxccGkiLDAseyJjdXJ2ZSI6NH1dLFswLDIsIlxccGlfe1xceGknfSIsMix7ImN1cnZlIjotMX1dLFswLDMsIlxccGlfe1xceGl9IiwyLHsiY3VydmUiOjJ9XSxbMiwzLCJcXHBpX3tcXHhpLCBcXHhpJ30iXV0=
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\[\begin{tikzcd}
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X \\
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\\
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& {X_{\xi'}} \\
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& {X_\xi} \\
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\\
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Y
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\arrow["\pi", curve={height=24pt}, from=1-1, to=6-1]
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\arrow["{\pi_{\xi'}}"', curve={height=-6pt}, from=1-1, to=3-2]
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\arrow["{\pi_{\xi}}"', curve={height=12pt}, from=1-1, to=4-2]
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\arrow["{\pi_{\xi, \xi'}}", from=3-2, to=4-2]
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\end{tikzcd}\]
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Then we say that $(X,T)$ is a \vocab{quasi-isometric extension}
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of $(Y,T)$.
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\end{definition}
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\begin{definition}
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If $(Y,T)$ is trivial, i.e.~$|Y| = 1$,
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then a quasi-isometric extension $(X,T)$ of $(Y,T)$
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is called a \vocab{quasi-isometric flow}.
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\end{definition}
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\begin{corollary}
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Every quasi-isometric flow is distal.
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\end{corollary}
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\begin{proof}
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\todo{TODO}
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% The trivial flow is distal.
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\end{proof}
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\begin{theorem}[Fürstenberg]
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Every minimal distal flow is quasi-isometric.
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\end{theorem}
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Therefore one can talk about ranks of distal minimal flows.
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\begin{definition}
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Let $(X, \Z)$ be distal minimal.
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Then $\rank((X,\Z)) \coloneqq \min \{\eta : (X, \Z) \cong (X_\eta, \Z)\}$
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where $(X_{\eta}, \Z)$ is as from the definition of quasi-isometric flows,
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i.e.~$\rank((X,\Z))$ is the minimal height such
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that a tower as in the definition exists.
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\end{definition}
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\begin{theorem}[Beleznay-Foreman]
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Let $T = \Z$.
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\begin{itemize}
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\item For any $\alpha < \omega_1$,
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there is a distal minimal flow of rank $\alpha$.
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\item Distal flows form a $\Pi^1_1$-complete set:
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\todo{Move the explanations to a remark}
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For a fixed compact metric space $X$,
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view the flows $(X,\Z)$
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as a subset of $\cC(X,X)$.
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Note that $\cC(X,X)$ is Polish.
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Then the minimal flows on $X$ are a Borel subset of $\cC(X,X)$.
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But we want to look all flows at the same time.
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The Hilbert cube $[0,1]^{\N}$
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embeds all compact metric spaces.
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Thus we consider $K([0,1]^{\N})$,
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the space of compact subsets of $[0,1]^{\N}$.\todo{move definition}
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$K([0,1]^{\N})$ is a Polish space.
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Consider $K(([0,1]^\N)^2)$.
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A flow $\Z \acts X$ corresponds to the graph of
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\begin{IEEEeqnarray*}{rCl}
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X &\longrightarrow & X \\
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1&\longmapsto & 1 \cdot x
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\end{IEEEeqnarray*}
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and this graph is an element of $K(([0,1]^{\N})^2)$.
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\item Moreover, this rank is a $\Pi^1_1$-rank.
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\end{itemize}
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\end{theorem}
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\fi
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@ -40,6 +40,7 @@
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\input{inputs/lecture_13}
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\input{inputs/lecture_13}
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\input{inputs/lecture_14}
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\input{inputs/lecture_14}
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\input{inputs/lecture_15}
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\input{inputs/lecture_15}
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\input{inputs/lecture_16}
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