lecture 16
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@ -107,13 +107,12 @@ Recall:
\begin{remark} \begin{remark}
Note that a flow is minimal iff it has no proper subflows. Note that a flow is minimal iff it has no proper subflows.
\end{remark} \end{remark}
% \begin{example} \begin{example}
% Recall that $S_1 = \{z \in \C : |z| = 1\}$. Recall that $S_1 = \{z \in \C : |z| = 1\}$.
% Let $X = S_1$, $T = S_1$ Let $X = S_1$, $T = S_1$
% $(\alpha,\beta) \mapsto \alpha + \beta$ is isometric. $(\alpha,\beta) \mapsto \alpha + \beta$ is isometric.%
% % TODO: In the official notes it says \alpha + \beta, but this is no group action. \footnote{Note that this means $x \cdot \alpha$ in complex numbers, but we consider the abelian group structure of $S^1$}
% % Maybe \alpha * \beta ? \end{example}
% \end{example}
\begin{definition} \begin{definition}
Let $X,Y$ be compact metric spaces Let $X,Y$ be compact metric spaces

185
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@ -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

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@ -40,6 +40,7 @@
\input{inputs/lecture_13} \input{inputs/lecture_13}
\input{inputs/lecture_14} \input{inputs/lecture_14}
\input{inputs/lecture_15} \input{inputs/lecture_15}
\input{inputs/lecture_16}