lecture 16
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inputs/facts.tex Normal file
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\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}

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@ -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}

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@ -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
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
$G \cdot x = \{f(x) : f \in G\}$.
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 $\{(X_\alpha,T), \alpha < \xi\}$.
@ -138,10 +129,28 @@ $X$ 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 $\rank((X,\Z)) \coloneqq \min \{\eta : (X, \Z) \cong (X_\eta, \Z)\}$
@ -150,33 +159,61 @@ Therefore one can talk about ranks of distal minimal flows.
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

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@ -60,6 +60,10 @@
\input{inputs/tutorial_07}
\input{inputs/tutorial_08}
\section{Facts}
\input{inputs/facts}
\PrintVocabIndex