Josia Pietsch
76fb9b562f
Some checks failed
Build latex and deploy / checkout (push) Failing after 18m11s
239 lines
8.6 KiB
TeX
239 lines
8.6 KiB
TeX
\lecture{16}{2023-12-08}{}
|
|
|
|
$X$ 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 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{proof}
|
|
\gist{%
|
|
The action of $1$ determines $h$.
|
|
Consider
|
|
\[
|
|
\{h^n : n \in \Z\} \subseteq \cC(X,X)\gist{ = \{f\colon X \to X : f \text{ continuous}\}}{},
|
|
\]
|
|
where the topology is the uniform convergence topology.
|
|
Let $G = \overline{ \{h^n : n \in \Z\} } \subseteq \cC(X,X)$.
|
|
Since the family $\{h^n : n \in \Z\}$ is uniformly equicontinuous,
|
|
i.e.~
|
|
\[
|
|
\forall \epsilon > 0.~\exists \delta > 0.~d(x,y) < \delta \implies \forall n.~d(h^n(x), h^n(y)) < \epsilon,
|
|
% Here we use isometric
|
|
\]
|
|
we have by the Arzel\`a-Ascoli-Theorem % TODO REF
|
|
that $G$ is compact.
|
|
|
|
$G$ is a closure of a topological group,
|
|
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 $X$.
|
|
|
|
\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 $G \cdot x = X$
|
|
(compact subsets of Hausdorff spaces are closed).
|
|
|
|
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.
|
|
|
|
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.
|
|
|
|
For $\alpha = h$ we get that
|
|
a flow $\Z \acts X$ corresponds to $\Z \acts K$
|
|
with $(1,x) \mapsto x + \alpha$.
|
|
}{
|
|
\begin{itemize}
|
|
\item $G \coloneqq \overline{\{h^n : n \in \Z\} } \subseteq \cC(X,X)$.
|
|
\item $G$ is compact (Arzela-Ascoli), abelian topological group (closure of ab. top. group)
|
|
\item Take any $x \in G$.
|
|
\item $Gx$ is compact (since $g \mapsto gx$ is continuous and $G$ is compact)
|
|
\item Stabilizer $G_x$ is closed. $K \coloneqq \faktor{G}{G_x}$, $K \to X, fG_x \mapsto f(x)$.
|
|
\end{itemize}
|
|
}
|
|
\end{proof}
|
|
\begin{definition}
|
|
Let $(X,T)$ be a flow
|
|
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.~$\pi_\xi = \pi_{\xi, \xi'} \circ \pi_{\xi'}$,
|
|
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 $\{(X_\alpha,T), \alpha < \xi\}$.
|
|
\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}
|
|
The trivial flow is distal.
|
|
Apply \yaref{prop:isomextdistal} and \yaref{prop:limitdistal}.
|
|
\end{proof}
|
|
|
|
\begin{theorem}[Furstenberg]
|
|
\label{thm:furstenberg}
|
|
Every minimal distal flow is quasi-isometric.
|
|
\end{theorem}
|
|
By Zorn's lemma, this will follow from
|
|
\begin{theorem}[Furstenberg]
|
|
\label{thm:l16:3}
|
|
Let $(X, T)$ be a minimal distal flow
|
|
and let $(Y, T)$ be a proper factor.%
|
|
\footnote{i.e.~$(X,T)$ and $(Y,T)$ are not isomorphic}
|
|
Then there is another factor $(Z,T)$ of $(X,T)$
|
|
which is a proper isometric extension of $Y$.
|
|
|
|
% https://q.uiver.app/#q=WzAsMyxbMiwwLCIoWCxUKSJdLFswLDAsIihZLFQpIl0sWzEsMSwiKFosVCkiXSxbMCwxXSxbMCwyXSxbMiwxLCJcXHRleHR7aXNvbWV0cmljIGV4dGVuc2lvbn0iLDFdXQ==
|
|
\[\begin{tikzcd}
|
|
{(Y,T)} && {(X,T)} \\
|
|
& {(Z,T)}
|
|
\arrow[from=1-3, to=1-1]
|
|
\arrow[from=1-3, to=2-2]
|
|
\arrow["{\text{isometric extension}}"{description}, from=2-2, to=1-1]
|
|
\end{tikzcd}\]
|
|
|
|
\end{theorem}
|
|
\yaref{thm:furstenberg} allows us to talk about ranks of distal minimal flows:
|
|
\begin{definition}[{\cite[{}13.1]{Furstenberg}}]
|
|
\label{def:floworder}
|
|
Let $(X,T)$ be a quasi-isometric flow,
|
|
and let $\eta$ be the smallest ordinal
|
|
such that there exists a quasi-isometric system $\{(X_\xi, T), \xi \le \eta\}$
|
|
with $(X,T) = (X_\xi, T)$.
|
|
Then $\eta$ is called the \vocab{rank} or \vocab{order} of the flow
|
|
and is denoted by $\rank((X,T))$.
|
|
\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{cf.~\yaref{s9e2}}
|
|
of the form
|
|
$[U_0; U_1,\ldots, U_n]$,
|
|
for $U_0,\ldots, U_n \overset{\text{open}}{\subseteq} X$,
|
|
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.\footnote{cf.~\yaref{s1e4}}
|
|
Then the minimal flows on $X$ are a Borel subset of $\cC(X,X)$.\footnote{Exercise} % TODO
|
|
|
|
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.\footnote{cf.~\yaref{s9e2}, \yaref{s12e4}}
|
|
% TODO LEARN EXERCISES
|
|
Consider $K(\bH^2)$.
|
|
A flow $\Z \acts X$ corresponds to the graph of
|
|
\begin{IEEEeqnarray*}{rCl}
|
|
X &\longrightarrow & X \\
|
|
x&\longmapsto & 1 \cdot x
|
|
\end{IEEEeqnarray*}
|
|
and this graph is an element of $K(\bH^2)$.
|
|
|
|
\begin{theorem}[Beleznay-Foreman]
|
|
\label{thm:beleznay-foreman}
|
|
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,
|
|
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}
|