Josia Pietsch 76fb9b562f
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$X$ is always compact metrizable.
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 $x \in K$
% \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}
The action of $1$ determines $h$.
\{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,
\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 $X$.
$G \cdot x$ is compact.
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.
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
K &\longrightarrow & X \\
f G_x &\longmapsto & f(x).
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 $(1,x) \mapsto x + \alpha$.
\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. $K \coloneqq \faktor{G}{G_x}$, $K \to X, fG_x \mapsto f(x)$.
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
\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 $\{(X_\alpha,T), \alpha < \xi\}$.
% https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzAsNSwiWSJdLFsxLDIsIlhfe1xceGknfSJdLFsxLDMsIlhfXFx4aSJdLFswLDEsIlxccGkiLDAseyJjdXJ2ZSI6NH1dLFswLDIsIlxccGlfe1xceGknfSIsMix7ImN1cnZlIjotMX1dLFswLDMsIlxccGlfe1xceGl9IiwyLHsiY3VydmUiOjJ9XSxbMiwzLCJcXHBpX3tcXHhpLCBcXHhpJ30iXV0=
X \\
& {X_{\xi'}} \\
& {X_\xi} \\
\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]
Then we say that $(X,T)$ is a \vocab{quasi-isometric extension}
of $(Y,T)$.
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}.
Every quasi-isometric flow is distal.
The trivial flow is distal.
Apply \yaref{prop:isomextdistal} and \yaref{prop:limitdistal}.
Every minimal distal flow is quasi-isometric.
By Zorn's lemma, this will follow from
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==
{(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]
\yaref{thm:furstenberg} allows us to talk about ranks of distal minimal flows:
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 $\rank((X,T))$.
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
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.
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 $U_0,\ldots, U_n \overset{\text{open}}{\subseteq} X$,
[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\}.
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}}
Consider $K(\bH^2)$.
A flow $\Z \acts X$ corresponds to the graph of
X &\longrightarrow & X \\
x&\longmapsto & 1 \cdot x
and this graph is an element of $K(\bH^2)$.
Consider $\Z$-flows.
\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.