This commit is contained in:
parent
ab088dd877
commit
ca0594ea69
4 changed files with 169 additions and 71 deletions
36
inputs/facts.tex
Normal file
36
inputs/facts.tex
Normal file
|
@ -0,0 +1,36 @@
|
|||
\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}
|
||||
|
||||
|
|
@ -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}
|
||||
|
|
|
@ -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
|
||||
|
|
|
@ -60,6 +60,10 @@
|
|||
\input{inputs/tutorial_07}
|
||||
\input{inputs/tutorial_08}
|
||||
|
||||
\section{Facts}
|
||||
\input{inputs/facts}
|
||||
|
||||
|
||||
|
||||
\PrintVocabIndex
|
||||
|
||||
|
|
Loading…
Reference in a new issue