This commit is contained in:
parent
206b61941e
commit
4b9bbd2ef0
3 changed files with 172 additions and 5 deletions
|
@ -84,5 +84,3 @@ Let $X_n \coloneqq (S^1)^n$ and $X \coloneqq (S^1)^{\N}$.
|
||||||
\begin{corollary}
|
\begin{corollary}
|
||||||
The order of $(X,\tau)$ is $\omega$.
|
The order of $(X,\tau)$ is $\omega$.
|
||||||
\end{corollary}
|
\end{corollary}
|
||||||
\todo{I could not attend lecture 21 as I was sick. The official notes on the lecture are very short.
|
|
||||||
Is something missing in the official notes?}
|
|
||||||
|
|
|
@ -127,19 +127,21 @@ For this we define
|
||||||
% \end{example}
|
% \end{example}
|
||||||
|
|
||||||
\begin{theorem}[Beleznay Foreman]
|
\begin{theorem}[Beleznay Foreman]
|
||||||
|
\label{thm:distalminimalofallranks}
|
||||||
Whenever $I = \eta$ for some $\eta < \omega_1$,
|
Whenever $I = \eta$ for some $\eta < \omega_1$,
|
||||||
then
|
then
|
||||||
\[
|
\[
|
||||||
\{\overline{f} \in \mathbb{K}_I : E_I(\overline{f}) \text{ is distal, minimal and of rank$\eta$}\} % TODO rank = order
|
\{\overline{f} \in \mathbb{K}_I : E_I(\overline{f}) \text{ is distal, minimal and of rank $\eta$}\} % TODO rank = order
|
||||||
\]
|
\]
|
||||||
is comeager in $\mathbb{K}_I$.
|
is comeager in $\mathbb{K}_I$.
|
||||||
In particular such flows exist.
|
In particular such flows exist.
|
||||||
\end{theorem}
|
\end{theorem}
|
||||||
\begin{proof}[sketch]
|
\begin{proof}[sketch]
|
||||||
|
\leavevmode
|
||||||
\begin{itemize}
|
\begin{itemize}
|
||||||
\item Distality:
|
\item Distality:
|
||||||
For all $\overline{f} \in \mathbb{K}_I$,
|
For all $\overline{f} \in \mathbb{K}_I$,
|
||||||
the flow $E_I \overline{f}$is distal.
|
the flow $E_I \overline{f}$ is distal.
|
||||||
This is the same as for iterated skew shifts.
|
This is the same as for iterated skew shifts.
|
||||||
% TODO since for $\overline{x}, \overline{y} \in \mathbb{K}^I$,
|
% TODO since for $\overline{x}, \overline{y} \in \mathbb{K}^I$,
|
||||||
% $d(x_\alpha, y_\alpha) = d((f(\overline{x})_\alpha, (f(\overline{y})_\alpha))$.
|
% $d(x_\alpha, y_\alpha) = d((f(\overline{x})_\alpha, (f(\overline{y})_\alpha))$.
|
||||||
|
@ -176,7 +178,7 @@ For this we define
|
||||||
Fix a countable dense set $(\overline{x_n})$ in $\mathbb{K}^I$.
|
Fix a countable dense set $(\overline{x_n})$ in $\mathbb{K}^I$.
|
||||||
For $\epsilon \in \Q$ let
|
For $\epsilon \in \Q$ let
|
||||||
\begin{IEEEeqnarray*}{rCl}
|
\begin{IEEEeqnarray*}{rCl}
|
||||||
V_{j,m,n,\epsilon} &\coloneqq & \{\overline{f} \in \mathbb{K}_I : \\
|
V_{j,m,n,\epsilon} &\coloneqq \{\overline{f} \in \mathbb{K}_I :& \\
|
||||||
&&\text{if } \Pi_{j+1}(\overline{x}_n) = \Pi_{j+1}(\overline{x_m}),\\
|
&&\text{if } \Pi_{j+1}(\overline{x}_n) = \Pi_{j+1}(\overline{x_m}),\\
|
||||||
&&\text{then there are $k_m$, $k_n$, $\overline{z}$ such that}\\
|
&&\text{then there are $k_m$, $k_n$, $\overline{z}$ such that}\\
|
||||||
&&\pi_j(\overline{x_n}) = \pi_j(\overline{z}), \forall k> j+1.~z_k = 1,\\
|
&&\pi_j(\overline{x_n}) = \pi_j(\overline{z}), \forall k> j+1.~z_k = 1,\\
|
||||||
|
|
167
inputs/lecture_23.tex
Normal file
167
inputs/lecture_23.tex
Normal file
|
@ -0,0 +1,167 @@
|
||||||
|
\lecture{23}{2024-01-19}{More sketches of ideas of Beleznay and Foreman}
|
||||||
|
|
||||||
|
\begin{notation}
|
||||||
|
Let $X$ be a Polish space and $\cP$ a property of elements of $X$,
|
||||||
|
then we say that $x_0 \in X$ is \vocab{generic}
|
||||||
|
if
|
||||||
|
\[
|
||||||
|
A_\cP \coloneqq \{x \in X \colon \cP(x)\}
|
||||||
|
\]
|
||||||
|
is comeager
|
||||||
|
and $x_0 \in A_\cP$.
|
||||||
|
\end{notation}
|
||||||
|
For example let $X = \mathbb{K}_I$
|
||||||
|
and $\cP$ the property of being a distal minimal flow.
|
||||||
|
\begin{abuse}
|
||||||
|
We will usually omit $\cP$.
|
||||||
|
\end{abuse}
|
||||||
|
|
||||||
|
|
||||||
|
Let $I$ be a linear order
|
||||||
|
|
||||||
|
\begin{theorem}[Beleznay and Foreman]
|
||||||
|
The set of distal minimal flows is $\Pi_1^1$-complete.
|
||||||
|
\end{theorem}
|
||||||
|
|
||||||
|
\begin{proof}[sketch]
|
||||||
|
Consider $\WO(\N) \subset \LO(\N)$.
|
||||||
|
We know that this is $\Pi_1^1$-complete. % TODO ref
|
||||||
|
|
||||||
|
Let
|
||||||
|
\begin{IEEEeqnarray*}{rCll}
|
||||||
|
S & \coloneqq & \{ x \in \LO(\N) :& x \text{ has a least element},\\
|
||||||
|
&&& \text{for any $t$, there is $t \oplus 1$, the successor of $t$.}\}
|
||||||
|
\end{IEEEeqnarray*}
|
||||||
|
\todo{Exercise sheet 12}
|
||||||
|
$S$ is Borel.
|
||||||
|
|
||||||
|
We will % TODO ?
|
||||||
|
construct a reduction
|
||||||
|
\begin{IEEEeqnarray*}{rCl}
|
||||||
|
M \colon S &\longrightarrow & C(\mathbb{K}^\N,\mathbb{K})^\N. %\\
|
||||||
|
% \alpha &\longmapsto & M(\alpha)
|
||||||
|
\end{IEEEeqnarray*}
|
||||||
|
We want that $\alpha \in \WO(\N) \iff M(\alpha)$
|
||||||
|
codes a distal minimal flow of rank $\alpha$.
|
||||||
|
|
||||||
|
\begin{enumerate}[1.]
|
||||||
|
\item For any $\alpha \in S$, $M(\alpha)$ is a code for
|
||||||
|
a flow which is coded by a generic $(f_i)_{i \in I}$.
|
||||||
|
Specifically we will take a flow
|
||||||
|
corresponding to some $(f_i)_{i \in I}$
|
||||||
|
which is in the intersection of all
|
||||||
|
$U_n$, $V_{j,m,n,\frac{p}{q}}$
|
||||||
|
(cf.~proof of \yaref{thm:distalminimalofallranks}).
|
||||||
|
|
||||||
|
\item If $\alpha \in \WO(\N)$,
|
||||||
|
then additionally $(f_i)_{i \in I}$ will code
|
||||||
|
a distal minimal flow of ordertype $\alpha$.
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
One can get a Borel map $S \ni \alpha \mapsto \{T_n^{\alpha} : n \in \N\}$,
|
||||||
|
such that $T^{\alpha}_n$ is closed,
|
||||||
|
$T^{\alpha}_n \neq \emptyset$, $\diam(T^\alpha_n) \xrightarrow{n \to \infty} 0$,
|
||||||
|
$T^\alpha_{n+1} \subseteq T^\alpha_n$,
|
||||||
|
$T^{\alpha}_n \subseteq W^{\alpha}_n$,
|
||||||
|
where $W^{\alpha}_n$ is an enumeration of $U_m^\alpha$,$V^\alpha_{j,m,n,\frac{p}{q}}$.
|
||||||
|
Then $(f_i)_{i \in I} \in \bigcap_{n} T_{n}^\alpha$.
|
||||||
|
\end{proof}
|
||||||
|
\begin{lemma}
|
||||||
|
Let $\{(X_\xi, T) : \xi \le \eta\}$ be a normal
|
||||||
|
quasi-isometric system
|
||||||
|
and $\{(Y_i, T) : i \in I\}$
|
||||||
|
such that
|
||||||
|
\begin{enumerate}[(i)]
|
||||||
|
\item $I \in S$ and additionally $I$ has a largest element.
|
||||||
|
\item $Y_0$ is the trivial flow and $Y_\infty = X_\eta$,
|
||||||
|
where $0$ and $\infty$ denote the minimal
|
||||||
|
resp.~maximal element of $I$.
|
||||||
|
\item $\forall i < j$
|
||||||
|
% https://q.uiver.app/#q=WzAsMyxbMCwwLCIoWF9cXGV0YSwgVCkiXSxbMSwwLCJZX2oiXSxbMSwxLCJZX2kiXSxbMCwxLCJcXHBpX2oiXSxbMCwyLCJcXHBpX2kiLDJdLFsxLDIsIlxccGleal9pIl1d
|
||||||
|
\[\begin{tikzcd}
|
||||||
|
{(X_\eta, T)} & {Y_j} \\
|
||||||
|
& {Y_i}
|
||||||
|
\arrow["{\pi_j}", from=1-1, to=1-2]
|
||||||
|
\arrow["{\pi_i}"', from=1-1, to=2-2]
|
||||||
|
\arrow["{\pi^j_i}", from=1-2, to=2-2]
|
||||||
|
\end{tikzcd}\]
|
||||||
|
\item If $i \in I$ is a limit (i.e.~there does not exist
|
||||||
|
an immediate predecessor),
|
||||||
|
then $(Y_i,T)$ is the inverse limit
|
||||||
|
of $\{(Y_j,T) : j < i\}$
|
||||||
|
with respect to the factor maps.
|
||||||
|
\item $(Y_{i\oplus 1}, T)$ is a maximal isometric
|
||||||
|
extension of $(Y_i, T)$
|
||||||
|
in $(X_\eta, T)$.
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
Then $I$ is well-ordered with $\otp(Y) = \eta + 1$.
|
||||||
|
\end{lemma}
|
||||||
|
|
||||||
|
\begin{theorem}[Beleznay Foreman]
|
||||||
|
The order %TODO (Furstenberg rank)
|
||||||
|
is a $\Pi^1_1$-rank.
|
||||||
|
\end{theorem}
|
||||||
|
For the proof one shows that $\le^\ast$ and $<^\ast$
|
||||||
|
are $\Pi^1_1$, where
|
||||||
|
\begin{enumerate}[(1)]
|
||||||
|
\item $p_1 \le^\ast p_2$ iff $p_1$ codes
|
||||||
|
a distal minimal flow and if
|
||||||
|
$p_2$ also codes a distal minimal flow,
|
||||||
|
then $\mathop{order}(p_1) \le \mathop{order}(p_2)$.
|
||||||
|
\item $p_1 <^\ast p_2$ iff $p_1$ codes
|
||||||
|
a distal minimal flow and if
|
||||||
|
$p_2$ also codes a distal minimal flow,
|
||||||
|
then $\mathop{order}(p_1) < \mathop{order}(p_2)$.
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
One uses that $(Y_{i+1}, T)$ is a maximal
|
||||||
|
isometric extension of $(Y_i,T)$
|
||||||
|
ind $(X,T)$
|
||||||
|
iff for all $x_1,x_2$ from a fixed countable dense set
|
||||||
|
in $X$,
|
||||||
|
for all $i$ with $\pi_{i\oplus 1}(x_1) = \pi_{i \oplus 1}(x_2)$,
|
||||||
|
there is a sequence $(z_k)$ such that $\pi_i(z_k) = \pi_i(x_1)$,
|
||||||
|
$F(z_k, x_1) \to 0$, $F(z_k, x_2) \to 0$.
|
||||||
|
|
||||||
|
\begin{proposition}
|
||||||
|
The order of a minimal distal flow on a separable,
|
||||||
|
metric space is countable.
|
||||||
|
\end{proposition}
|
||||||
|
\begin{proof}
|
||||||
|
Let $(X,\Z)$ be such a flow,
|
||||||
|
i.e.~ $X$ is separable, metric and compact.
|
||||||
|
|
||||||
|
Produce a normal quasi-isometric system
|
||||||
|
\[
|
||||||
|
\{(X_\alpha, \Z) : \alpha \le \beta\}
|
||||||
|
\]
|
||||||
|
with $(X_\beta, \Z) = (X,\Z)$.
|
||||||
|
We need to show that $\beta < \omega_1$.
|
||||||
|
|
||||||
|
Let $\pi_\alpha\colon (X,\Z) \to (X_\alpha, \Z)$.
|
||||||
|
Fix $x_0 \in X$.
|
||||||
|
For every $\alpha$
|
||||||
|
consider $\pi_\alpha^{-1}\left( \pi_\alpha(x_0) \right)
|
||||||
|
= F_\alpha \overset{\text{closed}}{\subseteq} X$.
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item For $\alpha_1 < \alpha_2 \le \beta$
|
||||||
|
we have that $F_{\alpha_1} \supseteq F_{\alpha_2}$.
|
||||||
|
\item For limits $\gamma \le \beta$,
|
||||||
|
we have that $F_\gamma = \bigcap_{\alpha < \gamma} F_\alpha$,
|
||||||
|
since $(X_\gamma,\Z)$ is the inverse limit of
|
||||||
|
$\{(X_{\alpha}, \Z):\alpha < \gamma\}$.
|
||||||
|
\item For all $\alpha < \beta$, $F_{\alpha+1} \subsetneq F_\alpha$,
|
||||||
|
because $\pi^{\alpha+1}_\alpha \colon (X_{\alpha+1},\Z) \to (X_\alpha,\Z)$
|
||||||
|
is not a bijection
|
||||||
|
and all the fibers are isomorphic.
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
So $(F_\alpha)_{\alpha \le \beta}$ is a strictly
|
||||||
|
increasing chain of closed subsets.
|
||||||
|
But $X$ is second countable,
|
||||||
|
so $\beta$ is countable.
|
||||||
|
\end{proof}
|
||||||
|
|
||||||
|
|
Loading…
Reference in a new issue