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Josia Pietsch 2024-02-06 01:46:55 +01:00
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commit ba609dfed0
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5 changed files with 29 additions and 28 deletions

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@ -226,8 +226,8 @@ i.e.~show that if $(Z,T)$ is a proper factor of a minimal distal flow
\] \]
$X \setminus B_q = \{a \in X : \Gamma(a) < q\}$ $X \setminus B_q = \{a \in X : \Gamma(a) < q\}$
is open, i.e.~$B_q$ is closed. is open, i.e.~$B_q$ is closed.
Note that $x \in F_q \coloneqq B_q \setminus B_q^\circ$ Note that $x \in F_q \coloneqq B_q \setminus \inter(B_q)$
and $B_q \setminus B_q^\circ$ is nwd and $B_q \setminus \inter(B_q)$ is nwd
as it is closed and has empty interior, as it is closed and has empty interior,
so $\bigcup_{q \in \Q} F_q$ is meager. so $\bigcup_{q \in \Q} F_q$ is meager.
} }

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@ -1,9 +1,6 @@
\subsection{The Order of a Flow} \subsection{The Order of a Flow}
\lecture{19}{2023-12-19}{Orders of Flows} \lecture{19}{2023-12-19}{Orders of Flows}
% TODO ANKI-MARKER
See also \cite[\href{https://terrytao.wordpress.com/2008/01/24/254a-lecture-6-isometric-systems-and-isometric-extensions/}{Lecture 6}]{tao}. See also \cite[\href{https://terrytao.wordpress.com/2008/01/24/254a-lecture-6-isometric-systems-and-isometric-extensions/}{Lecture 6}]{tao}.
@ -195,8 +192,8 @@ More generally we can show:
on the fibers of $Y$ over $Z_2$ on the fibers of $Y$ over $Z_2$
and invariant under $T$. and invariant under $T$.
$\sigma$ is a metric, since if $\sigma$ is a metric,
if $\pi_2(y) = \pi_2(y')$ and $\sigma(y,y') = 0$, since if $\pi_2(y) = \pi_2(y')$ and $\sigma(y,y') = 0$,
then $\pi_1(y) = \pi_1(y')$ or $y = y'$. then $\pi_1(y) = \pi_1(y')$ or $y = y'$.
\end{proof} \end{proof}
@ -223,6 +220,7 @@ More generally we can show:
For this, we show that for all $\xi < \eta$, For this, we show that for all $\xi < \eta$,
$(X_\xi', T)$ is a factor of $(X_\xi ,T)$ $(X_\xi', T)$ is a factor of $(X_\xi ,T)$
using transfinite induction. using transfinite induction.
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\[\begin{tikzcd} \[\begin{tikzcd}
{X'_{\eta'}} & \dots & {X'_3} & {X'_2} & {X_1'} \\ {X'_{\eta'}} & \dots & {X'_3} & {X'_2} & {X_1'} \\
@ -244,7 +242,8 @@ More generally we can show:
\arrow["{\pi'_2}", curve={height=-18pt}, from=1-1, to=1-4] \arrow["{\pi'_2}", curve={height=-18pt}, from=1-1, to=1-4]
\arrow["{\pi'_1}", curve={height=-30pt}, from=1-1, to=1-5] \arrow["{\pi'_1}", curve={height=-30pt}, from=1-1, to=1-5]
\end{tikzcd}\] \end{tikzcd}\]
% TODO: induction start?
We'll only show the successor step:
Suppose we have Suppose we have
$(X'_\xi, T) = \theta((X_\xi, T)$. $(X'_\xi, T) = \theta((X_\xi, T)$.
@ -253,18 +252,21 @@ More generally we can show:
\[Y \coloneqq \{(\pi_\xi(x), \pi'_{\xi+1}(x)) \in X_\xi \times X'_{\xi+ 1}: x \in X\}.\] \[Y \coloneqq \{(\pi_\xi(x), \pi'_{\xi+1}(x)) \in X_\xi \times X'_{\xi+ 1}: x \in X\}.\]
Then Then
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\[\begin{tikzcd} \begin{tikzcd}
{(X_{\xi+1},T)} && {(Y,T)} \\ {(X_{\xi+1},T)} && {(Y,T)} \\
& {(X_\xi,T)} && {(X'_{\xi+1},T)} \\ & {(X_\xi,T)} && {(X'_{\xi+1},T)} \\
&& {(X'_\xi,T)} && {(X'_\xi,T)}
\arrow["{\text{max.~iso}}"{description}, from=1-1, to=2-2] \arrow["{\text{max.~iso}}"{description}, from=1-1, to=2-2]
\arrow["\theta"{description}, from=2-2, to=3-3] \arrow["\theta"{description}, from=2-2, to=3-3]
\arrow["{{\color{orange}\text{iso}}}"{description}, from=2-4, to=3-3] \arrow[""{name=0, anchor=center, inner sep=0}, "{{\color{orange}\text{iso}}}"{description}, from=2-4, to=3-3]
\arrow["{{\color{orange}\text{iso}}}"{description}, from=1-3, to=2-2] \arrow[""{name=1, anchor=center, inner sep=0}, "{{\color{orange}\text{iso}}}"{description}, from=1-3, to=2-2]
\arrow[from=1-3, to=2-4] \arrow[from=1-3, to=2-4]
\arrow[curve={height=-6pt}, dashed, from=1-1, to=1-3] \arrow[curve={height=-6pt}, dashed, from=1-1, to=1-3]
\end{tikzcd}\] \arrow["{\pi'}", draw=none, from=1-3, to=2-4]
\arrow["{\theta'}", draw=none, from=0, to=3-3]
\arrow["\pi"', draw=none, from=1-3, to=1]
\end{tikzcd}
The diagram commutes, since all maps are the induced maps. The diagram commutes, since all maps are the induced maps.
By definition of $Y$ is clear that $\pi$ and $\pi'$ separate points in $Y$. By definition of $Y$ is clear that $\pi$ and $\pi'$ separate points in $Y$.
@ -277,6 +279,9 @@ More generally we can show:
In particular, In particular,
$(X'_{\xi+1}, T)$ is a factor of $(X_{\xi+1}, T)$. $(X'_{\xi+1}, T)$ is a factor of $(X_{\xi+1}, T)$.
\end{proof} \end{proof}
% TODO ANKI-MARKER
\begin{example}[{\cite[p. 513]{Furstenberg}}] \begin{example}[{\cite[p. 513]{Furstenberg}}]
\label{ex:19:inftorus} \label{ex:19:inftorus}
Let $X$ be the infinite torus Let $X$ be the infinite torus

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@ -86,7 +86,7 @@ Let $\mathbb{K} = S^1$
and $I$ a countable linear order. and $I$ a countable linear order.
Let $\mathbb{K}^I$ be the product of $|I|$ many $\mathbb{K}$, Let $\mathbb{K}^I$ be the product of $|I|$ many $\mathbb{K}$,
$\mathbb{K}^{<i} \coloneqq \mathbb{K}^{\{j : j < i\}}$ $\mathbb{K}^{<i} \coloneqq \mathbb{K}^{\{j : j < i\}}$
and $\pi_{i}\colon \mathbb{K}^{I} \to \mathbb{K}^{<i}$\todo{maybe call it $\pi_{<i}$?} and $\pi_{i}\colon \mathbb{K}^{I} \to \mathbb{K}^{<i}$% \todo{maybe call it $\pi_{<i}$?}
the projection. the projection.
Let $\mathbb{K}_I \coloneqq \prod_{i \in I} C(\mathbb{K}^{<i}, \mathbb{K})$. Let $\mathbb{K}_I \coloneqq \prod_{i \in I} C(\mathbb{K}^{<i}, \mathbb{K})$.

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@ -17,17 +17,19 @@
we have $X \in \cU \lor \N \setminus X \in \cU$. we have $X \in \cU \lor \N \setminus X \in \cU$.
\end{enumerate} \end{enumerate}
\end{definition} \end{definition}
\gist{
\begin{remark} \begin{remark}
\begin{itemize} \begin{itemize}
\item If $X \cup Y \in \cU$ then $X \in \cU \lor Y$ or $Y \in \cU$: \item If $X \cup Y \in \cU$ then $X \in \cU$ or $Y \in \cU$:
Consider $((\N \setminus X) \cap (\N \setminus Y) = \N \setminus (X \cup Y)$. Consider $((\N \setminus X) \cap (\N \setminus Y) = \N \setminus (X \cup Y)$.
\item Every filter can be extended to an ultrafilter. \item Every filter can be extended to an ultrafilter.
(Zorn's lemma) (Zorn's lemma)
\end{itemize} \end{itemize}
\end{remark} \end{remark}
}{}
\begin{definition} \begin{definition}
An ultrafilter is called \vocab[Ultrafilter!principal]{principal} or \vocab[Ultrafilter!trivial]{trivial} An ultrafilter is called \vocab[Ultrafilter!principal]{principal} or \vocab[Ultrafilter!trivial]{trivial}
if it is of the form iff it is of the form
\[ \[
\hat{n} = \{X \subseteq \N : n \in X\}. \hat{n} = \{X \subseteq \N : n \in X\}.
\] \]
@ -46,9 +48,9 @@
Let $\phi(\cdot )$, $\psi(\cdot )$ be formulas. Let $\phi(\cdot )$, $\psi(\cdot )$ be formulas.
\begin{enumerate}[(1)] \begin{enumerate}[(1)]
\item $(\cU n) (\phi(n) \land \psi(m)) \iff (\cU n) \phi(n) \land (\cU n) \psi(n)$. \item $(\cU n) ~ (\phi(n) \land \psi(m)) \iff (\cU n) \phi(n) \land (\cU n) \psi(n)$.
\item $(\cU n) (\phi(n) \lor \psi(m)) \iff (\cU n) \phi(n) \lor (\cU n) \psi(n)$. \item $(\cU n) ~ (\phi(n) \lor \psi(m)) \iff (\cU n) ~ \phi(n) \lor (\cU n) ~ \psi(n)$.
\item $(\cU n) \lnot \phi(n) \iff \lnot (\cU n) \phi(n)$. \item $(\cU n) ~\lnot \phi(n) \iff \lnot (\cU n)~ \phi(n)$.
\end{enumerate} \end{enumerate}
\end{observe} \end{observe}
\begin{lemma} \begin{lemma}
@ -59,7 +61,7 @@
there is a unique $x \in X$, there is a unique $x \in X$,
such that such that
\[ \[
(\cU n) (x_n \in G) (\cU n)~(x_n \in G)
\] \]
for every neighbourhood% for every neighbourhood%
\footnote{$G \subseteq X$ is a neighbourhood iff $x \in \inter G$.} \footnote{$G \subseteq X$ is a neighbourhood iff $x \in \inter G$.}

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@ -17,18 +17,12 @@ Then $t_n y \to (0) = t_n x$.
% TODO this is redundant
\begin{refproof}{fact:isometriciffequicontinuous}. \begin{refproof}{fact:isometriciffequicontinuous}.
$d$ and $d'(x,y) \coloneqq \sup_{t \in T} d(tx,ty)$ $d$ and $d'(x,y) \coloneqq \sup_{t \in T} d(tx,ty)$
induce the same topology. induce the same topology.
Let $\tau, \tau'$ be the corresponding topologies. Let $\tau, \tau'$ be the corresponding topologies.
$\tau \subseteq \tau'$ easy, $\tau \subseteq \tau'$ easy,
$\tau' \subseteq \tau'$ : use equicontinuity. $\tau' \subseteq \tau'$ : use equicontinuity.
\end{refproof} \end{refproof}