w23-logic-3/inputs/lecture_27.tex
Josia Pietsch de89e2dc1d
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\lecture{27}{2024-02-02}{}
\begin{refproof}{thm:unifrprox:helper}
\gist{%
\begin{subproof}[(2), $\impliedby$, sketch]
Assume that $x $ is not uniformly recurrent.
Then there is a neighbourhood $G \ni x$
such that for all $M \in \N$
\[
Y_M = \{ n \in \N : \forall k < M.~T^{n+k}(x) \not\in G\} \neq \emptyset.
\]
Note that $Y_1 \supseteq Y_2 \supseteq Y_3 \supseteq \ldots$
Take $\cV \in \beta\N$ containing all $Y_n$.
We aim to show that there is no $\cU\in \beta\N$ such that $T_\cU(T_\cV(x)) = x$.
Towards a contradiction suppose that such $\cU$ exists.
For every $k + 1$ we have $Y_{k+1} \in \cV$.
In particular
\[
\{n \in \N : T^{n+k}(x) \not\in G\} \supseteq Y_{k+1},
\]
so
\[
(\cV n) T^{n+k}(x) \not\in G,
\]
i.e.~
\[
(\cV n) T^n(x) \not\in \underbrace{T^{-k}(G)}_{\text{open}}.
\]
Thus
\[
\underbrace{\ulim{\cV}_n T^n(x)}_{T^\cV(x)} \not\in T^{-k}(G).
\]
We get that
\[
\forall k.~T^k(T^\cV(x)) \not\in G.
\]
It follows that
$\forall \cU \in \beta\N.~T^{\cU}(T^\cV(x)) \not\in G$.
% TODO Why? Think about this.
\end{subproof}
}{%
\begin{itemize}
\item 2, $\implies$:
\begin{itemize}
\item TODO
\end{itemize}
\item 2, $\impliedby$:
\begin{itemize}
\item TODO
\end{itemize}
\end{itemize}
}
\end{refproof}
Take $X = \beta\N$,
$S \colon \beta\N \to \beta\N$,
$S(\cU ) = \hat{1}+ \cU$.
Then
\[
S^\cV(\cU) = \ulim{\cV}_n S^n(\cU) = \ulim{\cV}_n(\hat{n} + \cU) =
\ulim{\cV}_n \hat{n} + \cU = \cV + \cU.
\]
\begin{corollary}
$\cU$ is recurrent
iff
\[
\exists \cV \in \beta\N \setminus \N .~S^\cV(\cU) = \cU.
\]
$\cU$ is uniformly recurrent iff
\[
\forall \cV.~\exists \cW.~\cW + \cV + \cU = \cU.
\]
$\cU_1$ and $\cU_2$ are proximal
iff $\exists \cV.~\cV + \cU_1 = \cV + \cU_2$.
\end{corollary}
\begin{definition}
We say that $I \subseteq \beta\N$
is a \vocab{left ideal} ,
if
\[
\forall \cU \in I.~\forall \cV \in \beta\N.~\cV + \cU \in I.
\]
\end{definition}
\begin{theorem}
\label{thm:unifrprox:helper2}
\begin{enumerate}[(1)]
\item $\cU$ is uniformly recurrent in $\beta\N$
iff $\cU$ belongs to a minimal\footnote{wrt.~$\subseteq $} (closed)
left ideal in $\beta\N$.
\item $\cU$ is an idempotent in $\beta\N$
iff $\cU$ belong to a minimal closed
subsemigroup of $\beta\N$.
\end{enumerate}
\end{theorem}
\begin{proof}
\begin{enumerate}[(1)]
\item
\gist{
Note that any $\cU \in \beta\N$ yields
%gives rise to
a left ideal $\beta\N + \cU$.
It is closed, since it is the image
of $\beta\N$ under the continuous maps
$\cV \mapsto \cV + \cU$
and $\beta\N$ is compact.
}{%
Note that $\beta\N + \cU$ is closed,
since $\beta\N$ is compact and $\cdot + \cU$ continuous.
}
$\cU$ belongs to a minimal left ideal
iff $\beta\N + \cU$ is minimal%
\gist{,
since every ideal containing $\cU$
contains $\beta\N + \cU$.
}{.}
\gist{%
Note that $\beta\N + \cV + \cU \subseteq \beta\N + \cU$
and if $I \subsetneq \beta\N + \cU$,
we have $\cV_0 = \cV + \cU \in I$
and $\beta\N + \cV + \cU \subseteq \beta\N + \cU$.
So $\cU$ belongs to a minimal left ideal iff
}{Equivalently}
\[
\forall \cV \in \beta\N .~\beta\N + \cV + \cU = \beta\N + \cU.
\]
This is the case iff
\[
\underbrace{\forall \cV .~\exists \cW.~ \cW + \cV + \cU = \cU.}_%
{\cV \text{ uniformly recurrent}}
\]
\gist{(For one direction take $\cW$ such that $\cW + \cV + \cU= \hat{0} +\cU$.
For the other direction note that
for every $\cV_0 $, $\cV_0 + \cU$
can be written as $\cV_0 + \cW + (\cV + \cU)$.
Where we take $\cW$ such that $\cW + \cV + \cU = \cU$.
}{}
\item This is very similar to the proof of the \yaref{lem:ellisnumakura}.
If $\cU$ is idempotent, then $\{\cU\}$
is a semigroup.
Let $C$ be a minimal closed subsemigroup of $\beta\N$.
Then $C + \cU$ is a closed subsemigroup.
By minimality, we get $C = C + \cU$.
Let $D = \{ \cV \in C .~ \cV + \cU = \cU\}$.
We have $D \neq \emptyset$.
$D$ is a closed semigroup,
so $D = C$ be minimality.
Hence $\cU + \cU = \cU$.
\end{enumerate}
\end{proof}
\begin{corollary}
Idempotent and uniformly recurrent elements exist.
\end{corollary}
\begin{proof}
Use \yaref{thm:unifrprox:helper2}
and Zorn's lemma.
\end{proof}
\begin{theorem}
(1) $\implies$ (2) $\implies$ (3)
where
\begin{enumerate}[(1)]
\item $\cU$ is uniformly recurrent and proximal to $\hat{0}$.
\item $\cU$ is an idempotent.
\item $\cU$ is recurrent and proximal to $\hat{0}$.
\end{enumerate}
\end{theorem}
\begin{proof}
(1) $\implies$ (2):
Let $\cU$ be uniformly recurrent and proximal to $ \hat{0}$.
Take $\cV$ such that $\cV + \cU = \cV + \hat{0} = \cV$.
% TODO REF beginning of lecture
Since $\cU$ is uniformly recurrent,
there exists $\cW$ such that $\cW + \cV + \cU = \cU$,
i.e.~$\cW + \cV = \cU$.
Then $\cU + \cU = \cW + \cV + \cU = \cU$.
(2) $\implies$ (3):
Let $\cU$ be an idempotent.
Then $\cV + \cU = \cV$ (proximal to $0$)
and $\cV + \cU = \cU$ (recurrent)
are satisfied for $\cV \coloneqq \cU$.
\end{proof}
\begin{corollary}
$\cU$ is uniformly recurrent and proximal to $0$
iff $\cU$ is an idempotent
and belongs to some minimal left ideal of $\beta\N$.
\end{corollary}
Finally:
\begin{refproof}{thm:unifrprox}
Let $T\colon X \to X$ and $x \in X$.
We want to find $y \in X$
such that $y$ is uniformly recurrent
and proximal to $x$.
We first prove a version for ultrafilters and then
transfer it to $X$.
There exists a uniformly recurrent $\cV \in \beta\N$.
So for any $\cW$,
$\cW + \cV$ is also uniformly recurrent\gist{:
Take $\cV_0$.
We need to find $\cX$ such that $\cX + \cV_0 + \cW +\cV = \cW + \cV$.
By uniform recurrence of $\cV$ we find $\cX'$
such that $\cX' + (\cV_0 + \cW) + \cV = \cV$.
Then $\cX = \cW + \cX'$ works.
}{.}
So all elements of $\beta\N + \cV$
are uniformly recurrent.
It is a closed ideal and hence a closed semigroup.
So $\beta\N + \cV$ contains a minimal closed
semigroup.
In particular, there exists an idempotent $\cU \in \beta\N + \cV$.
$\cU$ is idempotent and uniformly recurrent
hence it is proximal to $0$.
Now let us consider $X$.
Take $y = T^\cU(x)$.
\begin{claim}
$y$ uniformly recurrent.
\end{claim}
\begin{subproof}
Recall that $T^{\cV_1 + \cV_2} = T^{\cV_1} \circ T^{\cV_2}$.
Since $\cU$ is uniformly recurrent,
$\forall \cV .~\exists \cW.~\cW+ \cV+\cU= \cU$,
i.e.~$T^{\cW + \cV + \cU} (x) = T^\cW(T^\cV(y)) = T^\cU(x) = y$.
\end{subproof}
\begin{claim}
$y$ is proximal to $x$.
\end{claim}
\begin{subproof}
$\cU$ is proximal to $0$.
So $\exists \cV.~\cV + \cU = \cV + \hat{0} = \cV$,
i.e.~$T^{\cV}(y) = T^{\cV + \cU}(x) = T^\cV(x)$.
Thus $x$ and $y$ are proximal.%TODO REF
\end{subproof}
\end{refproof}
% Office hours wednesday 15:30 - 18:30 office 805
% Exam: First question: present favorite theorem (7-8 minutes, moderate length proof)