Josia Pietsch
7746866373
Some checks failed
Build latex and deploy / checkout (push) Failing after 17m20s
246 lines
7.7 KiB
TeX
246 lines
7.7 KiB
TeX
\lecture{27}{2024-02-02}{}
|
|
|
|
|
|
\begin{refproof}{thm:unifrprox:helper}
|
|
\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}
|
|
|
|
\end{refproof}
|
|
Take $X = \beta\N$,
|
|
$S \colon \beta\N \to \beta\N$,
|
|
$S(\cU ) = \hat{1}+ \cU$.
|
|
Then
|
|
\[
|
|
S^\cV(\cU) = \ulim{\cV}_n S^n(\cU) = \ulim{\cV}_n(\hat{n} + \cU) =
|
|
\ulim{\cV}_n \hat{n} + \cU = \cV + \cU.
|
|
\]
|
|
% TODO check
|
|
|
|
\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):
|
|
\todo{TODO}
|
|
% TODO
|
|
% Let $\cU$ be an idempotent.
|
|
% We want to find $\cV$ such that $\cV + \cU = \cU$.
|
|
% $\cV'$ such that $\cV' + \cU = \cV' + 0$ proximal to $0$?
|
|
% TAKE $\cV = \cV' = \cU$.
|
|
\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)
|