diff --git a/inputs/lecture_26.tex b/inputs/lecture_26.tex index 0b1b5a8..f4342ac 100644 --- a/inputs/lecture_26.tex +++ b/inputs/lecture_26.tex @@ -89,7 +89,6 @@ We do a second proof of \yaref{thm:hindman}: (y(0), y(1), \ldots, y(n)) \text{ as a subsequence.} \] - \end{itemize} Consider $y(0)$. @@ -139,10 +138,14 @@ We do a second proof of \yaref{thm:hindman}: \end{proof} % TODO ultrafilter extension continuous - -\begin{refproof}{thm:unifrprox}[sketch] +In order to prove \yaref{thm:unifrprox}, +we need the following: +\begin{theorem} + \label{thm:unifrprox:helper} + Let $X$ be a compact Hausdorff space. % ? Let $T\colon X \to X$ be continuous. Let us rephrase the problem in terms of $\beta\N$: + \todo{remove duplicate} \begin{enumerate}[(1)] \item $x \in X$ is recurrent iff $T^\cU(x) = x$ for some $\cU \in \beta\N \setminus \N$. @@ -155,10 +158,15 @@ We do a second proof of \yaref{thm:hindman}: such that $T^\cU(x) = T^\cU(y)$. % TODO compare with the statement for the ellis semigroup. \end{enumerate} - We only to (2) here, as it is the most interesting point.% +\end{theorem} + + + +\begin{refproof}{thm:unifrprox:helper}[sketch] + We only prove (2) here, as it is the most interesting point.% \todo{other parts will be in the official notes} - \begin{subproof}[(2)] + \begin{subproof}[(2), $\implies$] Suppose that $ x$ is uniformly recurrent. Take some $\cV \in \beta\N$. Let $G_0$ be a neighbourhood of $x$. @@ -200,18 +208,7 @@ We do a second proof of \yaref{thm:hindman}: Since we get this for every neighbourhood, it follows that $T^\cU ( T^\cV(x)) = x$. - - - - - - - - \end{subproof} - - - - + \phantom\qedhere \end{refproof} diff --git a/inputs/lecture_27.tex b/inputs/lecture_27.tex new file mode 100644 index 0000000..b05f90d --- /dev/null +++ b/inputs/lecture_27.tex @@ -0,0 +1,248 @@ +\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 + \[ + \underbrace{\cV-\lim_n T^n(x)}_{T^\cV(x)} \not\in T^{-k}(G). + \] + 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 +\[ +S^\cV(\cU) = \cV-\lim_n S^n(\cU) = \cV-\lim_n(\hat{n} + \cU) = +\cV-\lim_n \hat{n} + \cU = \cV + \cU. +\] +% 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} + \yalabel{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) diff --git a/logic3.tex b/logic3.tex index 173b5f4..ce30abc 100644 --- a/logic3.tex +++ b/logic3.tex @@ -53,6 +53,7 @@ \input{inputs/lecture_24} \input{inputs/lecture_25} \input{inputs/lecture_26} +\input{inputs/lecture_27} \cleardoublepage @@ -80,10 +81,8 @@ \input{inputs/facts} - \PrintVocabIndex \printbibliography - \end{document}