lecture 27

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
 
-
+In order to prove \yaref{thm:unifrprox},
-\begin{refproof}{thm:unifrprox}[sketch]
+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}

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)

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}