diff --git a/inputs/lecture_24.tex b/inputs/lecture_24.tex new file mode 100644 index 0000000..7a0c2a8 --- /dev/null +++ b/inputs/lecture_24.tex @@ -0,0 +1,178 @@ +\lecture{24}{2024-01-23}{Combinatorics!} + + +\subsection{Applications to Combinatorics} % Ramsey Theory} + +% TODO Define Ultrafilter + +\begin{definition} + An \vocab{ultrafilter} on $\N$ (or any other set) + is a family $\cU \subseteq \cP(\N)$ + such that + \begin{enumerate}[(1)] + \item $X \in \cU \land X \subseteq Y \subseteq \N \implies Y \in \cU$. + \item $X,Y \in \cU \implies X \cap Y \in \cU$. + \item $\emptyset \not\in \cU$, $\N \in \cU$. + \item For all $X \subseteq \N$, + we have $X \in \cU \lor \N \setminus X \in \cU$. + \end{enumerate} +\end{definition} +\begin{remark} + \begin{itemize} + \item If $X \cup Y \in \cU$ then $X \in \cU \lor Y$ or $Y \in \cU$: + Consider $((\N \setminus X) \cap (\N \setminus Y) = \N \setminus (X \cup Y)$. + \item Every filter can be extended to an ultrafilter. + (Zorn's lemma) + \end{itemize} +\end{remark} +\begin{definition} + An ultrafilter is called \vocab[Ultrafilter!principal]{principal} or \vocab[Ultrafilter!trivial]{trivial} + if it is of the form + \[ + \hat{n} = \{X \subseteq \N : n \in X\}. + \] +\end{definition} +\begin{notation} + Let $\phi(\cdot )$ be a formula, where the argument is a natural number. + Let $\cU$ be an ultrafilter. + We write + \[ + (\cU n) ~ \phi(n) + \] + for $\{ n \in \N : \phi(n)\} \in \cU$. + We say that $\phi(n)$ holds for \vocab{$\cU$-almost all} $n$. +\end{notation} +\begin{observe} + Let $\phi(\cdot )$, $\psi(\cdot )$ be formulas. + + \begin{enumerate}[(1)] + \item $(\cU n) (\phi(n) \land \psi(m)) \iff (\cU n) \phi(n) \land (\cU n) \psi(n)$. + \item $(\cU n) (\phi(n) \lor \psi(m)) \iff (\cU n) \phi(n) \lor (\cU n) \psi(n)$. + \item $(\cU n) \lnot \phi(n) \iff \lnot (\cU n) \phi(n)$. + \end{enumerate} +\end{observe} +\begin{lemma} + \label{lem:ultrafilterlimit} + Let $X $ be a compact Hausdorff space. + Let $\cU$ be an ultrafilter. + Then for every sequence $(x_n)$ in $X$, + there is a unique $x \in X$, + such that + \[ + (\cU_n) (x_n \in G) + \] + for every neighbourhood% + \footnote{$G \subseteq X$ is a neighbourhood iff $x \in \inter G$.} + $G$ of $x$. +\end{lemma} +\begin{notation} + In this case we write $x = \cU-\lim_n x_n$. +\end{notation} +\begin{refproof}{lem:ultrafilterlimit}[sketch] + Whenever we write $X = Y \cup Z$ + we have $(\cU n) x_n \in Y$ + or $(\cU n) x_n \in Z$. + + So we can repeatedly chop the space in two pieces, + one of them contains $\cU$-almost all $x_n$, + Then we restrict to this piece and continue. + + For this to work, we need + a finite collection $\cP_n$ of closed sets for every $n$, + such that $\bigcup \cP_n = X$, + $C \in \cP_{n+1} \implies \exists C \subseteq D \in \cP_{n}$ + and + $C_1 \supseteq C_2 \supseteq \ldots$, $C_i \in \cP_i $ $\implies | \bigcap_{i} C_i| = 1$. + It is clear that we can do this for metric spaces, + but such partition can be found for compact Hausdorff spaces as well. +\end{refproof} + +Let $\beta \N$ be the Čech-Stone compactification of $\N$, +i.e.~the set of all ultrafilters on $\N$ +with the topology given by open sets $V_{A} = \{ p \in \beta\N : A \in P\} $ +for $A \subseteq \N$. + +This is a compact Hausdorff space.\todo{Homework} +We can turn it into a compact semigroup: +Consider $+ \colon \N \times \N \to \N$. +This gives an operation on principal ultrafilters +(we identify $n \in \N$ with the corresponding principal filter). +We want to extend this to all of $\beta\N$. +Fix the first argument to get a function $\N \to \N, n \mapsto k+n$. +For $\cU \in \beta\N$ consider $\cU-\lim_n (k+n)$. +So for a fixed $k \in \N$ we get $k+ \cdot \colon\beta\N \to \beta\N$, +i.e.~$+ \colon \N \times \beta\N \to \beta\N$. +Fixing the second coordinate to be $\cV \in \beta\N$, +we get a function $+\cV \colon \N \to \beta\N$. +For $ \cU \in \beta\N$ +consider $\cU-\lim_n n + \cV$. +This gives $+ \colon \beta\N \times \beta\N \to \beta\N$. +% TODO ? + +\[ +\cU + \cV = \{X \subseteq \N : \{m \colon \{n \colon m+n \in X\} \in \cV \} \in \cU \}. +\] + + +This is not commutative, +but associative and $a \mapsto a + b$ is continuous +for a fixed $b$. +This is called a left compact topological semigroup. + + + + +Let $X$ be a compact Hausdorff space +and let $T \colon X \to X$ be continuous.% +\footnote{Note that this need not be a homeomorphism, i.e.~we only get a $\N$-action + but not a $\Z$-action.} + +For any $\cU \in \beta\N$, we define $T^{\cU}$ by +$T^\cU(x) \coloneqq \cU-\lim_n T^n(x)$ for $x \in X$. + +For fixed $x$, the map $\cU \mapsto T^{\cU}(x)$ is continuous. + +(More generally, for every $f\colon \N \to X$ +the extension $\tilde{f}\colon \beta\N \to X$ is continuous). + +Note that for fixed $\cU$, the map $x \mapsto T^\cU(x)$ +is not necessarily continuous. + + +\begin{definition} + Let $X$ be a compact Hausdorff space + and $T\colon X\to X$ continuous. + A point $x \in X$ is \vocab{recurrent}, + iff for every neighbourhood $G$ of $x$, + infinitely many $n$ satisfy $T^n(x) \in G$. + + A point $x \in X$ is \vocab{uniformly recurrent}, + if for every neighbourhood $G$ of $x$, + there exists $M \in \N$, + such that + \[ + \forall n.~\exists k < M.~ T^{n+k}(x) \in G. + \] + +\end{definition} +\begin{fact} + Let $\cU, \cV \in \beta\N$ + and $T\colon X \to X$ continuous + for a compact Hausdorff space $X$. + Then $T^{\cU}(T^{\cV}(x)) = T^{\cU + \cV}(x)$. +\end{fact} +\begin{proof} + \begin{IEEEeqnarray*}{rCl} + T^{\cU + \cV}(x) &=& (\cU + \cV)-\lim_k T^k(x)\\ + &=& \cU-\lim_m \cV-\lim_n T^{m+n}(x)\\ + &\overset{T^m \text{ continuous}}{=}& \cU-\lim_m T^m (\cV-\lim_n T^n(x))\\ + &=& T^\cU(T^\cV(x)). + \end{IEEEeqnarray*} +\end{proof} + + +\todo{Homework: Check the details that were omitted during the lecture.} + + + + diff --git a/logic3.tex b/logic3.tex index 6127c23..b8548c2 100644 --- a/logic3.tex +++ b/logic3.tex @@ -50,6 +50,7 @@ \input{inputs/lecture_21} \input{inputs/lecture_22} \input{inputs/lecture_23} +\input{inputs/lecture_24} \cleardoublepage