From 59d595f9efe3f0f8b9f0161c7fc3e0f55b7911ac Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Mon, 5 Feb 2024 23:36:57 +0100 Subject: [PATCH] some small changes --- inputs/lecture_14.tex | 2 +- inputs/lecture_17.tex | 3 +-- inputs/lecture_18.tex | 24 ++++++++++++++---------- inputs/lecture_19.tex | 8 +++++--- 4 files changed, 21 insertions(+), 16 deletions(-) diff --git a/inputs/lecture_14.tex b/inputs/lecture_14.tex index 7b5989a..e7d2ce2 100644 --- a/inputs/lecture_14.tex +++ b/inputs/lecture_14.tex @@ -104,7 +104,7 @@ \forall x \in X.~(\exists n.~(x,n) \in R \iff \exists! n.~(x,n)\in R^\ast). \] We say that $R^\ast$ \vocab[uniformization]{uniformizes} $R$.% - \footnote{Wikimedia has a \href{https://upload.wikimedia.org/wikipedia/commons/4/4c/Uniformization_ill.png}{nice picture.}} + \footnote{Wikimedia has a nice \href{https://upload.wikimedia.org/wikipedia/commons/4/4c/Uniformization_ill.png}{picture}.} \end{theorem} \begin{proof} Let $\phi\colon R \to \Ord$ diff --git a/inputs/lecture_17.tex b/inputs/lecture_17.tex index 1891dd0..c9b265d 100644 --- a/inputs/lecture_17.tex +++ b/inputs/lecture_17.tex @@ -191,8 +191,7 @@ since $X^X$ has these properties. Since $f$ is injective, we get that $x = f(x)$, i.e.~$f = \id$. - Take $g' \in G$ such that $f = g' \circ g$.% - %\footnote{This exists since $f \in Gg$.} + Since $f \in Gg$, there exists $g' \in G$ such that $f = g' \circ g$. It is $g' = g'gg'$, so $\forall x .~g'(x) = g'(g g'(x))$. diff --git a/inputs/lecture_18.tex b/inputs/lecture_18.tex index 4c9f35d..0341da4 100644 --- a/inputs/lecture_18.tex +++ b/inputs/lecture_18.tex @@ -1,9 +1,6 @@ \subsection{Sketch of proof of \yaref{thm:l16:3}} \lecture{18}{2023-12-15}{Sketch of proof of \yaref{thm:l16:3}} -% TODO ANKI-MARKER - - The goal for this lecture is to give a very rough sketch of \yaref{thm:l16:3} in the case of $|Z| = 1$. @@ -21,7 +18,9 @@ F(x,x') \coloneqq \inf \{d(gx, gx') : g \in G\}. \item $F(x,x') = F(x', x)$, \item $F(x,x') \ge 0$ and $F(x,x') = 0$ iff $x = x'$. \item $F(gx, gx') = F(x,x')$ since $G$ is a group. - \item $F$ is an upper semi-continuous function on $X^2$, + \item $F$ is an \vocab{upper semi-continuous}\footnote{% + Wikimedia has a nice \href{https://upload.wikimedia.org/wikipedia/commons/c/c0/Upper_semi.svg}{picture}.} + function on $X^2$, i.e.~$\forall a \in R.~\{(x,x') \in X^2 : F(x,x') < a\} \overset{\text{open}}{\subseteq} X^2$. This holds because $F$ is the infimum of continuous functions @@ -47,23 +46,28 @@ This will follow from the following lemma: \begin{lemma} \label{lem:ftophelper} Let $F(x,x') < a$. + \gist{% Then there exists $\epsilon > 0$ such that whenever $F(x',x'') < \epsilon$, then $F(x,x'') < a$. + }{Then $\exists \epsilon > 0.~\forall x''.~F(x',x'') < \epsilon \implies F(x,x'') < a$.} \end{lemma} \begin{refproof}{def:ftop} +\gist{% We have to show that if $U_a(x_1) \cap U_b(x_2) \neq \emptyset$, then this intersection is the union of sets of this kind. - Let $x' \in U_a(x_1)$. +}{} + Let $x' \in U_a(x_1) \cap U_b(x_2)$. Then by \yaref{lem:ftophelper}, there exists $\epsilon_1 > 0$ with $U_{\epsilon_1}(x') \subseteq U_a(x_1)$. - Similarly there exists $\epsilon_2 > 0$ - such that $U_{\epsilon_2}(x') \subseteq U_b(x_2)$. + Similarly there exists $\epsilon_2 > 0$\gist{ + such that $U_{\epsilon_2}(x') \subseteq U_b(x_2)$.}{.} So for $\epsilon \le \epsilon_1, \epsilon_2$, we get $U_{\epsilon}(x') \subseteq U_a(x_1) \cap U_b(x_2)$. \end{refproof} \begin{refproof}{lem:ftophelper}% \notexaminable{\footnote{This was not covered in class.} + % TODO: maybe learn? Let $T = \bigcup_n T_n$,% TODO Why does this exist? $T_n$ compact, wlog.~$T_n \subseteq T_{n+1}$, and @@ -164,8 +168,7 @@ i.e.~show that if $(Z,T)$ is a proper factor of a minimal distal flow \item One can show that $H$ is a topological group and $(M,H)$ is a flow.\footnote{This is non-trivial.} \item Since $H$ is compact, - $(M,H)$ is equicontinuous, %\todo{We didn't define this} - i.e.~it is isometric. + $(M,H)$ is equicontinuous, i.e.~it is isometric. In particular, $(M,T)$ is isometric. \end{enumerate} \item $M \neq \{\star\}$, i.e.~$(M,T)$ is non-trivial: @@ -209,9 +212,9 @@ i.e.~show that if $(Z,T)$ is a proper factor of a minimal distal flow Let $X$ be a metric space and $\Gamma\colon X \to \R$ be upper semicontinuous. Then the set of continuity points of $\Gamma$ is comeager. - \todo{Missing figure: upper semicontinuous function} \end{theorem} \begin{proof} +\notexaminable{ Take $x$ such that $\Gamma$ is not continuous at $x$. Then there is an $\epsilon > 0$ and $x_n \to x$ such that @@ -227,6 +230,7 @@ i.e.~show that if $(Z,T)$ is a proper factor of a minimal distal flow and $B_q \setminus B_q^\circ$ is nwd as it is closed and has empty interior, so $\bigcup_{q \in \Q} F_q$ is meager. +} \end{proof} diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index 159746e..2230fdb 100644 --- a/inputs/lecture_19.tex +++ b/inputs/lecture_19.tex @@ -1,5 +1,8 @@ -\subsection{The order of a flow} -\lecture{19}{2023-12-19}{Orders of flows} +\subsection{The Order of a Flow} +\lecture{19}{2023-12-19}{Orders of Flows} + +% TODO ANKI-MARKER + See also \cite[\href{https://terrytao.wordpress.com/2008/01/24/254a-lecture-6-isometric-systems-and-isometric-extensions/}{Lecture 6}]{tao}. @@ -71,7 +74,6 @@ equicontinuity coincide. i.e.~if $(Y_\alpha, f_{\alpha,\beta})$, $\beta < \alpha \le \Theta$ are isometric, then the inverse limit $Y$ is isometric.% - \todo{Why does an inverse limit exist?} % https://q.uiver.app/#q=WzAsNCxbMSwwLCJZX1xcYWxwaGEiXSxbMSwxLCJZX1xcYmV0YSJdLFswLDAsIlkiXSxbMiwwLCJYIl0sWzAsMSwiZl97XFxhbHBoYSwgXFxiZXRhfSJdLFsyLDAsImZfXFxhbHBoYSJdLFsyLDEsImZfXFxiZXRhIiwyXSxbMywwLCJcXHBpX1xcYWxwaGEiLDJdLFszLDEsIlxccGlfXFxiZXRhIl1d \[\begin{tikzcd} Y & {Y_\alpha} & X \\