From d7809a884acefdd2d529b01cb5494a9d126857b5 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <git@jrpie.de>
Date: Wed, 7 Feb 2024 13:18:14 +0100
Subject: [PATCH] some changes

---
 inputs/lecture_15.tex |  4 ++++
 inputs/lecture_16.tex | 11 ++++++++++-
 2 files changed, 14 insertions(+), 1 deletion(-)

diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex
index fa9c486..77a74ef 100644
--- a/inputs/lecture_15.tex
+++ b/inputs/lecture_15.tex
@@ -273,6 +273,7 @@ Recall:
     A limit of distal flows is distal.
 \end{proposition}
 \begin{proof}
+\gist{%
     Let $(X,T)$ be a limit of $\Sigma = \{(X_i, T) : i \in I\}$.
     Suppose that each $(X_i, T)$ is distal.
     If $(X,T)$ was not distal,
@@ -283,4 +284,7 @@ Recall:
     But then $g_n \pi_i(x_1) \to \pi_i(z)$
     and $g_n \pi_i(x_2) \to \pi_i(z)$,
     which is a contradiction since $(X_i, T)$ is distal.
+}{Suppose there is a proximal pair $x_1,x_2$.
+    Take $i$ such that $\pi_i(x_1) \neq \pi_i(x_2) \lightning$.
+}
 \end{proof}
diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex
index 075f78c..352277d 100644
--- a/inputs/lecture_16.tex
+++ b/inputs/lecture_16.tex
@@ -17,7 +17,7 @@ $X$ is always compact metrizable.
 %     and $h\colon x \mapsto x + \alpha$.\footnote{Again $x + \alpha$ denotes $x \cdot \alpha$ in $\C$.}
 % \end{example}
 \begin{proof}
-    % TODO TODO TODO Think!
+\gist{%
     The action of $1$ determines $h$.
     Consider
     \[
@@ -82,6 +82,15 @@ $X$ is always compact metrizable.
     For $\alpha = h$ we get that
     a flow $\Z \acts X$ corresponds to $\Z \acts K$
     with $(1,x) \mapsto x + \alpha$.
+}{
+    \begin{itemize}
+        \item $G \coloneqq  \overline{\{h^n : n \in \Z\} } \subseteq \cC(X,X)$.
+        \item $G$ is compact (Arzela-Ascoli), abelian topological group (closure of ab. top. group)
+        \item Take any $x \in G$.
+        \item $Gx$ is compact (since $g \mapsto gx$ is continuous and $G$ is compact)
+        \item Stabilizer $G_x$ is closed. $K \coloneqq  \faktor{G}{G_x}$, $K \to X, fG_x \mapsto f(x)$.
+    \end{itemize}
+}
 \end{proof}
 \begin{definition}
     Let $(X,T)$ be a flow