some changes

inputs/lecture_15.tex

@@ -233,6 +233,7 @@ Recall:
     correspond to metrics witnessing that the flow is isometric.
 \end{remark}
 \begin{proposition}
+    \label{prop:isomextdistal}
     An isometric extension of a distal flow is distal.
 \end{proposition}
 \begin{proof}
@@ -263,11 +264,12 @@ Recall:
     % TODO THE inverse limit is A limit
     of $\Sigma$ iff
     \[
-    \forall x_1,x_2 \in X.~\exists i \in I.~\pi_i(x_1) \neq \pi_i(x_2).
+    \forall x_1 \neq x_2 \in X.~\exists i \in I.~\pi_i(x_1) \neq \pi_i(x_2).
     \]
 \end{definition}
 
 \begin{proposition}
+    \label{prop:limitdistal}
     A limit of distal flows is distal.
 \end{proposition}
 \begin{proof}

inputs/lecture_16.tex

@@ -1,5 +1,4 @@
 \lecture{16}{2023-12-08}{}
-% TODO ANKI-MARKER
 
 $X$ is always compact metrizable.
 
@@ -18,16 +17,19 @@ $X$ 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!
     The action of $1$ determines $h$.
     Consider
     \[
-        \{h^n : n \in \Z\}  \subseteq  \cC(X,X) = \{f\colon  X \to X : f \text{ continuous}\},
+        \{h^n : n \in \Z\}  \subseteq  \cC(X,X)\gist{ = \{f\colon  X \to X : f \text{ continuous}\}}{},
     \]
     where the topology is the uniform convergence topology. % TODO REF EXERCISE
     Let $G = \overline{ \{h^n : n \in \Z\} } \subseteq \cC(X,X)$.
-    Since
+    Since the family $\{h^n : n \in \Z\}$ is uniformly equicontinuous,
+    i.e.~
     \[
-    \forall \epsilon > 0.~\exists \delta > 0.~d(x,y) < \delta \implies \forall n.~d(h^n(x), h^n(y)) < \epsilon
+    \forall \epsilon > 0.~\exists \delta > 0.~d(x,y) < \delta \implies \forall n.~d(h^n(x), h^n(y)) < \epsilon,
+    % Here we use isometric
     \]
     we have by the Arzel\`a-Ascoli-Theorem % TODO REF
     that $G$ is compact.
@@ -126,8 +128,8 @@ $X$ is always compact metrizable.
     Every quasi-isometric flow is distal.
 \end{corollary}
 \begin{proof}
-    \todo{TODO}
+    The trivial flow is distal.
-    % The trivial flow is distal.
+    Apply \yaref{prop:isomextdistal} and \yaref{prop:limitdistal}.
 \end{proof}
 
 \begin{theorem}[Furstenberg]
@@ -199,7 +201,8 @@ The Hilbert cube $\bH = [0,1]^{\N}$
 embeds all compact metric spaces.
 Thus we can consider $K(\bH)$,
 the space of compact subsets of $\bH$.
-$K(\bH)$ is a Polish space.\todo{Exercise}
+$K(\bH)$ is a Polish space.\footnote{cf.~\yaref{s9e2}, \yaref{s12e4}}
+% TODO LEARN EXERCISES
 Consider $K(\bH^2)$.
 A flow $\Z \acts X$ corresponds to the graph of
 \begin{IEEEeqnarray*}{rCl}

inputs/lecture_17.tex

@@ -1,4 +1,5 @@
 \subsection{The Ellis semigroup}
+% TODO ANKI-MARKER
 \lecture{17}{2023-12-12}{The Ellis semigroup}
 
 Let $(X, d)$ be a compact metric space

inputs/lecture_18.tex

@@ -68,7 +68,7 @@ This will follow from the following lemma:
     we get $U_{\epsilon}(x') \subseteq U_a(x_1) \cap U_b(x_2)$.
 \end{refproof}
 \begin{refproof}{lem:ftophelper}%
-    \footnote{This was not covered in class.}
+    \notexaminable{\footnote{This was not covered in class.}
 
         Let $T = \bigcup_n  T_n$,% TODO Why does this exist?
         $T_n$ compact, wlog.~$T_n \subseteq T_{n+1}$, and
@@ -123,6 +123,7 @@ This will follow from the following lemma:
         i.e.~$d(t_1t_0x, t_1t_0x') < b$
         and therefore
         $F(x,x'') = d(t_1t_0x, t_1t_0x'') < a$.
+    }
 \end{refproof}
 
 Now assume $Z = \{\star\}$.

inputs/lecture_22.tex

@@ -145,7 +145,9 @@ For this we define
             This is the same as for iterated skew shifts.
             %  TODO since for $\overline{x}, \overline{y} \in \mathbb{K}^I$,
             % $d(x_\alpha, y_\alpha) = d((f(\overline{x})_\alpha, (f(\overline{y})_\alpha))$.
-        \item Minimality:
+        \item Minimality:%
+            \gist{%
+                \footnote{This is not relevant for the exam.}
 
                 Let $\langle E_n : n < \omega \rangle$
                 be an enumeration of a countable basis for $\mathbb{K}^I$.
@@ -163,8 +165,12 @@ For this we define
                 is dense in $\overline{x} \mapsto f(\overline{x})$.
                 Since the flow is distal, it suffices to show
                 that one orbit is dense (cf.~\yaref{thm:distalflowpartition}).
+            }{ Not relevant for the exam.}
 
-        \item Let $\overline{f} = (f_i)_{i \in I} \in \mathbb{K}_I$.
+        \item The order of the flow is $\eta$:%
+            \gist{%
+                \footnote{This is not relevant for the exam.}
+                Let $\overline{f} = (f_i)_{i \in I} \in \mathbb{K}_I$.
                 Consider the flows we get from $(f_i)_{i < j}$
                 resp.~$(f_i)_{i \le j}$
                 denoted by $X_{<j}$ resp.~$X_{\le j}$.
@@ -186,7 +192,7 @@ For this we define
                     &&\}
                 \end{IEEEeqnarray*}
                 Beleznay and Foreman show that this is open and dense.%
-            \footnote{This is not relevant for the exam.}
                 % TODO similarities to the lemma used today
+            }{ Not relevant for the exam.}
     \end{itemize}
 \end{proof}

inputs/lecture_23.tex

@@ -24,6 +24,7 @@ Let $I$ be a linear order
 \end{theorem}
 
 \begin{proof}[sketch]
+    \notexaminable{%
         Consider $\WO(\N) \subset \LO(\N)$.
         We know that this is $\Pi_1^1$-complete. % TODO ref
 
@@ -65,6 +66,7 @@ Let $I$ be a linear order
         $T^{\alpha}_n \subseteq  W^{\alpha}_n$,
         where $W^{\alpha}_n$ is an enumeration of $U_m^\alpha$,$V^\alpha_{j,m,n,\frac{p}{q}}$.
         Then $(f_i)_{i \in I} \in  \bigcap_{n} T_{n}^\alpha$.
+    }
 \end{proof}
 \begin{lemma}
     Let $\{(X_\xi, T) : \xi  \le \eta\}$ be a normal

logic.sty

@@ -156,5 +156,6 @@
 \newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}}
 \newcommand\tutorial[3]{\hrule{\color{darkgray}\hfill{\tiny[Tutorial #1, #2]}}}
 \newcommand\nr[1]{\subsubsection{Exercise #1}\yalabel{Sheet \arabic{subsection}, Exercise #1}{E \arabic{subsection}.#1}{s\arabic{subsection}e#1}}
+\newcommand\notexaminable[1]{\gist{\footnote{Not relevant for the exam.}#1}{Not relevant for the exam.}}
 
 \usepackage[bibfile=bibliography/references.bib, imagefile=bibliography/images.bib]{mkessler-bibliography}