gist

inputs/lecture_20.tex

@@ -135,6 +135,7 @@ coordinates.
     $x \mapsto x^{n}$.
 \end{definition}
 
+\gist{%
 \begin{remark}
     \label{rem:l20:sigma}
     Note that for
@@ -149,6 +150,7 @@ coordinates.
         (y, x_{d+1}) &\longmapsto & (\tau_d(y), \sigma(y) x_{d+1}).
     \end{IEEEeqnarray*}
 \end{remark}
+}{}
 \begin{theorem}
     \label{thm:taudminimal:help}
     For every $d$ if $\tau_d$\footnote{more formally $((S^1)^d, \langle \tau_d \rangle)$}
@@ -166,11 +168,14 @@ coordinates.
     $\tau$ is minimal.
 \end{corollary}
 \begin{proof}
+    \gist{%
+    We need to show that every orbit is dense.
     This follows from the definition of the product topology,
     since for a basic open set $U = U_1 \times \ldots \times U_d \times (S^1)^{\infty}$
     it suffices to analyze the first $d$ coordinates.
+    }{We need to show that every orbit is dense. For this it suffices to analyze finitely many coordinates.}
 \end{proof}
-
+\gist{%
 \begin{refproof}{thm:taudminimal:help}
     Let $S \coloneqq  \tau_d$, $T \coloneqq \tau_{d+1}$ and $Y \coloneqq (S^1)^d$.
     Consider
@@ -193,8 +198,5 @@ coordinates.
 
     \phantom\qedhere
 \end{refproof}
-
+}{}
-
-
-
 

inputs/lecture_21.tex

@@ -1,6 +1,7 @@
 \lecture{21}{2024-01-12}{Iterated Skew Shift}
 
 \begin{refproof}{thm:taudminimal:help}
+\gist{%
     Suppose towards a contradiction that
     $Y \times S^1$ contains a proper minimal subflow $Z$.
     Consider the projection $\pi\colon Y \times S^1 \to Y$.
@@ -77,6 +78,41 @@
     so $[\phi \circ S \circ \gamma] = [\phi \circ \gamma]$.
     Thus $[x \mapsto (\sigma(\gamma(x))^n] = 0$,
     but that is a contradiction to (b) $\lightning$
+}{
+    \begin{itemize}
+        \item $S \coloneqq  \tau_d$, $T \coloneqq  \tau_{d+1}$
+            ($T(y, \xi) = (S(y), \sigma(y) + \xi)$),
+            $Y \coloneqq  (S^1)^d$,
+            $\gamma: S^1 \to Y, x \mapsto (x,x,\ldots,x)$.
+        \item (a)  $\gamma \simeq S \circ \gamma$.
+        \item (b) $\forall m \in \Z.~ [m \cdot \sigma \circ \gamma] = m$
+            ($\sigma(\gamma(x)) = x$).
+        \item Suppose $Y \times S^1$ is not minimal.
+            Let $Z$ be a proper minimal subflow,
+            $\pi\colon Y \times S^1 \to Y$.
+        \item $\pi(Z) = Y$ ($Y$ minimal)
+        \item $Z + (0,\ldots,0,\theta)$ is minimal, so $\theta + Z = Z$ 
+            or $\theta + Z \cap Z = \emptyset$.
+        \item $H \coloneqq  \{\theta \in S^1 : \theta + Z = Z\}$ (closed subgroup of $S^1$).
+        \begin{itemize}
+            \item $H \neq S^1$ as $Z$ is a proper subflow.
+            \item $H = \langle \frac{1}{m} \rangle$.
+        \end{itemize}
+        \item $(y, \beta) \in Z$, then $(y, \beta + t) \in Z \iff m \cdot t = 0$.
+            Pick $\beta^{(y)}$ such that $(y, \xi) \in Z \iff \xi \in \{\beta^{(y)} + \frac{n}{m} : n \in \N\}$.
+        \item Consider $(y, \xi) \mapsto (y, m \cdot \xi)$.
+            Get
+            \begin{IEEEeqnarray*}{rCl}
+                \phi\colon Y &\longrightarrow & S^1 \\
+                 y &\longmapsto & m \beta^{(y)}
+            \end{IEEEeqnarray*}
+            continuous.
+        \item $\phi \circ S = m \cdot  \phi \circ \sigma$:
+            $Z$ is $T$-invariant.
+        \item $[\phi \circ S \circ \gamma] = [\phi \circ \gamma] + [m \sigma \circ \gamma]$
+            $\implies [m \sigma \circ \gamma] = 0 \lightning$.
+    \end{itemize}
+}
 \end{refproof}
 
 Let $X_n \coloneqq  (S^1)^n$ and $X \coloneqq (S^1)^{\N}$.

inputs/lecture_22.tex

@@ -103,7 +103,7 @@ For this we define
     \end{pmatrix*}
 \end{IEEEeqnarray*}
 
-
+\gist{%
 \begin{example}
     Consider the following flow:
     \begin{IEEEeqnarray*}{rCl}
@@ -121,10 +121,12 @@ For this we define
     \end{IEEEeqnarray*}
     we can write this as $\tau(x,y,z) = (x \cdot  f_1 \circ \pi_1(x,y,z), y \cdot f_2\pi_2(x,y,z), z \cdot f_3\pi_3(x,y,z))$
 \end{example}
-% \begin{example}
+\begin{example}
-%     The skew shift can be written in this form as well:
+    The skew shift can be written in this form as well.
-%     TODO
+    Consider $f_1\colon x \mapsto \alpha$
-% \end{example}
+    and $f_n\colon (x_0,\ldots, x_{n-2}) \mapsto x_{n-2}$.
+\end{example}
+}{}
 
 \begin{theorem}[Beleznay Foreman]
     \label{thm:distalminimalofallranks}
@@ -143,9 +145,8 @@ For this we define
             For all $\overline{f} \in \mathbb{K}_I$,
             the flow $E_I \overline{f}$ is distal.
             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:%
             \notexaminable{%
                 Let $\langle E_n : n < \omega \rangle$
                 be an enumeration of a countable basis for $\mathbb{K}^I$.
@@ -156,7 +157,7 @@ For this we define
                 \]
                 where $f = E_I \overline{f}$ and $\overline{1} = (1,1,1,\ldots)$.
 
-                Beleznay and Foreman showed that $U_n$ is open
+                \textsc{Beleznay} and \textsc{Foreman} showed that $U_n$ is open
                 and dense for all $n$.
 
                 So if $\overline{f} \in  \bigcap_{n} U_n$, then $\overline{1}$
@@ -165,7 +166,7 @@ For this we define
                 that one orbit is dense (cf.~\yaref{thm:distalflowpartition}).
             }
 
-        \item The order of the flow is $\eta$:%
+        \item The order of the flow is $\eta$:
             \notexaminable{%
                 Let $\overline{f} = (f_i)_{i \in I} \in \mathbb{K}_I$.
                 Consider the flows we get from $(f_i)_{i < j}$