gist
Some checks failed
Build latex and deploy / checkout (push) Failing after 17m51s

This commit is contained in:
Josia Pietsch 2024-02-07 19:44:21 +01:00
parent 4c0c7730f1
commit dd99d09b1c
Signed by: josia
GPG key ID: E70B571D66986A2D
3 changed files with 54 additions and 15 deletions

View file

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

View file

@ -1,6 +1,7 @@
\lecture{21}{2024-01-12}{Iterated Skew Shift} \lecture{21}{2024-01-12}{Iterated Skew Shift}
\begin{refproof}{thm:taudminimal:help} \begin{refproof}{thm:taudminimal:help}
\gist{%
Suppose towards a contradiction that Suppose towards a contradiction that
$Y \times S^1$ contains a proper minimal subflow $Z$. $Y \times S^1$ contains a proper minimal subflow $Z$.
Consider the projection $\pi\colon Y \times S^1 \to Y$. Consider the projection $\pi\colon Y \times S^1 \to Y$.
@ -77,6 +78,41 @@
so $[\phi \circ S \circ \gamma] = [\phi \circ \gamma]$. so $[\phi \circ S \circ \gamma] = [\phi \circ \gamma]$.
Thus $[x \mapsto (\sigma(\gamma(x))^n] = 0$, Thus $[x \mapsto (\sigma(\gamma(x))^n] = 0$,
but that is a contradiction to (b) $\lightning$ 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} \end{refproof}
Let $X_n \coloneqq (S^1)^n$ and $X \coloneqq (S^1)^{\N}$. Let $X_n \coloneqq (S^1)^n$ and $X \coloneqq (S^1)^{\N}$.

View file

@ -103,7 +103,7 @@ For this we define
\end{pmatrix*} \end{pmatrix*}
\end{IEEEeqnarray*} \end{IEEEeqnarray*}
\gist{%
\begin{example} \begin{example}
Consider the following flow: Consider the following flow:
\begin{IEEEeqnarray*}{rCl} \begin{IEEEeqnarray*}{rCl}
@ -121,10 +121,12 @@ For this we define
\end{IEEEeqnarray*} \end{IEEEeqnarray*}
we can write this as $\tau(x,y,z) = (x \cdot f_1 \circ \pi_1(x,y,z), y \cdot f_2\pi_2(x,y,z), z \cdot f_3\pi_3(x,y,z))$ 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} \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] \begin{theorem}[Beleznay Foreman]
\label{thm:distalminimalofallranks} \label{thm:distalminimalofallranks}
@ -143,9 +145,8 @@ For this we define
For all $\overline{f} \in \mathbb{K}_I$, For all $\overline{f} \in \mathbb{K}_I$,
the flow $E_I \overline{f}$ is distal. the flow $E_I \overline{f}$ is distal.
This is the same as for iterated skew shifts. 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{% \notexaminable{%
Let $\langle E_n : n < \omega \rangle$ Let $\langle E_n : n < \omega \rangle$
be an enumeration of a countable basis for $\mathbb{K}^I$. 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 $\overline{1} = (1,1,1,\ldots)$. 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 $n$. and dense for all $n$.
So if $\overline{f} \in \bigcap_{n} U_n$, then $\overline{1}$ 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}). 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{% \notexaminable{%
Let $\overline{f} = (f_i)_{i \in I} \in \mathbb{K}_I$. Let $\overline{f} = (f_i)_{i \in I} \in \mathbb{K}_I$.
Consider the flows we get from $(f_i)_{i < j}$ Consider the flows we get from $(f_i)_{i < j}$