Josia Pietsch
458dd9ab1f
Some checks failed
Build latex and deploy / checkout (push) Failing after 19m2s
31 lines
751 B
TeX
31 lines
751 B
TeX
\tutorial{12b}{2024-01-16T13:09:02}{}
|
|
|
|
\subsection{Sheet 10}
|
|
|
|
\todo{Copy from Abdelrahman and Shiguma}
|
|
|
|
\nr 2
|
|
|
|
\todo{Def skew shift flow (on $(\R / \Z)^2$!)}
|
|
The Bernoulli shift, $\Z \acts \{0,1\}^{\Z}$, is not distal.
|
|
Let $x = (0)$ and $y = (\delta_{0,i})_{i \in \Z}$.
|
|
Let $t_n \to \infty$.
|
|
Then $t_n y \to (0) = t_n x$.
|
|
|
|
% The skew shift flow is distal:
|
|
% This is tedious but probably not too hard.
|
|
%
|
|
% The skew shift flow is not equicontinuous:
|
|
|
|
|
|
|
|
% TODO this is redundant
|
|
\begin{refproof}{fact:isometriciffequicontinuous}.
|
|
$d$ and $d'(x,y) \coloneqq \sup_{t \in T} d(tx,ty)$
|
|
induce the same topology.
|
|
Let $\tau, \tau'$ be the corresponding topologies.
|
|
|
|
$\tau \subseteq \tau'$ easy,
|
|
$\tau' \subseteq \tau'$ : use equicontinuity.
|
|
\end{refproof}
|