Josia Pietsch 458dd9ab1f
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fixed some typesetting problems
2024-02-09 20:23:05 +01:00

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\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
$d$ 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.