w23-logic-3/inputs/tutorial_12b.tex

\tutorial{12b}{2024-01-16T13:09:02}{}

\subsection{Sheet 10}

\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:







\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}