Josia Pietsch
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34 lines
691 B
TeX
34 lines
691 B
TeX
\tutorial{12b}{2024-01-16T13:09:02}{}
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\subsection{Sheet 10}
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\nr 2
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\todo{Def skew shift flow (on $(\R / \Z)^2$!)}
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The Bernoulli shift, $\Z \acts \{0,1\}^{\Z}$, is not distal.
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Let $x = (0)$ and $y = (\delta_{0,i})_{i \in \Z}$.
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Let $t_n \to \infty$.
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Then $t_n y \to (0) = t_n x$.
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% The skew shift flow is distal:
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% This is tedious but probably not too hard.
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%
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% The skew shift flow is not equicontinuous:
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\begin{refproof}{fact:isometriciffequicontinuous}.
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$d$ and $d'(x,y) \coloneqq \sup_{t \in T} d(tx,ty)$
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induce the same topology.
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Let $\tau, \tau'$ be the corresponding topologies.
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$\tau \subseteq \tau'$ easy,
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$\tau' \subseteq \tau'$ : use equicontinuity.
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\end{refproof}
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