order / rank
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Josia Pietsch 2024-02-06 19:51:03 +01:00
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commit 227e2ac7b9
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2 changed files with 14 additions and 20 deletions

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@ -151,16 +151,17 @@ By Zorn's lemma, this will follow from
& {(Z,T)}
\arrow[from=1-1, to=1-3]
\arrow[from=2-2, to=1-3]
\arrow["{\text{isometric extension}}"{description}, from=1-1, to=2-2]
\arrow["{\text{isometric extension}}"{description}, from=1-1, to=2-2]
\end{tikzcd}\]
\end{theorem}
\yaref{thm:furstenberg} allows us to talk about ranks of distal minimal flows:
\begin{definition}
Let $(X, \Z)$ be distal minimal.
Then $\rank((X,\Z)) \coloneqq \min \{\eta : (X, \Z) \cong (X_\eta, \Z)\}$
where $(X_{\eta}, \Z)$ is as from the definition of quasi-isometric flows,
i.e.~$\rank((X,\Z))$ is the minimal height such
that a tower as in the definition exists.
\begin{definition}[{\cite[{}13.1]{Furstenberg}}]
Let $(X,T)$ be a quasi-isometric flow,
and let $\eta$ be the smallest ordinal
such that there exists a quasi-isometric system $\{(X_\xi, T), \xi \le \eta\}$
with $(X,T) = (X_\xi, T)$.
Then $\eta$ is called the \vocab{rank} or \vocab{order} of the flow
and is denoted by $\rank((X,T))$.
\end{definition}
\begin{definition}+

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@ -35,20 +35,13 @@ equicontinuity coincide.
By equicontinuity of $T$ we get that $\tilde{d}$ and $d$
induce the same topology on $X$.
\end{proof}
\gist{
Recall that we defined the order of a quasi-isometric flow
to be the minimal number of steps required when building the tower
to reach the flow with a quasi-isometric system (cf.~\yaref{thm:l16:3},
\yaref{def:floworder}).
}{}
\begin{question}
What is the minimal number of steps required
when building the tower to reach the flow
as in \yaref{thm:l16:3}?
\end{question}
\begin{definition}[{\cite[{}13.1]{Furstenberg}}]
Let $(X,T)$ be a quasi isometric flow,
and let $\eta$ be the smallest ordinal
such that there exists a quasi-isometric system $\{(X_\xi, T), \xi \le \eta\}$
with $(X,T) = (X_\xi, T)$.
Then $\eta$ is called the \vocab{order} of the flow.
\end{definition}
\begin{theorem}[Maximal isometric factor]
\label{thm:maxisomfactor}
For every flow $(X,T)$ there is a maximal factor $(Y,T)$, $\pi\colon X\to Y$,