order / rank

inputs/lecture_16.tex

@@ -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}
+\begin{definition}[{\cite[{}13.1]{Furstenberg}}]
-    Let $(X, \Z)$ be distal minimal.
+    Let $(X,T)$ be a quasi-isometric flow,
-    Then $\rank((X,\Z)) \coloneqq  \min \{\eta : (X, \Z) \cong (X_\eta, \Z)\}$
+    and let $\eta$ be the smallest ordinal
-    where $(X_{\eta}, \Z)$ is as from the definition of quasi-isometric flows,
+    such that there exists a quasi-isometric system $\{(X_\xi, T), \xi \le \eta\}$
-    i.e.~$\rank((X,\Z))$ is the minimal height such
+    with $(X,T) = (X_\xi, T)$.
-    that a tower as in the definition exists.
+    Then $\eta$ is called the \vocab{rank} or \vocab{order} of the flow
+    and is denoted by $\rank((X,T))$.
 \end{definition}
 
 \begin{definition}+

inputs/lecture_19.tex

@@ -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 $(Y,T)$, $\pi\colon X\to Y$,