maximal isometric extension

inputs/lecture_19.tex

@@ -47,12 +47,13 @@ to reach the flow with a quasi-isometric system (cf.~\yaref{thm:l16:3},
     For every flow $(X,T)$ there is a maximal factor $(Y,T)$, $\pi\colon X\to Y$,
     i.e.~if $(Y',T), \pi'\colon X \to Y'$ is any isometric factor of $(X,T)$,
     then  $(Y',T)$ is a factor of $(Y,T)$.
-    % https://q.uiver.app/#q=WzAsMyxbMCwwLCIoWCxUKSJdLFsxLDEsIihZLFQpXFxcXFxcdGV4dHttYXhpbWFsIGlzb21ldHJpY30iXSxbMCwyLCIoWScsVCkiXSxbMCwyLCJcXHRleHR7aXNvbWV0cmljfSIsMl0sWzAsMV0sWzEsMiwiXFxleGlzdHMiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=
+
+    % https://q.uiver.app/#q=WzAsMyxbMCwwLCIoWCxUKSJdLFsxLDEsIiBcXHN1YnN0YWNreyhZLFQpXFxcXFxcdGV4dHttYXhpbWFsIGlzb21ldHJpY319Il0sWzAsMiwiXFxzdWJzdGFja3soWScsVClcXFxcXFx0ZXh0e2lzb21ldHJpY319Il0sWzAsMl0sWzAsMV0sWzEsMiwiXFxleGlzdHMiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=
 \[\begin{tikzcd}
 	{(X,T)} \\
-	& {\substack{(Y,T)\\\text{maximal isometric}}} \\
-	{(Y',T)}
-	\arrow["{\text{isometric}}"', from=1-1, to=3-1]
+	& { \substack{(Y,T)\\\text{maximal isometric}}} \\
+	{\substack{(Y',T)\\\text{isometric}}}
+	\arrow[from=1-1, to=3-1]
 	\arrow[from=1-1, to=2-2]
 	\arrow["\exists", dashed, from=2-2, to=3-1]
 \end{tikzcd}\]
@@ -64,16 +65,18 @@ to reach the flow with a quasi-isometric system (cf.~\yaref{thm:l16:3},
     i.e.~if $(Y_\alpha, f_{\alpha,\beta})$,
     $\beta < \alpha \le  \Theta$
     are isometric, then the inverse limit $Y$ is isometric.%
-    % https://q.uiver.app/#q=WzAsNCxbMSwwLCJZX1xcYWxwaGEiXSxbMSwxLCJZX1xcYmV0YSJdLFswLDAsIlkiXSxbMiwwLCJYIl0sWzAsMSwiZl97XFxhbHBoYSwgXFxiZXRhfSJdLFsyLDAsImZfXFxhbHBoYSJdLFsyLDEsImZfXFxiZXRhIiwyXSxbMywwLCJcXHBpX1xcYWxwaGEiLDJdLFszLDEsIlxccGlfXFxiZXRhIl1d
+% https://q.uiver.app/#q=WzAsNCxbMSwxLCJZX1xcYWxwaGEiXSxbMSwyLCJZX1xcYmV0YSJdLFswLDEsIlkiXSxbMSwwLCJYIl0sWzAsMSwiZl97XFxhbHBoYSwgXFxiZXRhfSIsMV0sWzIsMCwiZl9cXGFscGhhIl0sWzIsMSwiZl9cXGJldGEiLDJdLFszLDAsIlxccGlfXFxhbHBoYSIsMV0sWzMsMSwiXFxwaV9cXGJldGEiLDEseyJjdXJ2ZSI6LTN9XV0=
 \[\begin{tikzcd}
-	Y & {Y_\alpha} & X \\
+	& X \\
+	Y & {Y_\alpha} \\
 	& {Y_\beta}
-	\arrow["{f_{\alpha, \beta}}", from=1-2, to=2-2]
-	\arrow["{f_\alpha}", from=1-1, to=1-2]
-	\arrow["{f_\beta}"', from=1-1, to=2-2]
-	\arrow["{\pi_\alpha}"', from=1-3, to=1-2]
-	\arrow["{\pi_\beta}", from=1-3, to=2-2]
+	\arrow["{f_{\alpha, \beta}}"{description}, from=2-2, to=3-2]
+	\arrow["{f_\alpha}", from=2-1, to=2-2]
+	\arrow["{f_\beta}"', from=2-1, to=3-2]
+	\arrow["{\pi_\alpha}"{description}, from=1-2, to=2-2]
+	\arrow["{\pi_\beta}"{description}, curve={height=-18pt}, from=1-2, to=3-2]
 \end{tikzcd}\]
+\gist{%
     Consider
     \[
         \Delta_\alpha \coloneqq  \{(y,y') \in Y^2 : f_{\alpha}(y) = f_\alpha(y')\}.
@@ -85,10 +88,10 @@ to reach the flow with a quasi-isometric system (cf.~\yaref{thm:l16:3},
     \[
     \bigcap_{\alpha \le \theta}\Delta_\alpha = \{(y,y) : y \in Y\}.
     \]
-    Consider
-    \[\{(y,y') \in \Delta_\alpha : d(y,y') \ge  \epsilon\}\]
-    for any $\epsilon > 0$.
-    By the finite intersection property % TODO WHY? TODO what is this TODO for compact?
+    Fix $\epsilon > 0$ and
+    consider
+    \[\{(y,y') \in \Delta_\alpha : d(y,y') \ge  \epsilon\}.\]
+    By the finite intersection property
     we get
     \[
     \exists  \alpha.~f_\alpha(y) = f_\alpha(y') \implies d(y,y') < \epsilon,
@@ -126,26 +129,53 @@ to reach the flow with a quasi-isometric system (cf.~\yaref{thm:l16:3},
     \[
     d(f^{-1}_\alpha(t_jz_j), f^{-1}_\alpha(t_jz'_j)) < \frac{\epsilon}{2} \lightning.
     \]
+}{
+    \begin{itemize}
+        \item Consider $\Delta_\alpha = \{(y,y') \in Y \times Y : f_\alpha(y) = f_\alpha(y')\} = Y \times_{Y_\alpha} Y$.
+            \begin{itemize}
+                \item $\beta < \alpha \implies \Delta_\beta \supseteq \Delta_\alpha$,
+                \item $\bigcap_{\alpha < \theta} \Delta_\alpha = \{(y,y) : y \in Y\}$.
+            \end{itemize}
+        \item Fix $\epsilon > 0$. Consider $M^{\epsilon}_{\alpha}\{(y,y') \in \Delta_\alpha : d(y,y') \ge \epsilon\}$.
+            FIP  $\implies$ $\exists \alpha.~M^{\epsilon}_{\alpha} = \emptyset$,
+            i.e.~$\forall z \in Y_\alpha.~\diam(f^{-1}_\alpha(z)) \le \epsilon$.
+        \item Suppose $Y$ is not isometric (i.e.~not equicontinuous).
+            Then $\exists (y_j), (y'_j)$ in $Y$ with $d(y_j,y_j') \to 0$
+            and $\epsilon > 0, t_j \in T$ s.t.~$d(t_jy_j, t_jy_j') > \epsilon$.
+        \item Wlog.~$y_j \to y$, $y'_j \to y$.
+            Fix $\alpha$ s.t.~$M^{\frac{\epsilon}{4}}_\alpha = \emptyset$.
+        \item $f_\alpha(y_j), f_{\alpha}(y'_j)$ converge $ z \in Y_\alpha$,
+            equicontinuity of $Y_{\alpha} \implies d_{Y_\alpha}(t_jf_\alpha(y_j),  t_jf_\alpha(y'_j)) \to 0$.
+            Wlog.~$t_jf_\alpha(y_j^{(')})$ converge to same point.
+        \item Consider $d(f^{-1}_\alpha(t_jf_\alpha(y_j)), f^{-1}_\alpha(t_jf_{\alpha}(y_{j}'))) \lessgtr \frac{\epsilon}{2}$ $\lightning$
+    \end{itemize}
+}
 \end{proof}
 
 More generally we can show:
-\begin{theorem}[{\cite[13.1]{Furstenberg}}]
+\begin{theorem}[{\cite[Prop.~13.1]{Furstenberg}}]
     Let $(X,T)$ be a distal flow
-    and $(Y,T) = \pi(X,T)$ a factor.
+    and $(Z,T) = \pi(X,T)$ a factor.
     Then there exists an isometric extension $(Y,T)$ of $(Z,T)$
     which is a factor of  $(X,T)$,
     such that $(Y,T)$ is maximal among such extensions,
     i.e.~if $(Y',T)$ is any flow with these two properties,
     then $(Y',T)$ is a factor of $(Y,T)$.
-    % https://q.uiver.app/#q=WzAsMyxbMCwwLCIoWCxUKSJdLFswLDIsIihaLFQpIl0sWzEsMSwiKFksVCkiXSxbMCwyXSxbMCwxLCJcXHBpIl0sWzIsMV1d
+    % https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXHN1YnN0YWNreyhYLFQpXFxcXFxcdGV4dHtkaXN0YWx9fSJdLFswLDMsIihaLFQpIl0sWzEsMSwiKFksVCkiXSxbMiwyLCIoWScsVCkiXSxbMCwyXSxbMCwxLCJcXHBpIl0sWzIsMSwiXFx0ZXh0e21heC5+aXNvLn0iLDFdLFswLDMsIiIsMix7ImN1cnZlIjotM31dLFszLDEsIlxcdGV4dHtpc28ufSIsMV0sWzIsMywiIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d
 \[\begin{tikzcd}
-	{(X,T)} \\
+	{\substack{(X,T)\\\text{distal}}} \\
 	& {(Y,T)} \\
+	&& {(Y',T)} \\
 	{(Z,T)}
 	\arrow[from=1-1, to=2-2]
-	\arrow["\pi", from=1-1, to=3-1]
-	\arrow[from=2-2, to=3-1]
-\end{tikzcd}\]
+	\arrow["\pi", from=1-1, to=4-1]
+	\arrow["{\text{max.~iso.}}"{description}, from=2-2, to=4-1]
+	\arrow[curve={height=-18pt}, from=1-1, to=3-3]
+	\arrow["{\text{iso.}}"{description}, from=3-3, to=4-1]
+	\arrow[dashed, from=2-2, to=3-3]
+\end{tikzcd}
+\]
+Such a factor $(Y,T)$ is called a \vocab{maximal isometric extension} of $(Z,T)$.
 \end{theorem}