From af5ccd6c33bce5bebf5fdfa6c00c8f80a24974f5 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Thu, 8 Feb 2024 15:23:21 +0100 Subject: [PATCH] maximal isometric extension --- inputs/lecture_19.tex | 74 ++++++++++++++++++++++++++++++------------- 1 file changed, 52 insertions(+), 22 deletions(-) diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index 07a96ec..54f11a1 100644 --- a/inputs/lecture_19.tex +++ b/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}