diff --git a/bibliography/images.bib b/bibliography/images.bib new file mode 100644 index 0000000..e69de29 diff --git a/bibliography/references.bib b/bibliography/references.bib new file mode 100644 index 0000000..b41d85e --- /dev/null +++ b/bibliography/references.bib @@ -0,0 +1,27 @@ +@misc{tao, + ISSN = {0003486X}, + URL = {https://terrytao.wordpress.com/category/teaching/254a-ergodic-theory/}, + author = {Terence Tao}, + title = {254A Ergodic Theory}, + urldate = {2024-01-01}, + year = {2008}, +} +@MISC{801106, + TITLE = {Closure of a topological group}, + AUTHOR = {Mariano Suarez-Alvarez}, + HOWPUBLISHED = {Mathematics Stack Exchange}, + URL = {https://math.stackexchange.com/q/801106}, +} +@article{Furstenberg, +author = {Furstenberg, H.}, +journal = {American journal of mathematics}, +issn = {0002-9327}, +number = {3}, +keywords = {Continuous functions ; Eigenfunctions ; Equivalence relation ; Geometry ; Integers ; Mathematical functions ; Mathematics ; Myelinated nerve fibers ; Structure (category theory) ; Topological compactness ; Topological spaces ; Topological theorems ; Topology}, +language = {eng}, +pages = {477-515}, +publisher = {Johns Hopkins Press}, +volume = {85}, +year = {1963}, +title = {The Structure of Distal Flows}, +} diff --git a/inputs/facts.tex b/inputs/facts.tex index f749a4c..0523d70 100644 --- a/inputs/facts.tex +++ b/inputs/facts.tex @@ -1,6 +1,6 @@ \subsection{Topological Dynamics} -\begin{fact}[\url{https://math.stackexchange.com/a/801106}] +\begin{fact}[\cite{801106}] \label{fact:topsubgroupclosure} Let $H$ be a topological group and $G \subseteq H$ a subgroup. diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex index 4f64adc..ba7c57c 100644 --- a/inputs/lecture_15.tex +++ b/inputs/lecture_15.tex @@ -161,6 +161,7 @@ Recall: A flow is \vocab{distal} iff it has no proximal pair. \end{definition} + \begin{definition}+ Let $(T,X)$ and $(T,Y)$ be flows. A \vocab{factor map} $\pi\colon (T,X) \to (T,Y)$ diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index f6f2f00..c9ee8f7 100644 --- a/inputs/lecture_16.tex +++ b/inputs/lecture_16.tex @@ -135,9 +135,10 @@ $X$ is always compact metrizable. \end{theorem} By Zorn's lemma, this will follow from \begin{theorem}[Furstenberg] + \label{thm:l16:3} Let $(X, T)$ be a minimal distal flow - and let $(Y, T)$ be a proper factor, - i.e.~$(X,T)$ and $(Y,T)$ are note isomorphic. + and let $(Y, T)$ be a proper factor. + \footnote{i.e.~$(X,T)$ and $(Y,T)$ are not isomorphic} Then there is another factor $(Z,T)$ of $(X,T)$ which is a proper isometric extension of $Y$. diff --git a/inputs/lecture_17.tex b/inputs/lecture_17.tex index 05cd27d..16d5e8b 100644 --- a/inputs/lecture_17.tex +++ b/inputs/lecture_17.tex @@ -1,7 +1,5 @@ -\lecture{17}{2023-12-12}{The Ellis semigroup} - - \subsection{The Ellis semigroup} +\lecture{17}{2023-12-12}{The Ellis semigroup} Let $(X, d)$ be a compact metric space and $(X, T)$ a flow. @@ -29,7 +27,7 @@ $X^{X}$ is a compact Hausdorff space. We have $ff_0 \in U_{\epsilon}(x,y)$ iff $f \in U_\epsilon(x,f_0(y))$. \item Fix $x_0 \in X$. - Then $f \mapsto f(x)$ is continuous. + Then $f \mapsto f(x_0)$ is continuous. \item In general $f \mapsto f_0 \circ f$ is not continuous, but if $f_0$ is continuous, then the map is continuous. \end{itemize} @@ -40,23 +38,24 @@ Let $(X,T)$ be a flow. Then the \vocab{Ellis semigroup} is defined by $E(X,T) \coloneqq \overline{T} \subseteq X^X$, -i.e.~identify $T$ with $x \mapsto tx$ +i.e.~identify $t \in T$ with $x \mapsto tx$ and take the closure in $X^X$. \end{definition} $E(X,T)$ is compact and Hausdorff, since $X^X$ has these properties. -Properties of $(X,T)$ translate to +Properties of $(X,T)$ translate to properties of $E(X,T)$: \begin{goal} We want to show that if $(X,T)$ is distal, then $E(X,T)$ is a group. \end{goal} \begin{proposition} - $G$ is a semigroup, + $E(X,T)$ is a semigroup, i.e.~closed under composition. \end{proposition} \begin{proof} + Let $G \coloneqq E(X,T)$. Take $t \in T$. We want to show that $tG \subseteq G$, i.e.~for all $h \in G$ we have $th \in G$. @@ -97,7 +96,7 @@ Properties of $(X,T)$ translate to such that $S \ni x \mapsto xs$ is continuous for all $s$. \end{definition} \begin{example} - Ellis semigroup is a compact semigroup. + The Ellis semigroup is a compact semigroup. \end{example} \begin{lemma}[Ellis–Numakura] @@ -181,10 +180,10 @@ However if we pick $y \in Y$, $Ty$ might not be dense. \begin{proof} Let $G = E(X,T)$. Note that for all $x \in X$, - we have $Gx \subseteq X$ is compact + we have that $Gx \subseteq X$ is compact and invariant under the action of $G$. - Since $G$ is a group, we have that the orbits partition $X$.% + Since $G$ is a group, the orbits partition $X$.% \footnote{Note that in general this does not hold for semigroups.} % Clearly the sets $Gx$ cover $X$. We want to show that they diff --git a/inputs/lecture_18.tex b/inputs/lecture_18.tex new file mode 100644 index 0000000..9acb47f --- /dev/null +++ b/inputs/lecture_18.tex @@ -0,0 +1,236 @@ +\subsection{Sketch of proof of \yaref{thm:l16:3}} +\lecture{18}{2023-12-15}{Sketch of proof of \yaref{thm:l16:3}} + +The goal for this lecture is to give a very rough +sketch of \yaref{thm:l16:3} in the case of $|Z| = 1$. +% \begin{theorem}[Furstenberg] +% Let $(X, T)$ be a minimal distal flow +% and let $(Z,T)$ be a proper factor of $X$% +% \footnote{i.e.~$(X,T)$ and $(Z,T)$ are not isomorphic.} +% Then three is another factor $(Y,T)$ of $(X,T)$ +% which is a proper isometric extension of $Z$. +% \end{theorem} + + +Let $(X,T)$ be a distal flow. +Then $G \coloneqq E(X,T)$ is a group. +\begin{definition} +For $x, x' \in X$ define +\[ +F(x,x') \coloneqq \inf \{d(gx, gx') : g \in G\}. +\] +\end{definition} +\begin{fact} +\begin{enumerate}[(a)] + \item $F(x,x') = F(x', x)$, + \item $F(x,x') \ge 0$ and $F(x,x') = 0$ iff $x = x'$. + \item $F(gx, gx') = F(x,x')$ since $G$ is a group. + \item $F$ is an upper semi-continuous function on $X^2$, + i.e.~$\forall a \in R.~\{(x,x') \in X^2 : F(x,x') < a\} \overset{\text{open}}{\subseteq} X^2$. + + This holds because $F$ is the infimum of continuous functions + \begin{IEEEeqnarray*}{rCl} + f_g\colon X^2 &\longrightarrow & \R \\ + (x,x') &\longmapsto & d(gx, gx') + \end{IEEEeqnarray*} + for $g \in G$. +\end{enumerate} +\end{fact} +\begin{theoremdef} + \label{def:ftop} + The sets + \[ + U_a(x) \coloneqq \{x' : F(x,x') < a\} + \] + form the basis of a topology in $X$. + This topology is called the \vocab{F-topology} on $X$. + In this setting, the original topology + is also called the \vocab{E-topology}. +\end{theoremdef} +This will follow from the following lemma: +\begin{lemma} + \label{lem:ftophelper} + Let $F(x,x') < a$. + Then there exists $\epsilon > 0$ such that + whenever $F(x',x'') < \epsilon$, then $F(x,x'') < a$. +\end{lemma} +\begin{refproof}{def:ftop} + We have to show that if $U_a(x_1) \cap U_b(x_2) \neq \emptyset$, + then this intersection is the union + of sets of this kind. + Let $x' \in U_a(x_1)$. + Then by \yaref{lem:ftophelper}, + there exists $\epsilon_1 > 0$ with $U_{\epsilon_1}(x') \subseteq U_a(x_1)$. + Similarly there exists $\epsilon_2 > 0$ + such that $U_{\epsilon_2}(x') \subseteq U_b(x_2)$. + So for $\epsilon \le \epsilon_1, \epsilon_2$, + we get $U_{\epsilon}(x') \subseteq U_a(x_1) \cap U_b(x_2)$. +\end{refproof} +\begin{refproof}{lem:ftophelper}% + \footnote{This was not covered in class.} + + Let $T = \bigcup_n T_n$,% TODO Why does this exist? + $T_n$ compact, wlog.~$T_n \subseteq T_{n+1}$, and + let $G(x,x') \coloneqq \{(gx,gx') : g \in G\} \subseteq X \times X$. + Take $b$ such that $F(x,x') < b < a$. + Then $U = \{(u,u') \in G(x,x') : d(u,u') < b\}$ + is open in $G(x,x')$ + and since $F(x,x') < b$ we have $U \neq \emptyset$. + \begin{claim} + There exists $n$ such that + \[ + \forall (u,u') \in G(x,x').~T_n(u,u')\cap U \neq \emptyset. + \] + \end{claim} + \begin{subproof} + Suppose not. + Then for all $n$, there is $(u_n, u_n') \in G(x,x')$ + with + \[T_n(u_n, u_n') \subseteq G(x,x') \setminus U.\] + Note that the RHS is closed. + For $m > n$ we have + $T_n(u_m, u'_m) \subseteq G(x,x') \setminus U$ + since $T_n \subseteq T_m$. + By compactness of $X$, + there exists $v,v'$ and some subsequence + such that $(u_{n_k}, u'_{n_k}) \to (v,v')$. + + So for all $n$ we have $T_n(v,v') \subseteq G(x,x') \setminus U$, + hence $T(v,v') \cap U = \emptyset$, + so $G(v,v') \cap U = \emptyset$. + But this is a contradiction as $\emptyset\neq U \subseteq G(v,v')$. + \end{subproof} + The map + \begin{IEEEeqnarray*}{rCl} + T\times X&\longrightarrow & X \\ + (t,x) &\longmapsto & tx + \end{IEEEeqnarray*} + is continuous. + Since $T_n$ is compact, + we have that $\{(x,t) \mapsto tx : t \in T_n\}$ + is equicontinuous for all $n$. + So there is $\epsilon > 0$ such that + $d(x_1,x_2) < \epsilon \implies d(tx_1, tx_2) < a -b$ + for all $t \in T_n$. + + Suppose now that $F(x', x'') < \epsilon$. + Then there is $t_0 \in T$ such that $d(t_0x', t_0x'') < \epsilon$, + hence $d(t t_0x', t t_0 x'') < a-b$ for all $t \in T_n$. + Since $(t_0x, t_0x') \in G(x,x')$, + there is $t_1 \in T_n$ + with $(t_1t_0x, t_1t_0x') \in U$, + i.e.~$d(t_1t_0x, t_1t_0x') < b$ + and therefore + $F(x,x'') = d(t_1t_0x, t_1t_0x'') < a$. +\end{refproof} + +Now assume $Z = \{\star\}$. +We want to sketch a proof of \yaref{thm:l16:3} in this case, +i.e.~show that if $(Z,T)$ is a proper factor of a minimal distal flow + $(X,T)$ then there is another factor $(Y,T)$ of $(X,T)$ + which is a proper isometric extension of $Z$. + +\begin{proof}[sketch] % TODO: Think about this +\leavevmode +\begin{enumerate}[1.] + \item For $x \in X$ define + \begin{IEEEeqnarray*}{rCl} + F_x\colon X &\longrightarrow & \R \\ + x' &\longmapsto & F(x,x'). + \end{IEEEeqnarray*} + \item Define an equivalence relation on $X$, + by $x_1 \sim x_2 :\iff \{x \in X : F_{x_1}(x) = F_{x_2}(x)\}$ + is comeager in $X$\footnote{with respect to the E-topology}. + Then for all $g \in G$ we have + $x_1 \sim x_2 \implies gx_1 \sim ~ gx_2$. + + Let $M \coloneqq \{[x]_{\sim } : x \in X\} = \faktor{X}{\sim}$ + bet the quotient space. + It is compact, second countable and Hausdorff. + Let $\pi\colon X\to M$ denote the quotient map. + \item $(Y,T) \mathbin{\text{\reflectbox{$\coloneqq$}}} (M,T)$ + is an isometric flow: + \begin{enumerate} + \item For $a > 0$, $x,x' \in X$ let + \[ + W(x,x') \coloneqq \{g \in G : F(x, gx') < a\}. + \] + This turns out to be a subbasis of a topology + which is coarser than the original topology on $G$. + The new topology makes $G$ compact. + \item Let $\theta(g)$ be the transformation of $M$ + defined by $\theta(g) \pi(x) = \pi(gx)$. + This is well defined. + Let $H = \theta(G)$. + This is just a quotient of $G$, $g \mapsto \theta(g)$ + may not be injective. + \item One can show that $H$ is a topological group and $(M,H)$ + is a flow.\footnote{This is non-trivial.} + \item Since $H$ is compact, + $(M,H)$ is equicontinuous, %\todo{We didn't define this} + i.e.~it is isometric. + In particular, $(M,T)$ is isometric. + \end{enumerate} + \item $M \neq \{\star\}$, i.e.~$(M,T)$ is non-trivial: + + Suppose towards a contradiction that $M = \{\star\}$, + i.e.~$x_1 \sim x_2$ for all $x_1,x_2 \in X$. + Fix $x_2$. For every $x_1 \in X$ + we have that + \[ + \{x : F(x_1,x) = F(x_2,x)\} + \] + is comeager. + Let $x_1$ be a point of continuity of $F_{x_2}$. + Let $\langle a_n : n < \omega \rangle$ be a sequence + of elements that set, i.e.~$F(x_1, a_n) = F(x_2, a_n)$, + such that $a_n \to x_1$. + So by the continuity of $F_{x_2}$ at $x_1$ + \begin{IEEEeqnarray*}{rCl} + \lim_{n \to \infty} F(x_2, a_n) &=& F(x_2, x_1) + \end{IEEEeqnarray*} + and by the definition of $F$ + \begin{IEEEeqnarray*}{rCl} + \lim_{n \to \infty} F(x_1,a_n) &=& F(x_1,x_1) = 0. + \end{IEEEeqnarray*} + So + \[ + F(x_2,x_1) = \lim_{n \to \infty} F(x_2, a_n) = \lim_{n \to \infty} + F(x_1,a_n) = 0 + \] + and by distality we get $x_1 = x_2$. + Since almost all points of $X$ + are points of continuity of $F_{x_2}$ + (\yaref{thm:usccomeagercont}) + this implies that $X \setminus \{x_2\}$ is meager. + But then $X = \{\star\} \lightning$. +\end{enumerate} +\end{proof} + +\begin{theorem}\footnote{Not covered in class} + \label{thm:usccomeagercont} + Let $X$ be a metric space + and $\Gamma\colon X \to \R$ be upper semicontinuous. + Then the set of continuity points of $\Gamma$ is comeager. + \todo{Missing figure: upper semicontinuous function} +\end{theorem} +\begin{proof} + Take $x$ such that $\Gamma$ is not continuous at $x$. + Then there is an $\epsilon > 0$ + and $x_n \to x$ such that + $\Gamma(x_n) + \epsilon \le \Gamma(x)$. + Take $q \in \Q$ such that $\Gamma(x) - \epsilon < q < \Gamma(x)$. + Then let + \[ + B_q \coloneqq \{a \in X : \Gamma(a) \ge q\}. + \] + $X \setminus B_q = \{a \in X : \Gamma(a) < q\}$ + is open, i.e.~$B_q$ is closed. + Note that $x \in F_q \coloneqq B_q \setminus B_q^\circ$ + and $B_q \setminus B_q^\circ$ is nwd + as it is closed and has empty interior, + so $\bigcup_{q \in \Q} F_q$ is meager. +\end{proof} + + + diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex new file mode 100644 index 0000000..d8fb2f7 --- /dev/null +++ b/inputs/lecture_19.tex @@ -0,0 +1,292 @@ +\subsection{The order of a flow} +\lecture{19}{2023-12-19}{Orders of flows} + +See also \cite[\href{https://terrytao.wordpress.com/2008/01/24/254a-lecture-6-isometric-systems-and-isometric-extensions/}{Lecture 6}]{tao}. + + +\begin{definition}+ + Let $X,Y$ be metric spaces. A family $F$ of functions $X \to Y$ + is called \vocab{equicontinuous} at $x_0 \in X$ + iff + \[ + \forall \epsilon > 0.~\exists \delta > 0.~ \forall f \in F.~d_X(x_0, x) < \delta \implies d_Y(f(x_0),f(x)) < \epsilon. + \] + It is called equicontinuous iff it is equicontinuous at every point. + It is called \vocab{uniformly equicontinuous} + iff + \[ + \forall \epsilon > 0.~\exists \delta > 0.~ \forall x_0 \in X.~\forall f \in F.~d_X(x_0, x) < \delta \implies d_Y(f(x_0),f(x)) < \epsilon. + \] + A flow $(X,T)$ is called equicontinuous iff $T$ is equicontinuous. +\end{definition} +Note that since $X$ compact the notions of equicontinuity and uniform +equicontinuity coincide. + +\begin{fact}+[{\cite[Lecture 6, Exercise 1]{tao}}] + A flow $(X,T)$ is isometric iff it is equicontinuous. +\end{fact} +\begin{proof} + Clearly an isometric flow is equicontinuous. + On the other hand suppose that $T$ is uniformly equicontinuous. + Define a metric $\tilde{d}$ on $X$ by setting + $\tilde{d}(x,y) \coloneqq \sup_{t \in T} d(tx,ty)$. + By equicontinuity of $T$ we get that $\tilde{d}$ and $d$ + induce the same topology on $X$. +\end{proof} + + +\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 $(Y,T)$, $\pi\colon X\to Y$, + 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= +\[\begin{tikzcd} + {(X,T)} \\ + & {\substack{(Y,T)\\\text{maximal isometric}}} \\ + {(Y',T)} + \arrow["{\text{isometric}}"', from=1-1, to=3-1] + \arrow[from=1-1, to=2-2] + \arrow["\exists", dashed, from=2-2, to=3-1] +\end{tikzcd}\] +\end{theorem} +\begin{proof} + % TODO Think about this + We want to apply Zorn's lemma. + If suffices to show that isometric flows are closed under inverse limits, + i.e.~if $(Y_\alpha, f_{\alpha,\beta})$, + $\beta < \alpha \le \Theta$ + are isometric, then the inverse limit $Y$ is isometric.% + \todo{Why does an inverse limit exist?} + % https://q.uiver.app/#q=WzAsNCxbMSwwLCJZX1xcYWxwaGEiXSxbMSwxLCJZX1xcYmV0YSJdLFswLDAsIlkiXSxbMiwwLCJYIl0sWzAsMSwiZl97XFxhbHBoYSwgXFxiZXRhfSJdLFsyLDAsImZfXFxhbHBoYSJdLFsyLDEsImZfXFxiZXRhIiwyXSxbMywwLCJcXHBpX1xcYWxwaGEiLDJdLFszLDEsIlxccGlfXFxiZXRhIl1d +\[\begin{tikzcd} + Y & {Y_\alpha} & X \\ + & {Y_\beta} + \arrow["{f_{\alpha, \beta}}", from=1-2, to=2-2] + \arrow["{f_\alpha}", from=1-1, to=1-2] + \arrow["{f_\beta}"', from=1-1, to=2-2] + \arrow["{\pi_\alpha}"', from=1-3, to=1-2] + \arrow["{\pi_\beta}", from=1-3, to=2-2] +\end{tikzcd}\] + Consider + \[ + \Delta_\alpha \coloneqq \{(y,y') \in Y^2 : f_{\alpha}(y) = f_\alpha(y')\}. + \] + Let $d$ be a metric on $Y$ and $d_{\alpha}$ a metric on $Y_{\alpha}$, + wlog.~$d, d_\alpha \le 1$. + Note that $\beta < \alpha \implies \Delta_\beta \supseteq \Delta_\alpha$ + and + \[ + \bigcap_{\alpha \le \theta}\Delta_\alpha = \{(y,y) : y \in Y\}. + \] + 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? + we get + \[ + \exists \alpha.~f_\alpha(y) = f_\alpha(y') \implies d(y,y') < \epsilon, + \] + i.e.~$\forall z \in Y_\alpha.~\diam(f^{-1}_\alpha(z)) \le \epsilon$. + + Towards a contradiction assume that $Y$ is not isometric, + i.e.~not equicontinuous. + Then there are $(y_j), (y'_j) \in Y$ + such that $d(y_j,y'_j) \to 0$ + and $\epsilon > 0, t_j \in T$ + such that $d(t_jy_j, t_jy'_j) > \epsilon$. + + By compactness wlog.~$(y_j)$ and $(y'_j)$ + converge (to the same point). + Find $\alpha$ such that $f_\alpha(y) = f_{\alpha}(y') \implies d(y,y') < \frac{\epsilon}{4}$. + Let $z_j \coloneqq f_{\alpha}(y_j)$ and $z'_j \coloneqq f_\alpha(y'_j)$. + Then $(z_j)$ and $(z'_j)$ converge to the same point $z \in Y_\alpha$. + By equicontinuity of $(Y_\alpha, T)$, + $d_{Y_{\alpha}}(t_jz_j, t_jz'_j) \to 0$. + Wlog.~$(t_jz_j)$ and $(t_jz'_j)$ converge. + Let $z^\ast$ be their limit. + On the one hand, by the triangle inequality we get + \[ + d(f^{-1}_\alpha(t_jz_j), f^{-1}_\alpha(t_jz_j')) > \underbrace{\epsilon}_{\mathclap{< d(t_jy_j, t_jy_j')}} - \overbrace{\frac{\epsilon}{4}}^{\mathclap{\text{Diameter of fiber}}}- \frac{\epsilon}{4} = \frac{\epsilon}{2}. + \] + + On the other hand, from + \begin{IEEEeqnarray*}{rCl} + d(f^{-1}_\alpha(t_jz_j), f^{-1}_{\alpha}(z^\ast)) &\to & 0,\\ + d(f^{-1}_\alpha(t_jz'_j), f^{-1}_{\alpha}(z^{\ast})) &\to & 0,\\ + \diam f^{-1}_\alpha(\{z^\ast\}) & <& \frac{\epsilon}{4} + \end{IEEEeqnarray*} + we obtain + \[ + d(f^{-1}_\alpha(t_jz_j), f^{-1}_\alpha(t_jz'_j)) < \frac{\epsilon}{2} \lightning. + \] +\end{proof} + +More generally we can show: +\begin{theorem}[{\cite[13.1]{Furstenberg}}] + Let $(X,T)$ be a distal flow + and $(Y,T) = \pi(X,T)$ 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 +\[\begin{tikzcd} + {(X,T)} \\ + & {(Y,T)} \\ + {(Z,T)} + \arrow[from=1-1, to=2-2] + \arrow["\pi", from=1-1, to=3-1] + \arrow[from=2-2, to=3-1] +\end{tikzcd}\] +\end{theorem} + + +\begin{lemma} + \label{lec19:lem1} + Let four flows be given as in + + % https://q.uiver.app/#q=WzAsNCxbMSwwLCIoWSxUKSJdLFsyLDEsIihaXzIsIFQpIl0sWzAsMSwiKFpfMSwgVCkiXSxbMSwyLCIoVyxUKSJdLFsyLDMsIndfMSJdLFsxLDMsIndfMiIsMl0sWzAsMiwiXFxwaV8xIl0sWzAsMSwiXFxwaV8yIiwyXV0= +\[\begin{tikzcd} + & {(Y,T)} \\ + {(Z_1, T)} && {(Z_2, T)} \\ + & {(W,T)} + \arrow["{w_1}", from=2-1, to=3-2] + \arrow["{w_2}"', from=2-3, to=3-2] + \arrow["{\pi_1}", from=1-2, to=2-1] + \arrow["{\pi_2}"', from=1-2, to=2-3] +\end{tikzcd}\] + + Suppose that whenever $y \neq y' \in Y$, + then either % TODO REALLY? + $\pi_1(y) \neq \pi(y')$ + or $\pi_2(y) \neq \pi_2(y')$. + + If $(Z_1,T)$ is an isometric extension of $(W,T)$, + then $(Y,T)$ is an isometric extension of $(Z_2, T)$. +\end{lemma} +\begin{proof} + % TODO TODO TODO Think about this + For $z_1,z_1' \in Z_1$ with + $w_1(z_1) = w_1(z_1')$ let + $\rho(z_1,z_1')$ be the metric on the fiber of $Z_1$ over $W$. + Set $\sigma(y,y') \coloneqq \rho(\pi_1(y), \pi_1(y'))$ whenever $\pi_2(y) = \pi_2(y')$. + In this case $w_2 \circ \pi_2(y) = w_2 \circ \pi_2(y')$ + and $w_1 \circ \pi_1(y) = w_1 \circ \pi_1(y')$, + so $\sigma$ is well defined. + $\sigma$ is a semi-metric\footnote{Like a metric, but the distinct points can have distance $0$.} + on the fibers of $Y$ over $Z_2$ + and invariant under $T$. + + $\sigma$ is a metric, since if + if $\pi_2(y) = \pi_2(y')$ and $\sigma(y,y') = 0$, + then $\pi_1(y) = \pi_1(y')$ or $y = y'$. +\end{proof} + + +\begin{definition} + A quasi-isometric system + $\{(X_\xi, T) : \xi \le \eta\}$ + is called \vocab{normal} if $(X_{\xi+1}, T)$ is the maximal + isometric extension of $(X_\xi,T)$ in $(X_\eta, T)$ + for all $\xi < \eta$. +\end{definition} +\begin{theorem}[{\cite[{}13.2]{Furstenberg}}] + If $\{(X_\xi, T), \xi \le \eta\}$ + is a normal quasi-isometric + system, then $(X_\eta, T)$ has order $\eta$. +\end{theorem} +\begin{proof} + We only sketch the proof here. + Details can be found in \cite{Furstenberg}, section 13. + Let $\{(X_\xi', T), \xi \le \eta'\} $ be + another quasi-isometric system + terminating with $(X_\eta, T) = (X'_{\eta'}, T)$. + We want to show that $\eta' \ge \eta$. + For this, we show that for all $\xi < \eta$, + $(X_\xi', T)$ is a factor of $(X_\xi ,T)$ + using transfinite induction. +% https://q.uiver.app/#q=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 +\[\begin{tikzcd} + {X'_{\eta'}} & \dots & {X'_3} & {X'_2} & {X_1'} \\ + X \\ + {X_\eta} & \dots & {X_3} & {X_2} & {X_1} + \arrow[Rightarrow, no head, from=1-1, to=2-1] + \arrow[Rightarrow, no head, from=2-1, to=3-1] + \arrow[from=3-3, to=3-4] + \arrow[from=3-4, to=3-5] + \arrow[from=1-4, to=1-5] + \arrow[from=1-3, to=1-4] + \arrow[dotted, from=3-3, to=1-3] + \arrow[dotted, from=3-4, to=1-4] + \arrow[dotted, from=3-5, to=1-5] + \arrow["{\pi_3}"', curve={height=6pt}, from=3-1, to=3-3] + \arrow["{\pi_2}"', curve={height=18pt}, from=3-1, to=3-4] + \arrow["{\pi_1}"', curve={height=30pt}, from=3-1, to=3-5] + \arrow["{\pi'_3}", curve={height=-6pt}, from=1-1, to=1-3] + \arrow["{\pi'_2}", curve={height=-18pt}, from=1-1, to=1-4] + \arrow["{\pi'_1}", curve={height=-30pt}, from=1-1, to=1-5] +\end{tikzcd}\] + % TODO: induction start? + + Suppose we have + $(X'_\xi, T) = \theta((X_\xi, T)$. + Let $\pi_\xi$ and $\pi'_\xi$ denote the maps from $X$ to $X_\xi$ resp.~$X'_\xi$. + Set + \[Y \coloneqq \{(\pi_\xi(x), \pi'_{\xi+1}(x)) \in X_\xi \times X'_{\xi+ 1}: x \in X\}.\] + Then + + % https://q.uiver.app/#q=WzAsNSxbMCwwLCIoWF97XFx4aSsxfSxUKSJdLFsyLDAsIihZLFQpIl0sWzMsMSwiKFgnX3tcXHhpKzF9LFQpIl0sWzIsMiwiKFgnX1xceGksVCkiXSxbMSwxLCIoWF9cXHhpLFQpIl0sWzAsNCwiXFx0ZXh0e21heC5+aXNvfSIsMV0sWzQsMywiXFx0aGV0YSIsMV0sWzIsMywie1xcY29sb3J7b3JhbmdlfVxcdGV4dHtpc299fSIsMV0sWzEsNCwie1xcY29sb3J7b3JhbmdlfVxcdGV4dHtpc299fSIsMV0sWzEsMl0sWzAsMSwiIiwwLHsiY3VydmUiOi0xLCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0= +\[\begin{tikzcd} + {(X_{\xi+1},T)} && {(Y,T)} \\ + & {(X_\xi,T)} && {(X'_{\xi+1},T)} \\ + && {(X'_\xi,T)} + \arrow["{\text{max.~iso}}"{description}, from=1-1, to=2-2] + \arrow["\theta"{description}, from=2-2, to=3-3] + \arrow["{{\color{orange}\text{iso}}}"{description}, from=2-4, to=3-3] + \arrow["{{\color{orange}\text{iso}}}"{description}, from=1-3, to=2-2] + \arrow[from=1-3, to=2-4] + \arrow[curve={height=-6pt}, dashed, from=1-1, to=1-3] +\end{tikzcd}\] + + The diagram commutes, since all maps are the induced maps. + By definition of $Y$ is clear that $\pi$ and $\pi'$ separate points in $Y$. + Thus \yaref{lec19:lem1} can be applied. + Since $\theta'$ is an isometric extension, so is $\pi$. + Then $(Y,T)$ is a factor of $(X_{\xi+1}, T)$ by + the maximality of the isometric extension + $(X_{\xi+1 }, T) \to (X_\xi, T)$. + + In particular, + $(X'_{\xi+1}, T)$ is a factor of $(X_{\xi+1}, T)$. +\end{proof} +\begin{example}[{\cite[p. 513]{Furstenberg}}] + Let $X$ be the infinite torus + \[ + X \coloneqq \{(\xi_1, \xi_2, \ldots) : \xi_i \in \C, |\xi_i| = 1\}. + \] + Let $\pi_n$ be the projection to the first $n$ coordinates + and $X_n \coloneqq \pi_n(X)$. + + Let $\tau_1(\xi_1,\xi_2, \ldots, \xi_n, \ldots) = (e^{\i \alpha} \xi_1, \xi_1\xi_2, \ldots, \xi_{n-1}\xi_n, \ldots)$ + where $\frac{\alpha}{\pi}$ is irrational. + Let $T = \langle \tau_1 \rangle \cong \Z$. + + We will show that $(X_n,T)$ is minimal for all $n$, + and so $(X,T)$ is minimal. + Furthermore $(X_{n+1},T)$ is the maximal isometric extension of $(X_n,T)$ + so $(X,T)$ has order $\omega$. +\end{example} diff --git a/inputs/tutorial_09.tex b/inputs/tutorial_09.tex index ee5ef79..2acc229 100644 --- a/inputs/tutorial_09.tex +++ b/inputs/tutorial_09.tex @@ -16,9 +16,8 @@ Material on topological dynamics: \begin{itemize} - \item Terence Tao's notes on ergodic theory 254A: - \url{https://terrytao.wordpress.com/category/teaching/254a-ergodic-theory/} - \item Furstenberg (uses very different notation!). + \item Terence Tao's notes on ergodic theory 254A: \cite{tao} + \item \cite{Furstenberg} (uses very different notation!). \end{itemize} diff --git a/logic.sty b/logic.sty index 22104a7..fda9c2f 100644 --- a/logic.sty +++ b/logic.sty @@ -56,6 +56,7 @@ \usepackage{imakeidx} \makeindex[name = ccode, title = \texttt{C} functions and macros] +\PassOptionsToPackage{hyphens}{url}% \usepackage{hyperref} \usepackage[quotation]{knowledge}[22/02/12] @@ -153,3 +154,5 @@ \newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}} \newcommand\tutorial[3]{\hrule{\color{darkgray}\hfill{\tiny[Tutorial #1, #2]}}} \newcommand\nr[1]{\subsubsection{Exercise #1}} + +\usepackage[bibfile=bibliography/references.bib, imagefile=bibliography/images.bib]{mkessler-bibliography} diff --git a/logic3.tex b/logic3.tex index 52f3cab..01247c9 100644 --- a/logic3.tex +++ b/logic3.tex @@ -42,6 +42,8 @@ \input{inputs/lecture_15} \input{inputs/lecture_16} \input{inputs/lecture_17} +\input{inputs/lecture_18} +\input{inputs/lecture_19} @@ -70,5 +72,7 @@ \PrintVocabIndex +\printbibliography + \end{document}