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bibliography/images.bib
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bibliography/images.bib
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bibliography/references.bib
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bibliography/references.bib
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@misc{tao,
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ISSN = {0003486X},
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URL = {https://terrytao.wordpress.com/category/teaching/254a-ergodic-theory/},
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author = {Terence Tao},
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title = {254A Ergodic Theory},
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urldate = {2024-01-01},
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year = {2008},
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}
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@MISC{801106,
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TITLE = {Closure of a topological group},
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AUTHOR = {Mariano Suarez-Alvarez},
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HOWPUBLISHED = {Mathematics Stack Exchange},
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URL = {https://math.stackexchange.com/q/801106},
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}
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@article{Furstenberg,
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author = {Furstenberg, H.},
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journal = {American journal of mathematics},
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issn = {0002-9327},
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number = {3},
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keywords = {Continuous functions ; Eigenfunctions ; Equivalence relation ; Geometry ; Integers ; Mathematical functions ; Mathematics ; Myelinated nerve fibers ; Structure (category theory) ; Topological compactness ; Topological spaces ; Topological theorems ; Topology},
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language = {eng},
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pages = {477-515},
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publisher = {Johns Hopkins Press},
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volume = {85},
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year = {1963},
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title = {The Structure of Distal Flows},
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}
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@ -1,6 +1,6 @@
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\subsection{Topological Dynamics}
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\begin{fact}[\url{https://math.stackexchange.com/a/801106}]
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\begin{fact}[\cite{801106}]
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\label{fact:topsubgroupclosure}
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Let $H$ be a topological group
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and $G \subseteq H$ a subgroup.
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@ -161,6 +161,7 @@ Recall:
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A flow is \vocab{distal} iff
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it has no proximal pair.
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\end{definition}
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\begin{definition}+
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Let $(T,X)$ and $(T,Y)$ be flows.
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A \vocab{factor map} $\pi\colon (T,X) \to (T,Y)$
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@ -135,9 +135,10 @@ $X$ is always compact metrizable.
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\end{theorem}
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By Zorn's lemma, this will follow from
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\begin{theorem}[Furstenberg]
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\label{thm:l16:3}
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Let $(X, T)$ be a minimal distal flow
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and let $(Y, T)$ be a proper factor,
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i.e.~$(X,T)$ and $(Y,T)$ are note isomorphic.
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and let $(Y, T)$ be a proper factor.
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\footnote{i.e.~$(X,T)$ and $(Y,T)$ are not isomorphic}
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Then there is another factor $(Z,T)$ of $(X,T)$
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which is a proper isometric extension of $Y$.
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@ -1,7 +1,5 @@
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\lecture{17}{2023-12-12}{The Ellis semigroup}
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\subsection{The Ellis semigroup}
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\lecture{17}{2023-12-12}{The Ellis semigroup}
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Let $(X, d)$ be a compact metric space
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and $(X, T)$ a flow.
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@ -29,7 +27,7 @@ $X^{X}$ is a compact Hausdorff space.
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We have $ff_0 \in U_{\epsilon}(x,y)$
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iff $f \in U_\epsilon(x,f_0(y))$.
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\item Fix $x_0 \in X$.
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Then $f \mapsto f(x)$ is continuous.
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Then $f \mapsto f(x_0)$ is continuous.
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\item In general $f \mapsto f_0 \circ f$ is not continuous,
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but if $f_0$ is continuous, then the map is continuous.
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\end{itemize}
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@ -40,23 +38,24 @@ Let $(X,T)$ be a flow.
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Then the \vocab{Ellis semigroup}
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is defined by
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$E(X,T) \coloneqq \overline{T} \subseteq X^X$,
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i.e.~identify $T$ with $x \mapsto tx$
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i.e.~identify $t \in T$ with $x \mapsto tx$
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and take the closure in $X^X$.
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\end{definition}
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$E(X,T)$ is compact and Hausdorff,
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since $X^X$ has these properties.
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Properties of $(X,T)$ translate to
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Properties of $(X,T)$ translate to properties of $E(X,T)$:
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\begin{goal}
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We want to show that if $(X,T)$ is distal,
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then $E(X,T)$ is a group.
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\end{goal}
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\begin{proposition}
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$G$ is a semigroup,
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$E(X,T)$ is a semigroup,
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i.e.~closed under composition.
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\end{proposition}
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\begin{proof}
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Let $G \coloneqq E(X,T)$.
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Take $t \in T$. We want to show that $tG \subseteq G$,
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i.e.~for all $h \in G$ we have $th \in G$.
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@ -97,7 +96,7 @@ Properties of $(X,T)$ translate to
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such that $S \ni x \mapsto xs$ is continuous for all $s$.
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\end{definition}
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\begin{example}
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Ellis semigroup is a compact semigroup.
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The Ellis semigroup is a compact semigroup.
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\end{example}
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\begin{lemma}[Ellis–Numakura]
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@ -181,10 +180,10 @@ However if we pick $y \in Y$, $Ty$ might not be dense.
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\begin{proof}
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Let $G = E(X,T)$.
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Note that for all $x \in X$,
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we have $Gx \subseteq X$ is compact
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we have that $Gx \subseteq X$ is compact
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and invariant under the action of $G$.
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Since $G$ is a group, we have that the orbits partition $X$.%
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Since $G$ is a group, the orbits partition $X$.%
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\footnote{Note that in general this does not hold for semigroups.}
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% Clearly the sets $Gx$ cover $X$. We want to show that they
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236
inputs/lecture_18.tex
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inputs/lecture_18.tex
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@ -0,0 +1,236 @@
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\subsection{Sketch of proof of \yaref{thm:l16:3}}
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\lecture{18}{2023-12-15}{Sketch of proof of \yaref{thm:l16:3}}
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The goal for this lecture is to give a very rough
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sketch of \yaref{thm:l16:3} in the case of $|Z| = 1$.
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% \begin{theorem}[Furstenberg]
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% Let $(X, T)$ be a minimal distal flow
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% and let $(Z,T)$ be a proper factor of $X$%
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% \footnote{i.e.~$(X,T)$ and $(Z,T)$ are not isomorphic.}
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% Then three is another factor $(Y,T)$ of $(X,T)$
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% which is a proper isometric extension of $Z$.
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% \end{theorem}
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Let $(X,T)$ be a distal flow.
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Then $G \coloneqq E(X,T)$ is a group.
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\begin{definition}
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For $x, x' \in X$ define
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\[
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F(x,x') \coloneqq \inf \{d(gx, gx') : g \in G\}.
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\]
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\end{definition}
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\begin{fact}
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\begin{enumerate}[(a)]
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\item $F(x,x') = F(x', x)$,
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\item $F(x,x') \ge 0$ and $F(x,x') = 0$ iff $x = x'$.
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\item $F(gx, gx') = F(x,x')$ since $G$ is a group.
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\item $F$ is an upper semi-continuous function on $X^2$,
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i.e.~$\forall a \in R.~\{(x,x') \in X^2 : F(x,x') < a\} \overset{\text{open}}{\subseteq} X^2$.
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This holds because $F$ is the infimum of continuous functions
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\begin{IEEEeqnarray*}{rCl}
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f_g\colon X^2 &\longrightarrow & \R \\
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(x,x') &\longmapsto & d(gx, gx')
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\end{IEEEeqnarray*}
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for $g \in G$.
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\end{enumerate}
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\end{fact}
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\begin{theoremdef}
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\label{def:ftop}
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The sets
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\[
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U_a(x) \coloneqq \{x' : F(x,x') < a\}
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\]
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form the basis of a topology in $X$.
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This topology is called the \vocab{F-topology} on $X$.
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In this setting, the original topology
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is also called the \vocab{E-topology}.
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\end{theoremdef}
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This will follow from the following lemma:
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\begin{lemma}
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\label{lem:ftophelper}
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Let $F(x,x') < a$.
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Then there exists $\epsilon > 0$ such that
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whenever $F(x',x'') < \epsilon$, then $F(x,x'') < a$.
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\end{lemma}
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\begin{refproof}{def:ftop}
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We have to show that if $U_a(x_1) \cap U_b(x_2) \neq \emptyset$,
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then this intersection is the union
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of sets of this kind.
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Let $x' \in U_a(x_1)$.
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Then by \yaref{lem:ftophelper},
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there exists $\epsilon_1 > 0$ with $U_{\epsilon_1}(x') \subseteq U_a(x_1)$.
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Similarly there exists $\epsilon_2 > 0$
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such that $U_{\epsilon_2}(x') \subseteq U_b(x_2)$.
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So for $\epsilon \le \epsilon_1, \epsilon_2$,
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we get $U_{\epsilon}(x') \subseteq U_a(x_1) \cap U_b(x_2)$.
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\end{refproof}
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\begin{refproof}{lem:ftophelper}%
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\footnote{This was not covered in class.}
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Let $T = \bigcup_n T_n$,% TODO Why does this exist?
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$T_n$ compact, wlog.~$T_n \subseteq T_{n+1}$, and
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let $G(x,x') \coloneqq \{(gx,gx') : g \in G\} \subseteq X \times X$.
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Take $b$ such that $F(x,x') < b < a$.
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Then $U = \{(u,u') \in G(x,x') : d(u,u') < b\}$
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is open in $G(x,x')$
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and since $F(x,x') < b$ we have $U \neq \emptyset$.
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\begin{claim}
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There exists $n$ such that
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\[
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\forall (u,u') \in G(x,x').~T_n(u,u')\cap U \neq \emptyset.
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\]
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\end{claim}
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\begin{subproof}
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Suppose not.
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Then for all $n$, there is $(u_n, u_n') \in G(x,x')$
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with
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\[T_n(u_n, u_n') \subseteq G(x,x') \setminus U.\]
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Note that the RHS is closed.
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For $m > n$ we have
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$T_n(u_m, u'_m) \subseteq G(x,x') \setminus U$
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since $T_n \subseteq T_m$.
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By compactness of $X$,
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there exists $v,v'$ and some subsequence
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such that $(u_{n_k}, u'_{n_k}) \to (v,v')$.
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So for all $n$ we have $T_n(v,v') \subseteq G(x,x') \setminus U$,
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hence $T(v,v') \cap U = \emptyset$,
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so $G(v,v') \cap U = \emptyset$.
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But this is a contradiction as $\emptyset\neq U \subseteq G(v,v')$.
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\end{subproof}
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The map
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\begin{IEEEeqnarray*}{rCl}
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T\times X&\longrightarrow & X \\
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(t,x) &\longmapsto & tx
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\end{IEEEeqnarray*}
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is continuous.
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Since $T_n$ is compact,
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we have that $\{(x,t) \mapsto tx : t \in T_n\}$
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is equicontinuous for all $n$.
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So there is $\epsilon > 0$ such that
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$d(x_1,x_2) < \epsilon \implies d(tx_1, tx_2) < a -b$
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for all $t \in T_n$.
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Suppose now that $F(x', x'') < \epsilon$.
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Then there is $t_0 \in T$ such that $d(t_0x', t_0x'') < \epsilon$,
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hence $d(t t_0x', t t_0 x'') < a-b$ for all $t \in T_n$.
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Since $(t_0x, t_0x') \in G(x,x')$,
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there is $t_1 \in T_n$
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with $(t_1t_0x, t_1t_0x') \in U$,
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i.e.~$d(t_1t_0x, t_1t_0x') < b$
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and therefore
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$F(x,x'') = d(t_1t_0x, t_1t_0x'') < a$.
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\end{refproof}
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Now assume $Z = \{\star\}$.
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We want to sketch a proof of \yaref{thm:l16:3} in this case,
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i.e.~show that if $(Z,T)$ is a proper factor of a minimal distal flow
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$(X,T)$ then there is another factor $(Y,T)$ of $(X,T)$
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which is a proper isometric extension of $Z$.
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\begin{proof}[sketch] % TODO: Think about this
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\leavevmode
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\begin{enumerate}[1.]
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\item For $x \in X$ define
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\begin{IEEEeqnarray*}{rCl}
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F_x\colon X &\longrightarrow & \R \\
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x' &\longmapsto & F(x,x').
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\end{IEEEeqnarray*}
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\item Define an equivalence relation on $X$,
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by $x_1 \sim x_2 :\iff \{x \in X : F_{x_1}(x) = F_{x_2}(x)\}$
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is comeager in $X$\footnote{with respect to the E-topology}.
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Then for all $g \in G$ we have
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$x_1 \sim x_2 \implies gx_1 \sim ~ gx_2$.
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Let $M \coloneqq \{[x]_{\sim } : x \in X\} = \faktor{X}{\sim}$
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bet the quotient space.
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It is compact, second countable and Hausdorff.
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Let $\pi\colon X\to M$ denote the quotient map.
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\item $(Y,T) \mathbin{\text{\reflectbox{$\coloneqq$}}} (M,T)$
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is an isometric flow:
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\begin{enumerate}
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\item For $a > 0$, $x,x' \in X$ let
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\[
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W(x,x') \coloneqq \{g \in G : F(x, gx') < a\}.
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\]
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This turns out to be a subbasis of a topology
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which is coarser than the original topology on $G$.
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The new topology makes $G$ compact.
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\item Let $\theta(g)$ be the transformation of $M$
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defined by $\theta(g) \pi(x) = \pi(gx)$.
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This is well defined.
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Let $H = \theta(G)$.
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This is just a quotient of $G$, $g \mapsto \theta(g)$
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may not be injective.
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\item One can show that $H$ is a topological group and $(M,H)$
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is a flow.\footnote{This is non-trivial.}
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\item Since $H$ is compact,
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$(M,H)$ is equicontinuous, %\todo{We didn't define this}
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i.e.~it is isometric.
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In particular, $(M,T)$ is isometric.
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\end{enumerate}
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\item $M \neq \{\star\}$, i.e.~$(M,T)$ is non-trivial:
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Suppose towards a contradiction that $M = \{\star\}$,
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i.e.~$x_1 \sim x_2$ for all $x_1,x_2 \in X$.
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Fix $x_2$. For every $x_1 \in X$
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we have that
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\[
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\{x : F(x_1,x) = F(x_2,x)\}
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\]
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is comeager.
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Let $x_1$ be a point of continuity of $F_{x_2}$.
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Let $\langle a_n : n < \omega \rangle$ be a sequence
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of elements that set, i.e.~$F(x_1, a_n) = F(x_2, a_n)$,
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such that $a_n \to x_1$.
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So by the continuity of $F_{x_2}$ at $x_1$
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\begin{IEEEeqnarray*}{rCl}
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\lim_{n \to \infty} F(x_2, a_n) &=& F(x_2, x_1)
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\end{IEEEeqnarray*}
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and by the definition of $F$
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\begin{IEEEeqnarray*}{rCl}
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\lim_{n \to \infty} F(x_1,a_n) &=& F(x_1,x_1) = 0.
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\end{IEEEeqnarray*}
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So
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\[
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F(x_2,x_1) = \lim_{n \to \infty} F(x_2, a_n) = \lim_{n \to \infty}
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F(x_1,a_n) = 0
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\]
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and by distality we get $x_1 = x_2$.
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Since almost all points of $X$
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are points of continuity of $F_{x_2}$
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(\yaref{thm:usccomeagercont})
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this implies that $X \setminus \{x_2\}$ is meager.
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But then $X = \{\star\} \lightning$.
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\end{enumerate}
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\end{proof}
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\begin{theorem}\footnote{Not covered in class}
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\label{thm:usccomeagercont}
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Let $X$ be a metric space
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and $\Gamma\colon X \to \R$ be upper semicontinuous.
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Then the set of continuity points of $\Gamma$ is comeager.
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\todo{Missing figure: upper semicontinuous function}
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\end{theorem}
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\begin{proof}
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Take $x$ such that $\Gamma$ is not continuous at $x$.
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Then there is an $\epsilon > 0$
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and $x_n \to x$ such that
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$\Gamma(x_n) + \epsilon \le \Gamma(x)$.
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Take $q \in \Q$ such that $\Gamma(x) - \epsilon < q < \Gamma(x)$.
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Then let
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\[
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B_q \coloneqq \{a \in X : \Gamma(a) \ge q\}.
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\]
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$X \setminus B_q = \{a \in X : \Gamma(a) < q\}$
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is open, i.e.~$B_q$ is closed.
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Note that $x \in F_q \coloneqq B_q \setminus B_q^\circ$
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and $B_q \setminus B_q^\circ$ is nwd
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as it is closed and has empty interior,
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so $\bigcup_{q \in \Q} F_q$ is meager.
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\end{proof}
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292
inputs/lecture_19.tex
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292
inputs/lecture_19.tex
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|
@ -0,0 +1,292 @@
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\subsection{The order of a flow}
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\lecture{19}{2023-12-19}{Orders of flows}
|
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|
||||
See also \cite[\href{https://terrytao.wordpress.com/2008/01/24/254a-lecture-6-isometric-systems-and-isometric-extensions/}{Lecture 6}]{tao}.
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|
||||
|
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\begin{definition}+
|
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Let $X,Y$ be metric spaces. A family $F$ of functions $X \to Y$
|
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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}
|
||||
|
|
@ -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}
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -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}
|
||||
|
|
|
|||
|
|
@ -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}
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user