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12 changed files with 34 additions and 358 deletions
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@ -12,9 +12,6 @@ If you find errors or want to improve something,
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please send me a message:\\
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please send me a message:\\
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\texttt{lecturenotes@jrpie.de}.
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\texttt{lecturenotes@jrpie.de}.
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Many thanks to \textsc{Mirko Bartsch} for providing notes on lectures
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I could not attend!
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This notes follow the way the material was presented in the lecture rather
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This notes follow the way the material was presented in the lecture rather
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closely. Additions (e.g.~from exercise sheets)
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closely. Additions (e.g.~from exercise sheets)
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and slight modifications have been marked with $\dagger$.
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and slight modifications have been marked with $\dagger$.
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@ -61,9 +61,7 @@ However the converse of this does not hold.
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Then $\{x_0\}$ is dense in $X$, but $X$ is not second countable.
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Then $\{x_0\}$ is dense in $X$, but $X$ is not second countable.
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\end{example}
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\end{example}
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\begin{example}[Sorgenfrey line]
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\begin{example}[Sorgenfrey line]
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Consider $\R$ with the topology given by the basis $\{[a,b) : a,b \in \R\}$.
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\todo{Counterexamples in Topology}
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This is T3, but not second countable
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and not metrizable.
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\end{example}
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\end{example}
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\begin{fact}
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\begin{fact}
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@ -100,7 +98,6 @@ However the converse of this does not hold.
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then $X$ is metrisable.
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then $X$ is metrisable.
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\end{theorem}
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\end{theorem}
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\begin{absolutelynopagebreak}
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\begin{fact}
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\begin{fact}
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If $X$ is a compact Hausdorff space,
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If $X$ is a compact Hausdorff space,
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the following are equivalent:
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the following are equivalent:
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@ -110,7 +107,6 @@ However the converse of this does not hold.
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\item $X$ is second countable.
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\item $X$ is second countable.
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\end{itemize}
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\end{itemize}
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\end{fact}
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\end{fact}
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\end{absolutelynopagebreak}
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\subsection{Some facts about polish spaces}
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\subsection{Some facts about polish spaces}
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@ -179,7 +175,6 @@ suffices to show that open balls in one metric are unions of open balls in the o
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\end{proof}
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\end{proof}
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\begin{definition}[Our favourite Polish spaces]
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\begin{definition}[Our favourite Polish spaces]
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\leavevmode
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\begin{itemize}
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\begin{itemize}
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\item $2^{\omega}$ is called the \vocab{Cantor set}.
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\item $2^{\omega}$ is called the \vocab{Cantor set}.
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(Consider $2$ with the discrete topology)
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(Consider $2$ with the discrete topology)
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@ -190,7 +185,7 @@ suffices to show that open balls in one metric are unions of open balls in the o
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\end{itemize}
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\end{itemize}
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\end{definition}
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\end{definition}
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\begin{proposition}
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\begin{proposition}
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Let $X$ be a separable, metrisable topological space\footnote{e.g.~Polish, but we don't need completeness.}.
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Let $X$ be a separable, metrisable topological space.
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Then $X$ topologically embeds into the
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Then $X$ topologically embeds into the
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\vocab{Hilbert cube},
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\vocab{Hilbert cube},
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i.e. there is an injective $f: X \hookrightarrow [0,1]^{\omega}$
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i.e. there is an injective $f: X \hookrightarrow [0,1]^{\omega}$
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@ -232,9 +227,7 @@ suffices to show that open balls in one metric are unions of open balls in the o
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$f^{-1}$ is continuous.
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$f^{-1}$ is continuous.
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\end{claim}
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\end{claim}
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\begin{subproof}
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\begin{subproof}
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Consider $B_{\epsilon}(x_n) \subseteq X$ for some $n \in \N$, $\epsilon > 0$.
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\todo{Exercise!}
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Then $f(U) = f(X) \cap [0,1]^{n} \times [0,\epsilon) \times [0,1]^{\omega}$
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is open\footnote{as a subset of $f(X)$!}.
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\end{subproof}
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\end{subproof}
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\end{proof}
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\end{proof}
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\begin{proposition}
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\begin{proposition}
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@ -13,6 +13,7 @@
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\begin{proof}
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\begin{proof}
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Let $C \subseteq X$ be closed.
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Let $C \subseteq X$ be closed.
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Let $U_{\frac{1}{n}} \coloneqq \{x | d(x, C) < \frac{1}{n}\}$.
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Let $U_{\frac{1}{n}} \coloneqq \{x | d(x, C) < \frac{1}{n}\}$.
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\todo{Exercise}
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Clearly $C \subseteq \bigcap U_{\frac{1}{n}}$.
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Clearly $C \subseteq \bigcap U_{\frac{1}{n}}$.
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Let $x \in \bigcap U_{\frac{1}{n}}$.
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Let $x \in \bigcap U_{\frac{1}{n}}$.
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Then $\forall n .~ \exists x_n\in C.~d(x,x_n) < \frac{1}{n}$.
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Then $\forall n .~ \exists x_n\in C.~d(x,x_n) < \frac{1}{n}$.
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@ -53,7 +54,7 @@
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then there exists a complete metric on $Y$.
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then there exists a complete metric on $Y$.
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\end{claim}
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\end{claim}
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\begin{refproof}{psubspacegdelta:c1}
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\begin{refproof}{psubspacegdelta:c1}
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Let $Y = U$ be open in $X$.
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Let $Y = U$be open in $X$.
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Consider the map
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Consider the map
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\begin{IEEEeqnarray*}{rCl}
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\begin{IEEEeqnarray*}{rCl}
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f_U\colon U &\longrightarrow &
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f_U\colon U &\longrightarrow &
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@ -1,6 +1,6 @@
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\lecture{03}{2023-10-17}{Embedding of the cantor space into polish spaces}
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\lecture{03}{2023-10-17}{Embedding of the cantor space into polish spaces}
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\subsection{Trees}
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% ? \subsection{Trees} TODO
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\begin{notation}
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\begin{notation}
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@ -255,7 +255,9 @@
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[To be continued]
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\phantom\qedhere
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\phantom\qedhere
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\end{refproof}
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\end{refproof}
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@ -1,7 +1,6 @@
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\lecture{04}{2023-10-20}{}
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\lecture{04}{2023-10-20}{}
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\begin{remark}
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\begin{remark}
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Some of the $F_s$ might be empty.
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Some of $F_s$ might be empty.
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\end{remark}
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\end{remark}
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\begin{refproof}{thm:bairetopolish}
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\begin{refproof}{thm:bairetopolish}
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@ -30,6 +29,7 @@
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\begin{refproof}{thm:bairetopolish:c1}
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\begin{refproof}{thm:bairetopolish:c1}
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Let $x_n$ be a series in $D$
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Let $x_n$ be a series in $D$
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converging to $x$ in $\cN$.
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converging to $x$ in $\cN$.
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Then $x \in \cN$.
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\begin{claim}
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\begin{claim}
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$(f(x_n))$ is Cauchy.
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$(f(x_n))$ is Cauchy.
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\end{claim}
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\end{claim}
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@ -40,11 +40,12 @@
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$x_m\defon{N} = x\defon{N}$.
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$x_m\defon{N} = x\defon{N}$.
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Then for all $m, n \ge M$,
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Then for all $m, n \ge M$,
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we have that $f(x_m), f(x_n) \in F_{x\defon{N}}$.
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we have that $f(x_m), f(x_n) \in F_{x\defon{N}}$.
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So $d(f(x_m), f(x_n)) < \epsilon$, i.e.~$(f(x_n))$ is Cauchy.
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So $d(f(x_m), f(x_n)) < \epsilon$
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we have that $(f(x_n))$ is Cauchy.
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\end{subproof}
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Since $(X,d)$ is complete,
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Since $(X,d)$ is complete,
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there exists $y = \lim_n f(x_n)$.
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there exists $y = \lim_n f(x_n)$.
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Since for all $m \ge M$, $f(x_m) \in F_{x\defon{N}}$,
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Since for all $m \ge M$, $f(x_m) \in F_{x\defon{N}}$,
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we get that $y \in \overline{F_{x\defon{N}}}$.
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we get that $y \in \overline{F_{x\defon{N}}}$.
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@ -53,6 +54,7 @@
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Hence
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Hence
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\[y \in \bigcap_{n} \overline{F_{x\defon{n}}} = \bigcap_{n} F_{x\defon{n}},\]
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\[y \in \bigcap_{n} \overline{F_{x\defon{n}}} = \bigcap_{n} F_{x\defon{n}},\]
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i.e.~$y \in D$ and $y = f(x)$.
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i.e.~$y \in D$ and $y = f(x)$.
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\end{subproof}
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\end{refproof}
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\end{refproof}
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We extend $f$ to $g\colon\cN \to X$
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We extend $f$ to $g\colon\cN \to X$
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@ -68,7 +70,7 @@
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(i.e.~$r = \id$ on $D$ and $r$ is a continuous surjection).
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(i.e.~$r = \id$ on $D$ and $r$ is a continuous surjection).
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Then $g \coloneqq f \circ r$.
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Then $g \coloneqq f \circ r$.
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To construct $r$, we will define
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To construct $r$, we will define by induction
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$\phi\colon \N^{<\N} \to S$ by induction on the length
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$\phi\colon \N^{<\N} \to S$ by induction on the length
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such that
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such that
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\begin{itemize}
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\begin{itemize}
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@ -15,7 +15,6 @@ sketch of \yaref{thm:l16:3} in the case of $|Z| = 1$.
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Let $(X,T)$ be a distal flow.
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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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Then $G \coloneqq E(X,T)$ is a group.
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\begin{definition}
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\begin{definition}
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\label{def:F}
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For $x, x' \in X$ define
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For $x, x' \in X$ define
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\[
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\[
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F(x,x') \coloneqq \inf \{d(gx, gx') : g \in G\}.
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F(x,x') \coloneqq \inf \{d(gx, gx') : g \in G\}.
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@ -274,7 +274,6 @@ More generally we can show:
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$(X'_{\xi+1}, T)$ is a factor of $(X_{\xi+1}, T)$.
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$(X'_{\xi+1}, T)$ is a factor of $(X_{\xi+1}, T)$.
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\end{proof}
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\end{proof}
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\begin{example}[{\cite[p. 513]{Furstenberg}}]
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\begin{example}[{\cite[p. 513]{Furstenberg}}]
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\label{ex:19:inftorus}
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Let $X$ be the infinite torus
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Let $X$ be the infinite torus
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\[
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\[
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X \coloneqq \{(\xi_1, \xi_2, \ldots) : \xi_i \in \C, |\xi_i| = 1\}.
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X \coloneqq \{(\xi_1, \xi_2, \ldots) : \xi_i \in \C, |\xi_i| = 1\}.
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@ -1,196 +0,0 @@
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\lecture{20}{2024-01-09}{The infinite Torus}
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\begin{example}
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\footnote{This is the same as \yaref{ex:19:inftorus},
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but with new notation.}
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Let $X = (S^1)^{\N}$\footnote{We identify $S^1$ and $\faktor{\R}{\Z}$.}
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and consider $\left(X, \Z \right)$
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where the action is generated by
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\[
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\tau\colon (x_1,x_2,x_3,\ldots) \mapsto(x_1 + \alpha, x_1 + x_2, x_2 + x_3, \ldots)
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\]
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for some irrational $\alpha$.
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\end{example}
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\begin{remark}+
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Note that we can identify $S^1$ with a subset of $\C$ (and use multiplication)
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or with $\faktor{\R}{\Z}$ (and use addition).
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In the lecture both notations were used.% to make things extra confusing.
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Here I'll try to only use multiplicative notation.
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\end{remark}
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We will be studying projections to the first $d$ coordinates,
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i.e.
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\[
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\tau_d \colon (x_1,\ldots,x_d) \mapsto (e^{\i \alpha} x_1, x_1x_2, \ldots, x_{d-1}x_d).
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\]
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$\tau_d$ is called the \vocab{$d$-skew shift}.
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For $d = 1$ we get the circle rotation $x \mapsto e^{\i \alpha} x$.
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\begin{fact}
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\label{fact:tau1minimal}
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The circle rotation $x \mapsto e^{\i \alpha} x$ is minimal.
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In fact, every subgroup of $S^1$ is either dense in $S^1$
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or it is of the form
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\[
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H_m \coloneqq \{x \in S^1 : x^m = 0\}
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\]
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for some $m \in \Z$.
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\end{fact}
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\todo{Homework!}
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We will show that $\tau_d$ is minimal for all $d$,
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i.e.~every orbit is dense.
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From this it will follow that $\tau$ is minimal.
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Let $\pi_n\colon X \to (S^1)^n$ be the projection to the first $n$
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coordinates.
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\begin{lemma}
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Let $x,x' \in X$ with $\pi_n(x) = \pi_n(x')$
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for some $n$.
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Then there is a sequence of points $x_k$ with
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\[\pi_{n-1}(x_k) = \pi_{n-1}(x) = \pi_{n-1}(x')\]
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for all $k$
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and
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\[
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F(x_k, x) \xrightarrow{k \to \infty} 0,
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F(x_k, x') \xrightarrow{k \to \infty} 0,
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\]
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where $F$ is as in \yaref{def:F},
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i.e.~$F(a,b) = \inf_{n \in \Z} d(\tau^n a, \tau^n b)$,
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where $d$ is the metric on $X$,
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$d((x_i), (y_i)) = \max_n \frac{1}{2^n} | x_n - y_n|$.% TODO use multiplicative notation
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\end{lemma}
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\begin{proof}
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Let
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\begin{IEEEeqnarray*}{rCl}
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x &=& (\alpha^0_1, \alpha^0_2, \ldots, \alpha^0_{n-1}, \alpha_n, \alpha_{n+1}, \alpha_{n+2},\ldots)\\
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x' &=& (\alpha^0_1, \alpha^0_2, \ldots, \alpha^0_{n-1}, \alpha_n, \alpha'_{n+1}, \alpha'_{n+2},\ldots).\\
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\end{IEEEeqnarray*}
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We will choose $x_k$ of the form
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\[
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(\alpha^0_1, \alpha^0_2, \ldots, \alpha^0_{n-1} \alpha_n e^{\i \beta_k}, \alpha_{n+1}, \alpha_{n+2}, \ldots),
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\]
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where $\beta_k$ is such that $\frac{\beta_k}{\pi}$ is irrational
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and $|\beta_k| < 2^{-k}$.
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Fix a sequence of such $\beta_k$.
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Then
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\[d(x_k,x) = 2^{-n} |e^{\i \beta_k} - 1| < 2^{-n-k} \xrightarrow{k\to \infty} 0.\]
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In particular $F(x_k, x) \to 0$.
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We want to show that $F(x_k, x') < 2^{-n-k}$.
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For $u, u' \in X$,
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$u = (\xi_n)_{n \in \N}$,
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$u' = (\xi'_n)_{n \in \N}$,
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let $\frac{u}{u'} = (\frac{\xi_n}{\xi'_n})_{n \in \N}$
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($X$ is a group).
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We are interested in $F(x_k, x') = \inf_m d(\tau^m x_k, \tau^m x')$,
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but it is easier to consider the distance between
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their quotient and $1$.
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Consider
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\[
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w_k \coloneqq \frac{x_k}{x'} = (\underbrace{1,\ldots,1}_{n-1}, e^{\i \beta_k}, \overbrace{\frac{\alpha_{n+1}}{\alpha'_{n+1}}, \frac{\alpha_{n+2}}{\alpha'_{n+2}}, \ldots}^{\mathclap{\text{not interesting}}}).
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\]
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\begin{claim}
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$F(x_k, x') = \inf_m d(\sigma^m(w_k), 1)$,
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where $\sigma(\xi_1, \xi_2, \ldots) = (\xi_1, \xi_1\xi_2, \xi_2\xi_3, \ldots)$.
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\end{claim}
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\begin{subproof}
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We have
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\begin{IEEEeqnarray*}{rCl}
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F(u,u') &=& \inf_m d(\tau^m u, \tau^m u')\\
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&=& \inf_m d(\frac{\tau^m u}{\tau^m u'}, 1)\\
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&=& \inf_m d(\sigma^m\left( \frac{u}{u'} \right), 1).
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\end{IEEEeqnarray*}
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\end{subproof}
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Fix $k$. Let $w^\ast = (1,\ldots,1, e^{\i \beta_k}, 1, \ldots)$.
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By minimality of $(X,T)$ for any $\epsilon >0$,
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there exists $m \in \Z$ such that
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$d(\sigma^m w_k, w^\ast) < \epsilon$.
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% TODO Think about this
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|
||||||
Then
|
|
||||||
\begin{IEEEeqnarray*}{rCl}
|
|
||||||
\inf_m d(\sigma^m w_k, 1) &\le & \inf_m d(\sigma^m w_k, w^\ast) + d(w^\ast, 1)\\
|
|
||||||
&\le & 2^{-n} | e^{\i \beta_k}- 1|\\
|
|
||||||
&<& 2^{-n-k}.
|
|
||||||
\end{IEEEeqnarray*}
|
|
||||||
\end{proof}
|
|
||||||
|
|
||||||
|
|
||||||
\begin{definition}
|
|
||||||
For every continuous $f\colon S^1 \to S^1$, the
|
|
||||||
\vocab{winding number} $[f] \in \Z$
|
|
||||||
is the unique integer such that $f$ is homotopic%
|
|
||||||
\footnote{$f\colon Y \to Z$ and $g\colon Y \to Z$ are homotopic
|
|
||||||
iff there is $H\colon Y \times [0,1] \to \Z$
|
|
||||||
continuous such that $H(\cdot ,0) = f$ and $H(\cdot ,1) = g$.}
|
|
||||||
to the map
|
|
||||||
$x \mapsto x^{n}$.
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\begin{remark}
|
|
||||||
Note that for
|
|
||||||
\begin{IEEEeqnarray*}{rCl}
|
|
||||||
\sigma\colon (S^1)^d &\longrightarrow & S^1 \\
|
|
||||||
(x_1,\ldots,x_d) &\longmapsto & x_d
|
|
||||||
\end{IEEEeqnarray*}
|
|
||||||
we have that $T = \tau_{d+1}$,
|
|
||||||
where
|
|
||||||
\begin{IEEEeqnarray*}{rCl}
|
|
||||||
T\colon (S^1)^d \times S^1 &\longrightarrow & (S^1)^d \times S^1 \\
|
|
||||||
(y, x_{d+1}) &\longmapsto & (\tau_d(y), \sigma(y) x_{d+1}).
|
|
||||||
\end{IEEEeqnarray*}
|
|
||||||
\end{remark}
|
|
||||||
\begin{theorem}
|
|
||||||
\label{thm:taudminimal:help}
|
|
||||||
For every $d$ if $\tau_d$\footnote{more formally $((S^1)^d, \langle \tau_d \rangle)$}
|
|
||||||
is minimal, then $\tau_{d+1}$ is minimal.
|
|
||||||
\end{theorem}
|
|
||||||
\begin{corollary}
|
|
||||||
$\tau_d$ is minimal for all $d$.
|
|
||||||
\end{corollary}
|
|
||||||
\begin{proof}
|
|
||||||
$\tau_1$ is minimal (\yaref{fact:tau1minimal}).
|
|
||||||
Apply \yaref{thm:taudminimal:help}.
|
|
||||||
\end{proof}
|
|
||||||
\begin{corollary}
|
|
||||||
Since all the $\tau_d$ are minimal,
|
|
||||||
$\tau$ is minimal.
|
|
||||||
\end{corollary}
|
|
||||||
\begin{proof}
|
|
||||||
This follows from the definition of the product topology,
|
|
||||||
since for a basic open set $U = U_1 \times \ldots \times U_d \times (S^1)^{\infty}$
|
|
||||||
it suffices to analyze the first $d$ coordinates.
|
|
||||||
\end{proof}
|
|
||||||
|
|
||||||
\begin{refproof}{thm:taudminimal:help}
|
|
||||||
Let $s \coloneqq \tau_d$ and $Y \coloneqq (S^1)^d$.
|
|
||||||
Consider
|
|
||||||
\begin{IEEEeqnarray*}{rCl}
|
|
||||||
\gamma\colon S^1 &\longrightarrow & Y \\
|
|
||||||
x &\longmapsto & (x,x,\ldots,x)
|
|
||||||
\end{IEEEeqnarray*}
|
|
||||||
\begin{enumerate}[(a)]
|
|
||||||
\item $\gamma$ and $s \circ \gamma$ are homotopic
|
|
||||||
via
|
|
||||||
\begin{IEEEeqnarray*}{rCl}
|
|
||||||
H\colon S^1 \times [0,1] &\longrightarrow & (S^1)^d \\
|
|
||||||
(x, t)&\longmapsto & (x e^{\i t \alpha}, x^{t+1}, x^{t+1}, x^{t+1},\ldots, x^{t+1})
|
|
||||||
\end{IEEEeqnarray*}
|
|
||||||
\item For all $m \in \Z \setminus \{0\}$, we have
|
|
||||||
$[x \mapsto \left(\sigma(\gamma(x))\right)^m] = m \neq 0$,
|
|
||||||
since $\sigma(\gamma(x)) = \sigma((x,\ldots,x)) = x$.
|
|
||||||
\end{enumerate}
|
|
||||||
|
|
||||||
[to be continued]
|
|
||||||
\phantom\qedhere
|
|
||||||
|
|
||||||
\end{refproof}
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
|
@ -13,9 +13,9 @@
|
||||||
\begin{fact}
|
\begin{fact}
|
||||||
\begin{itemize}
|
\begin{itemize}
|
||||||
\item Let $X$ be a topological space.
|
\item Let $X$ be a topological space.
|
||||||
Then $X$ \nth{2} countable $\implies$ X separable.
|
Then $X$ 2nd countable $\implies$ X separable.
|
||||||
\item If $X$ is a metric space and separable,
|
\item If $X$ is a metric space and separable,
|
||||||
then $X$ is \nth{2} countable.
|
then $X$ is 2nd countable.
|
||||||
\end{itemize}
|
\end{itemize}
|
||||||
\end{fact}
|
\end{fact}
|
||||||
\begin{proof}
|
\begin{proof}
|
||||||
|
@ -32,18 +32,18 @@
|
||||||
\begin{fact}
|
\begin{fact}
|
||||||
Let $X$ be a metric space.
|
Let $X$ be a metric space.
|
||||||
If $X$ is Lindelöf,
|
If $X$ is Lindelöf,
|
||||||
then it is \nth{2} countable.
|
then it is 2nd countable.
|
||||||
\end{fact}
|
\end{fact}
|
||||||
\begin{proof}
|
\begin{proof}
|
||||||
For all $q \in \Q$
|
For all $q \in \Q$
|
||||||
consider the cover $B_q(x), x \in X$
|
Consider the cover $B_q(x), x \in X$
|
||||||
and choose a countable subcover.
|
and choose a countable subcover.
|
||||||
The union of these subcovers is
|
The union of these subcovers is
|
||||||
a countable base.
|
a countable base.
|
||||||
\end{proof}
|
\end{proof}
|
||||||
\begin{fact}
|
\begin{fact}
|
||||||
Let $X$ be a topological space.
|
Let $X$ be a topological space.
|
||||||
If $X$ is \nth{2} countable,
|
If $X$ is 2nd countable,
|
||||||
then it is Lindelöff.
|
then it is Lindelöff.
|
||||||
\end{fact}
|
\end{fact}
|
||||||
\begin{proof}
|
\begin{proof}
|
||||||
|
@ -60,7 +60,7 @@
|
||||||
\end{proof}
|
\end{proof}
|
||||||
\begin{remark}
|
\begin{remark}
|
||||||
For metric spaces the notions
|
For metric spaces the notions
|
||||||
of being \nth{2} countable, separable
|
of being 2nd countable, separable
|
||||||
and Lindelöf coincide.
|
and Lindelöf coincide.
|
||||||
|
|
||||||
In arbitrary topological spaces,
|
In arbitrary topological spaces,
|
||||||
|
|
|
@ -1,113 +0,0 @@
|
||||||
\tutorial{11}{2024-01-09}{}
|
|
||||||
|
|
||||||
An equivalent definition of subflows can be given as follows:
|
|
||||||
\begin{definition}
|
|
||||||
Let $(X,T)$ be a flow with action $\alpha_x$.
|
|
||||||
Let $Y \subseteq X$ be a compact subspace of $X$.
|
|
||||||
If $Y$ is invariant under $\alpha_x$, we say that
|
|
||||||
$(Y,T)$ (with action $\alpha_x\defon{T \times Y}$
|
|
||||||
is a subflow of $(X,T)$.
|
|
||||||
\end{definition}
|
|
||||||
|
|
||||||
\begin{example}[Flows with a non-closed orbit]
|
|
||||||
\begin{enumerate}[1.]
|
|
||||||
\item Consider $(S^1, \Z)$
|
|
||||||
with action given by $1 \cdot x = x + c$ for
|
|
||||||
a fixed $c \in \R\setminus\Q$.\footnote{We identify $S^1$ and $\faktor{\R}{\Z}$.}
|
|
||||||
Then the orbit of $0$, $\{nc : n \in \Z\}$ is dense but consists only of irrationals
|
|
||||||
(except $0$),
|
|
||||||
so it is not closed.
|
|
||||||
\item Consider $(S^1, \Q)$ with action $qx \coloneqq x + q$.
|
|
||||||
The orbit of $0$, $\faktor{\Q}{\Z} \subseteq S^1$,
|
|
||||||
is dense but not closed.
|
|
||||||
|
|
||||||
$(S^1,\Q)$ is minimal.
|
|
||||||
\end{enumerate}
|
|
||||||
\end{example}
|
|
||||||
\begin{example}[\vocab{Left Bernoulli shift}]
|
|
||||||
Consider $(\{0,1\}^{\Z}, T)$,
|
|
||||||
where $T = \Z$ and the action is given by
|
|
||||||
\begin{IEEEeqnarray*}{rCl}
|
|
||||||
\Z \times \{0,1\}^{\Z}&\longrightarrow & \Z \\
|
|
||||||
(m, (x_n)_{n \in \Z})&\longmapsto & (x_{n+m})_{n \in \Z}.
|
|
||||||
\end{IEEEeqnarray*}
|
|
||||||
|
|
||||||
The orbit of $z \coloneqq (0)_{n \in \Z}$ consist of only on point.
|
|
||||||
In particular it is closed.
|
|
||||||
|
|
||||||
Let $x \coloneqq ( [n = 0])_{n \in \Z}$.
|
|
||||||
Then $Tx = \{([n = m])_{n \in \Z} | m \in \Z\}$.
|
|
||||||
Clearly $z \not\in Tx$.
|
|
||||||
\begin{claim}
|
|
||||||
$z \in \overline{Tx}$
|
|
||||||
\end{claim}
|
|
||||||
\begin{proof}
|
|
||||||
Consider a basic open $z \in U_I = \{y : y_i = 0, i \in I\}$
|
|
||||||
where $I \subseteq \Z$ is finite.
|
|
||||||
Then $U_I \cap Tx \neq \emptyset$
|
|
||||||
as we can shift the $1$ out of $I$,
|
|
||||||
i.e.~$(\max I + 1) x \in U_I$.
|
|
||||||
\end{proof}
|
|
||||||
\end{example}
|
|
||||||
|
|
||||||
Flows are always on non-empty spaces $X$.
|
|
||||||
\begin{fact}
|
|
||||||
Consider a flow $(X,T)$.
|
|
||||||
The following are equivalent:
|
|
||||||
\begin{enumerate}[(i)]
|
|
||||||
\item Every $T$-orbit is dense.
|
|
||||||
\item There is no proper subflow,
|
|
||||||
\end{enumerate}
|
|
||||||
If these conditions hold, the flow is called \vocab{minimal}.
|
|
||||||
\end{fact}
|
|
||||||
\begin{proof}
|
|
||||||
(i) $\implies$ (ii):
|
|
||||||
Let $(Y,T)$ be a subflow of $(X,T)$.
|
|
||||||
take $y \in Y$. Then $Ty$ is dense in mKX.
|
|
||||||
But $Ty \subseteq Y$, so $Y$ is dense in $X$.
|
|
||||||
Since $Y$ is closed, we get $Y = X$.
|
|
||||||
|
|
||||||
(ii) $\implies$ (i):
|
|
||||||
Take $x \in X$. Consider $Tx$.
|
|
||||||
It suffices to show that $\overline{Tx}$ is a subflow.
|
|
||||||
Clearly $\overline{Tx}$ is closed,
|
|
||||||
so it suffices to show that it is $T$-invariant.
|
|
||||||
Let $y \in \overline{Tx}$ and $t \in T$.
|
|
||||||
Take $ty \in U \overset{\text{open}}{\subseteq} X$.
|
|
||||||
Since $t^{-1}$ acts as a homeomorphism
|
|
||||||
we have $y \in t^{-1} U \overset{\text{open}}{\subseteq} X$.
|
|
||||||
We find some $t'x \in t^{-1}U$ since $y \in \overline{Tx}$.
|
|
||||||
So $tt'x \in Tx \cap U$.
|
|
||||||
\end{proof}
|
|
||||||
|
|
||||||
\begin{fact}
|
|
||||||
Every flow $(X,T)$ contains a minimal subflow.
|
|
||||||
\end{fact}
|
|
||||||
\begin{proof}
|
|
||||||
We use Zorn's lemma:
|
|
||||||
Let $S$ be the set of all subflows of $(X,T)$
|
|
||||||
ordered by $Y \le Y' :\iff Y \supseteq Y'$.
|
|
||||||
We need to show that for a chain $\langle Y_i : i \in I \rangle$,
|
|
||||||
there exists a lower bound.
|
|
||||||
Consider $\bigcap_{i \in I} Y_i$. This a subflow:
|
|
||||||
\begin{itemize}
|
|
||||||
\item It is closed as it is an intersection of closed sets.
|
|
||||||
\item It is $T$-invariant, since each of the $Y_i$ is.
|
|
||||||
\item It is non-empty by \yaref{tut10fact}.
|
|
||||||
\end{itemize}
|
|
||||||
\end{proof}
|
|
||||||
\begin{fact}
|
|
||||||
\label{tut10fact}
|
|
||||||
Let $X$ be a topological space.
|
|
||||||
Then $X$ is compact iff every family of closed sets with
|
|
||||||
FIP\footnote{finite intersection property, i.e.~the intersection of every finite sub-family is non-empty}
|
|
||||||
has non-empty intersection.
|
|
||||||
\end{fact}
|
|
||||||
\begin{proof}
|
|
||||||
Note that families of
|
|
||||||
closed sets correspond to families of open sets by taking complements.
|
|
||||||
A family of open sets is a cover iff the corresponding family
|
|
||||||
has empty intersection,
|
|
||||||
and is admits a finite subcover iff the corresponding family
|
|
||||||
has the FIP.
|
|
||||||
\end{proof}
|
|
|
@ -50,13 +50,3 @@
|
||||||
\RequirePackage{mkessler-mathfixes} % Load this last since it renews behaviour
|
\RequirePackage{mkessler-mathfixes} % Load this last since it renews behaviour
|
||||||
\DeclareMathOperator{\inter}{int} % interior
|
\DeclareMathOperator{\inter}{int} % interior
|
||||||
\newcommand{\defon}[1]{|_{#1}} % TODO
|
\newcommand{\defon}[1]{|_{#1}} % TODO
|
||||||
|
|
||||||
\RequirePackage[super]{nth}
|
|
||||||
|
|
||||||
% TODO MOVE
|
|
||||||
% https://tex.stackexchange.com/a/94702
|
|
||||||
\newenvironment{absolutelynopagebreak}
|
|
||||||
{\par\nobreak\vfil\penalty0\vfilneg
|
|
||||||
\vtop\bgroup}
|
|
||||||
{\par\xdef\tpd{\the\prevdepth}\egroup
|
|
||||||
\prevdepth=\tpd}
|
|
||||||
|
|
|
@ -44,7 +44,10 @@
|
||||||
\input{inputs/lecture_17}
|
\input{inputs/lecture_17}
|
||||||
\input{inputs/lecture_18}
|
\input{inputs/lecture_18}
|
||||||
\input{inputs/lecture_19}
|
\input{inputs/lecture_19}
|
||||||
\input{inputs/lecture_20}
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
\cleardoublepage
|
\cleardoublepage
|
||||||
|
|
||||||
|
@ -61,7 +64,6 @@
|
||||||
\input{inputs/tutorial_08}
|
\input{inputs/tutorial_08}
|
||||||
\input{inputs/tutorial_09}
|
\input{inputs/tutorial_09}
|
||||||
\input{inputs/tutorial_10}
|
\input{inputs/tutorial_10}
|
||||||
\input{inputs/tutorial_11}
|
|
||||||
|
|
||||||
\section{Facts}
|
\section{Facts}
|
||||||
\input{inputs/facts}
|
\input{inputs/facts}
|
||||||
|
|
Loading…
Reference in a new issue