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12 changed files with 74 additions and 126 deletions
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@ -185,11 +185,11 @@ suffices to show that open balls in one metric are unions of open balls in the o
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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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\item $2^{\N}$ 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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\item $\cN \coloneqq \N^{\N}$ is called the \vocab{Baire space}.
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($\N$ with descrete topology)
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\item $\mathbb{H} \coloneqq [0,1]^{\N}$ is called the \vocab{Hilbert cube}.
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\item $\omega^{\omega}$ is called the \vocab{Baire space}.
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($\omega$ with descrete topology)
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\item $[0,1]^{\omega}$ is called the \vocab{Hilbert cube}.
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($[0,1] \subseteq \R$ with the usual topology)
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\end{itemize}
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\end{definition}
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@ -78,17 +78,18 @@ where $X$ is a metrizable, usually second countable space.
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we also have
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$A = \bigcup_n A_n'$ with $A'_n \in \Pi^0_{\xi_n}(X)$.
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We construct a $(2^{\omega \times \omega}) \cong 2^\omega$-universal set
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We construct a $(2^\omega)^\omega \cong 2^\omega$-universal set
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for $\Sigma^0_\xi(X)$.
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For $(y_{m,n}) \in (2^{\omega \times \omega})$
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For $(y_n) \in (2^\omega)^\omega$
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and $x \in X$
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we set $((y_{m,n}), x) \in \cU$
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iff $\exists n.~((y_{m,n})_{m < \omega}, x) \in U_{\xi_n}$,
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i.e.~iff $\exists n.~x \in (U_{\xi_n})_{(y_{m,n})_{m < \omega}}$.
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we set $((y_n), x) \in \cU$
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iff $\exists n.~(y_n, x) \in U_{\xi_n}$,
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i.e.~iff $\exists n.~x \in (U_{\xi_n})_{y_n}$.
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Let $A \in \Sigma^0_\xi(X)$.
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Then $A = \bigcup_{n} B_n$ for some $B_n \in \Pi^0_{\xi_n}(X)$.
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Furthermore $\cU \in \Sigma^0_{\xi}((2^{\omega \times \omega} \times X)$.
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% TODO
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Furthermore $\cU \in \Sigma^0_{\xi}((2^\omega)^\omega \times X)$.
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\end{proof}
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\begin{remark}
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Since $2^{\omega}$ embeds
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@ -154,7 +154,7 @@ We will see later that $\Sigma^1_1(X) \cap \Pi^1_1(X) = \cB(X)$.
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\begin{proof}
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Take $\cU \subseteq Y \times X \times \cN$
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which is $Y$-universal for $\Pi^0_1(X \times \cN)$.
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Let $\cV \coloneqq \proj_{Y \times X}(\cU)$.
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Let $\cV \coloneqq \proj_{Y \times Y}(\cU)$.
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Then $\cV$ is $Y$-universal for $\Sigma^1_1(X)$:
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\begin{itemize}
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\item $\cV \in \Sigma^1_1(Y \times X)$
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@ -34,8 +34,7 @@ we need the following definition:
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\begin{lemma}
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\label{lem:lusinsephelp}
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If $P = \bigcup_{m < \omega} P_m$, $Q = \bigcup_{n < \omega} Q_n$ are such that
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for any $m, n$ the sets $P_m$ and $Q_n$ are Borel separable,
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then $P$ and $Q$ are Borel separable.
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for any $m, n$ the sets $P_m$ and $Q_n$ are Borel separable.
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\end{lemma}
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\begin{proof}
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For all $m, n$ pick $R_{m,n}$ Borel,
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@ -43,7 +42,7 @@ we need the following definition:
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and $Q_n \cap R_{m,n} = \emptyset$.
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Then $R = \bigcup_m \bigcap_n R_{m,n}$
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has the desired property
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that $P \subseteq R$ and $R \cap Q = \emptyset$.
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that $R \subseteq R$ and $R \cap Q = \emptyset$.
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\end{proof}
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\begin{notation}
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@ -61,7 +60,7 @@ we need the following definition:
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Write $A_s \coloneqq f(\cN_s)$ and $B_s \coloneqq g(\cN_s)$.
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Note that $A_s = \bigcup_m A_{s\concat m}$
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and $B_s = \bigcup_{n < \omega} B_{s\concat n}$.
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and $B_ns = \bigcup_{n < \omega} B_{s\concat n}$.
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In particular
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$A = \bigcup_{m < \omega} A_{\underbrace{\langle m \rangle}_{\in \omega^1}}$
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@ -183,7 +183,7 @@ i.e.}{}
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Let $X$ be Polish and $C \subseteq X$ coanalytic.
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Then $\phi\colon C \to \Ord$
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is a \vocab[Rank!$\Pi^1_1$-rank]{$\Pi^1_1$-rank}
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provided that $\le^\ast_\phi$ and $<^\ast_\phi$ are coanalytic subsets of $X \times X$,
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provided that $\le^\ast$ and $<^\ast$ are coanalytic subsets of $X \times X$,
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where
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$x \le^\ast_{\phi} y$
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iff
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@ -55,7 +55,7 @@
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then $\xi = \alpha$
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and
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\[
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E_\xi = E_\alpha = \bigcup_{\eta < \alpha} E_\eta
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E_\xi = E_\alpha = \bigcup_{\eta < \alpha}
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\]
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is a countable union of Borel sets by the previous case.
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\end{itemize}
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@ -140,7 +140,7 @@ 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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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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@ -15,7 +15,7 @@ U_{\epsilon}(x,y) \coloneqq \{f \in X^X : d(x,f(y)) < \epsilon\}.
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\]
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for all $x,y \in X$, $\epsilon > 0$.
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$X^{X}$ is a compact Hausdorff space.\footnote{cf.~\href{https://en.wikipedia.org/wiki/Tychonoff's_theorem}{Tychonoff's theorem}}
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$X^{X}$ is a compact Hausdorff space.
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\begin{remark}%
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\footnote{cf.~\yaref{s11e1}}
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Let $f_0 \in X^X$ be fixed.
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@ -34,20 +34,16 @@ $X^{X}$ is a compact Hausdorff space.\footnote{cf.~\href{https://en.wikipedia.or
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\end{remark}
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\begin{definition}
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\gist{%
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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 $t \in T$ with $x \mapsto tx$
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and take the closure in $X^X$.
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}{%
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The \vocab{Ellis semigroup} of a flow $(X,T)$
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is $E(X,T) \coloneqq \overline{T} \subseteq X^X$.
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}
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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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% TODO THINK ABOUT THIS
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\gist{
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Properties of $(X,T)$ translate to properties of $E(X,T)$:
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@ -62,14 +58,15 @@ since $X^X$ has these properties.
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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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\gist{
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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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\gist{
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We have that $t^{-1}G$ is compact,
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since $t^{-1}$ is continuous
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and $G$ is compact.
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}{$t^{-1}G$ is compact.}
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It is $T \subseteq t^{-1}G$ since $T \ni s = t^{-1}\underbrace{(ts)}_{\in G}$.
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@ -96,21 +93,6 @@ since $X^X$ has these properties.
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Since $G$ compact,
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and $Tg \subseteq G$,
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we have $ \overline{Tg} \subseteq G$.
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}{
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$G \coloneqq E(X,T)$.
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\begin{itemize}
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\item $\forall t \in T. ~ tG \subseteq G$:
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\begin{itemize}
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\item $t^{-1}G$ is compact.
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\item $T \subseteq t^{-1}G$,
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\item $\leadsto G = \overline{T} \subseteq t^{-1}G$,
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i.e.~$tG \subseteq G$.
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\end{itemize}
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\item $\forall g \in G.~\overline{T}g = \overline{Tg}$ (cts.~map from compact to Hausdorff)
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\item $\forall g \in G.~Gg \subseteq G$ :
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$Gg = \overline{T}g = \overline{Tg} \overset{G \text{ compact}, Tg \subseteq G}{\subseteq} G$.
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\end{itemize}
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}
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\end{proof}
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\begin{definition}
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@ -119,11 +101,9 @@ since $X^X$ has these properties.
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with a compact Hausdorff topology,
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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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\gist{
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\begin{example}
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The Ellis semigroup is a compact semigroup.
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\end{example}
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}{}
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\begin{lemma}[Ellis–Numakura]
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\yalabel{Ellis-Numakura Lemma}{Ellis-Numakura}{lem:ellisnumakura}
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@ -132,7 +112,6 @@ since $X^X$ has these properties.
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i.e.~$f$ such that $f^2 = f$.
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\end{lemma}
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\begin{proof}
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\gist{
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Using Zorn's lemma, take a $\subseteq$-minimal
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compact subsemigroup $R$ of $S$
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and let $s \in R$.
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@ -149,39 +128,25 @@ since $X^X$ has these properties.
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Thus $P = R$ by minimality,
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so $s \in P$,
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i.e.~$s^2 = s$.
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}{
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\begin{itemize}
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\item Take $R \subseteq S$ minimal compact subsemigroup (Zorn),
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$s \in R$.
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\item $Rs \subseteq R \implies Rs = R$.
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\item $P \coloneqq \{x \in R : xs = s\}$:
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\begin{itemize}
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\item $P \neq \emptyset$, since $s \in Rs$
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\item $P$ compact, since $P = \alpha^{-1}(s) \cap R$,
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$\alpha: x \mapsto xs$ cts.
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\item $P = R \implies s^2 = s$.
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\end{itemize}
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\end{itemize}%
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}
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\end{proof}
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\gist{
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The \yaref{lem:ellisnumakura} is not very interesting for $E(X,T)$,
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since we already know that it has an identity.
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%in fact we might have chosen $R = \{1\}$ in the proof.
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But it is interesting for other semigroups.
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}{}
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\begin{theorem}[Ellis]
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$(X,T)$ is distal iff $E(X,T)$ is a group.
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\end{theorem}
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\begin{proof}
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Let $G \coloneqq E(X,T)$ and let $d$ be a metric on $X$.
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\gist{
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Let $G \coloneqq E(X,T)$.
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Let $d$ be a metric on $X$.
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For all $g \in G$ we need to show that $x \mapsto gx$ is bijective.
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If we had $gx = gy$, then $d(gx,gy) = 0$.
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Then $\inf_{t \in T} d(tx,ty) = 0$, but the flow is distal,
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Then $\inf d(tx,ty) = 0$, but the flow is distal,
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hence $x = y$.
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Let $g \in G$. Consider the compact semigroup $\Gamma \coloneqq Gg$.
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@ -199,39 +164,19 @@ since $X^X$ has these properties.
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Hence $g'$ is bijective
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and $x = gg'(x)$,
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i.e.~$g g' = \id$.
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}{
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\begin{itemize}
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\item $x \mapsto gx$ injective for all $g \in G$:
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\[gx = gy
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\implies d(gx,gy) = 0
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\implies \inf_{t \in T} d(tx, ty) = 0
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\overset{\text{distal}}{\implies} x = y.
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\]
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\item Fix $g \in G$.
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\begin{itemize}
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\item $\Gamma \coloneqq Gg$ is a compact semigroup.
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\item $\exists f\in \Gamma.~f^2 = f$ (\yaref{lem:ellisnumakura})
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\item $f$ is injective, hence $f = \id$.
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\item Take $g'$ such that $g'g = f$. $g'gg' = g' \implies gg' = \id$.
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\end{itemize}
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\end{itemize}
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}
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\gist{
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On the other hand if $(x_0,x_1)$ is proximal,
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then there exists $g \in G$ such that $gx_0 = gx_1$.%
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\footnote{cf.~\yaref{s11e1} (e)}
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It follows that an inverse to $g$ can not exist.
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}{If $(x_0,x_1)$ is proximal, there is $g \in G$ with $gx_0 = gx_1$, i.e.~no inverse to $g$.}
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\todo{The other direction is left as an easy exercise.}
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\end{proof}
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% Let $(X,T)$ be a flow.
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% Then by Zorn's lemma, there exists $X_0 \subseteq X$
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% such that $(X_0, T)$ is minimal.
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% In particular,
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% for $x \in X$ and $\overline{Tx} = Y$
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% we have that $(Y,T)$ is a flow.
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% However if we pick $y \in Y$, $Ty$ might not be dense.
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% TODO ANKI-MARKER
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Let $(X,T)$ be a flow.
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Then by Zorn's lemma, there exists $X_0 \subseteq X$
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such that $(X_0, T)$ is minimal.
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In particular,
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for $x \in X$ and $\overline{Tx} = Y$
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we have that $(Y,T)$ is a flow.
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However if we pick $y \in Y$, $Ty$ might not be dense.
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% TODO: question!
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% TODO: think about this!
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% We want to a minimal subflow in a nice way:
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@ -244,7 +189,6 @@ since $X^X$ has these properties.
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\end{theorem}
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\begin{proof}
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Let $G = E(X,T)$.
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\gist{
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Note that for all $x \in X$,
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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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@ -252,18 +196,17 @@ since $X^X$ has these properties.
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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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We need to show that $(Gx, T)$ is minimal.
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Suppose that $y \in Gx$, i.e.~$Gx = Gy$.
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Since $g \mapsto gy$ is continuous,
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we have $Gx = Gy = \overline{T}y = \overline{Ty}$,
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so $Ty$ is dense in $Gx$.
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}{
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\begin{itemize}
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\item $G$ is a group, so the $G$-orbits partition $X$.
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\item Let $y \in Gx$, then $Gx \overset{y \in Gx}{=} Gy \overset{\text{def}}{=} \overline{T}y \overset{g \mapsto gy \text{ cts.}}{=} \overline{Ty}$,
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i.e.~$(Gx,T)$ is minimal.
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\end{itemize}
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}
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% Clearly the sets $Gx$ cover $X$. We want to show that they
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% partition $X$.
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% It suffices to show that $y \in Gx \implies Gy = Gx$.
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% Take some $y \in Gx$.
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% Recall that $\overline{Ty} = \overline{T} y = Gy$.
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% We have $\overline{Ty} \subseteq Gx$,
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% so $Gy \subseteq Gx$.
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% Since $y = g_0 x \implies x = g_0^{-1}y$, we also have $x \in Gy$,
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% hence $Gx \subseteq Gy$
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% TODO: WHY?
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\end{proof}
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\begin{corollary}
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If $(X,T)$ is distal and minimal,
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|
|
|
@ -1,11 +1,16 @@
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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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% TODO ANKI-MARKER
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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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||||
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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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|
|
|
@ -78,7 +78,7 @@
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is Borel for all $\alpha < \omega_1$.
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\end{enumerate}
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\end{theorem}
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||||
\todo{This was already stated as \yaref{thm:beleznay-foreman} in lecture 16
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\todo{This was already stated as \yaref{thm:beleznayforeman} in lecture 16
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and should not have two numbers.}
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A few words on the proof:
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|
|
|
@ -108,13 +108,13 @@ This requires the use of the axiom of choice.
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\item Let $B = \{x \in \R | (F_i)_x \text{ is meager}\}$.
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Then $B$ is comeager in $\R$ and $|B| = \fc$.
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We have $|B| = \fc$:
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$B$ contains a comeager $G_\delta$ set, say $B'$.
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We have $|B| = \fc$, since $B$ contains a comeager
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$G_\delta$ set, $B'$:
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$B'$ is Polish,
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hence $B' = P \cup C$
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for $P$ perfect and $C$ countable,
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and $|P| \in \{\fc, 0\}$.
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But $B'$ can't contain an isolated point.
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But $B'$ can't contain isolated point.
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\item We use $B$ to find a suitable point $a_i$:
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||||
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||||
To ensure that (i) holds, it suffices to chose
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|
|
|
@ -102,7 +102,7 @@ Let $d$ be a compatible metric on $X$.
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Consider $\ev_x \colon X^X \to X$.
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$X^X$ is compact and $X$ is Hausdorff.
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Hence we can apply
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\yaref{fact:t12:2}.
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\label{fact:t12:2}.
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\item Let $x_0 \neq x_1 \in X$.
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Then $(x_0,x_1)$ is a proximal pair iff $d(gx_0,gx_1) = 0$
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