diff --git a/inputs/lecture_01.tex b/inputs/lecture_01.tex index acab3f3..b78ae56 100644 --- a/inputs/lecture_01.tex +++ b/inputs/lecture_01.tex @@ -15,7 +15,7 @@ \end{definition} Note that Polishness is preserved under homeomorphisms, i.e.~it is really a topological property. - +\gist{% \begin{example} \begin{itemize} \item $\R$ is a Polish space, @@ -26,6 +26,7 @@ i.e.~it is really a topological property. considered as a topological space. \end{itemize} \end{example} +}{} \gist{% Polish spaces behave very nicely. We will see that uncountable polish spaces have size $2^{\aleph_0}$. % TODO: mathfrak c for continuum @@ -75,11 +76,13 @@ However the converse of this does not hold. \item \vocab{Lindelöf} (every open cover has a countable subcover). \end{itemize} \end{fact} +\gist{% \begin{fact} Compact Hausdorff spaces are \vocab{normal} (T4) i.e.~two disjoint closed subsets can be separated by open sets. \end{fact} +}{} \begin{fact} For a metric space, the following are equivalent: \begin{itemize} @@ -172,7 +175,8 @@ suffices to show that open balls in one metric are unions of open balls in the o D\left( (x_n), (y_n) \right) \coloneqq \sum_{n< \omega} 2^{-(n+1)} d(x_n, y_n). \] - Clearly $D \le 1$. +\gist{% + Clearly $D \le 1$. It is also clear, that $D$ is a metric. We need to check that $D$ is complete: @@ -180,6 +184,7 @@ suffices to show that open balls in one metric are unions of open balls in the o Consider the pointwise limit $(a_n)$. This exists since $x_n^{(k)}$ is Cauchy for every fixed $n$. Then $(x_n)^{(k)} \xrightarrow{k \to \infty} (a_n)$. +}{Clearly $D$ is a complete metric.} \end{proof} \begin{definition}[Our favourite Polish spaces] diff --git a/inputs/lecture_03.tex b/inputs/lecture_03.tex index 88f85bc..88b1869 100644 --- a/inputs/lecture_03.tex +++ b/inputs/lecture_03.tex @@ -59,7 +59,6 @@ define extension, initial segments and concatenation of a finite sequence with an infinite one. \end{notation} -}{} \begin{definition} A \vocab{tree} @@ -87,6 +86,7 @@ \forall t\in T.\exists s \supsetneq t.~s \in T. \] \end{definition} +}{} \begin{definition} A \vocab{Cantor scheme} diff --git a/inputs/lecture_06.tex b/inputs/lecture_06.tex index 63709a9..cf333c0 100644 --- a/inputs/lecture_06.tex +++ b/inputs/lecture_06.tex @@ -139,7 +139,7 @@ \subsection{The hierarchy of Borel sets} -Let $\omega_1$ be the first uncountable ordinal. +\gist{Let $\omega_1$ be the first uncountable ordinal.}{} For every $d < \omega_1$, we define by transfinite recursion classes $\Sigma^0_\alpha$ @@ -201,6 +201,7 @@ i.e.~$\Delta^0_1$ is the set of clopen sets. \end{enumerate} \end{proposition} \begin{proof} +\gist{% \begin{enumerate}[(a)] \item \begin{observe} \label{ob:sigmasuffices} @@ -215,7 +216,7 @@ i.e.~$\Delta^0_1$ is the set of clopen sets. since $\Delta^0_\eta(X)$ is closed under complements. Furthermore, it suffices to show $\Sigma^0_\eta(X) \subseteq \Sigma^0_\xi(X)$, - by \yaref{ob:sigmasuffices} + by the observation (since $\Sigma^0_\eta(X) \subseteq \Pi^0_\xi(X)$ and $\Delta^0_\xi(X) = \Sigma^0_\xi(X) \cap \Pi^0_\xi(X)$). @@ -246,6 +247,17 @@ i.e.~$\Delta^0_1$ is the set of clopen sets. Then $\bigcup_{n < \omega} A_n \in \Sigma^0_\alpha(X)$. It is clear that $\cB_0$ is closed under complements. \end{enumerate} +}{ + \begin{enumerate}[(a)] + \item It suffices to show that $\Sigma^0_\eta(X) \subseteq \Sigma^0_\xi(X)$ + for all $1 \le \eta < \xi < \omega_1$. + For $\eta = 1, \xi = 2$ this holds, + since open sets of a metrizable space are $F_\sigma$. + Induction. + \item Let $\cB_0 \coloneqq \bigcup_{\alpha < \omega_1} \Delta^0_\alpha(X)$. + This is a $\sigma$-algebra containing all open sets. + \end{enumerate} +} \end{proof} diff --git a/inputs/lecture_07.tex b/inputs/lecture_07.tex index d350805..5b640e6 100644 --- a/inputs/lecture_07.tex +++ b/inputs/lecture_07.tex @@ -6,6 +6,7 @@ % $\fc := 2^{\aleph_0}$ \end{proposition} \begin{proof} +\gist{% We use strong induction on $\xi < \omega_1$. We have $\Sigma^0_1(X) \le \fc$ (for every element of the basis, we can decide @@ -27,6 +28,9 @@ \[ |\cB(X)| \le \omega_1 \cdot \fc = \fc. \] +}{Use strong induction. $|\Sigma^0_1(X)| \le \fc$, since $X$ is second countable. + \[|\Sigma^0_\xi(X)| \le (\overbrace{\aleph_0}^{\mathclap{\{\xi': \xi' < \xi\} \text{ is countable~ ~ ~ ~}}} \cdot \underbrace{\fc}_{\mathclap{\text{inductive assumption}}})^{\overbrace{\aleph_0}^{\mathclap{\text{countable unions}}}}\] +} \end{proof} \begin{proposition}[Closure properties] diff --git a/inputs/lecture_08.tex b/inputs/lecture_08.tex index c58fca2..1dcfc76 100644 --- a/inputs/lecture_08.tex +++ b/inputs/lecture_08.tex @@ -41,6 +41,7 @@ where $X$ is a metrizable, usually second countable space. similarly for $\Pi^0_\xi(X)$. \end{theorem} \begin{proof} +\gist{% Note that if $\cU$ is $2^{\omega}$ universal for $\Sigma^0_\xi(X)$, then $(2^{\omega} \times X) \setminus \cU$ is $2^{\omega}$-universal for $\Pi^0_\xi(X)$. @@ -89,6 +90,18 @@ where $X$ is a metrizable, usually second countable space. Let $A \in \Sigma^0_\xi(X)$. Then $A = \bigcup_{n} B_n$ for some $B_n \in \Pi^0_{\xi_n}(X)$. Furthermore $\cU \in \Sigma^0_{\xi}((2^{\omega \times \omega} \times X)$. +}{ + \begin{itemize} + \item Suffices for $\Sigma^0_\xi$ (complement is $\Pi^0_\xi$-universal). + \item $2^\omega$-universal set for $\Sigma^0_1(X)$, since + $X$ is second countable ($(y,x) \in \cU \iff x \in \bigcup_n \{V_n : y_n = 1\}$). + \item Induction: Take $\xi_k \to \xi$, + and $\cU_{\xi_k}$ $\Sigma^0_{\xi_k}$-universal. + Construct $2^{ \omega \times \omega}$-universal: + + $(y_{m,n}, x) \in \cU :\iff \exists n.~((y_{m,n}), x) \in \cU_{\xi_n}$. + \end{itemize} +} \end{proof} \begin{remark} Since $2^{\omega}$ embeds @@ -96,6 +109,6 @@ where $X$ is a metrizable, usually second countable space. % such that the image is closed, we can replace $2^{\omega}$ by $Y$ in the statement of the theorem.% - \footnote{By definition of the subspace topology - and transfinite induction, $\Sigma^0_\xi(Y)\defon{2^\omega} = \Sigma^0_\xi(2^\omega)$.} + \gist{\footnote{By definition of the subspace topology + and transfinite induction, $\Sigma^0_\xi(Y)\defon{2^\omega} = \Sigma^0_\xi(2^\omega)$.}}{} \end{remark} diff --git a/inputs/lecture_14.tex b/inputs/lecture_14.tex index 59914fa..d02c7be 100644 --- a/inputs/lecture_14.tex +++ b/inputs/lecture_14.tex @@ -108,14 +108,14 @@ \end{theorem} \begin{proof} Let $\phi\colon R \to \Ord$ - by a $\Pi^1_1$-rank. - Set + be a $\Pi^1_1$-rank. + \gist{Set \begin{IEEEeqnarray*}{rCl} (x,n) \in R^\ast &:\iff& (x,n) \in R\\ &&\land \forall m.~(x,n) \le^\ast_\phi (x,m)\\ &&\land \forall m.~\left( (x,n) <^\ast_\phi (x,m) \lor n \le m \right), \end{IEEEeqnarray*} - i.e.~take the element with minimal rank + i.e.~take}{Take} the element with minimal rank that has the minimal second coordinate among those elements. \end{proof} \gist{ @@ -196,8 +196,8 @@ Let $\rho(\prec) \coloneqq \sup \{\rho_{\prec}(x) + 1 : x \in X\}$. \[(s_0,u_1,s_1,\ldots, u_n,s_n) \succ^\ast (s_0',u_1', s_1', \ldots, u_m', s_m') :\iff\] \begin{itemize} \item $n < m$ and - \item $\forall i \le n.~s_i \subsetneq s_i' \land u_i \subsetneq u_i'$.% - \footnote{sic! (there is a typo in the official notes)} + \item $\forall i \le n.~s_i \subsetneq s_i' \land u_i \subsetneq u_i'$. + % \footnote{sic! (there was a typo in the official notes)} \end{itemize} \begin{claim} $\prec^\ast$ is well-founded. diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex index 77a74ef..22de86b 100644 --- a/inputs/lecture_15.tex +++ b/inputs/lecture_15.tex @@ -66,7 +66,7 @@ % \subsection*{Basic Definitions} % TODO: move to appendix? - +\gist{% Recall: \begin{definition}+ Let $X$ be a set. @@ -112,6 +112,7 @@ Recall: g&\longmapsto & (x \mapsto g \cdot x). \end{IEEEeqnarray*} \end{remark} +}{} \begin{definition}+ A group $G$ with a topology @@ -185,13 +186,13 @@ Recall: a homeomorphism $X \leftrightarrow Y$ commuting with the group action. \end{definition} + +\gist{% \begin{warning}+ What is called ``factor'' here is called ``subflow'' by Furstenberg. \end{warning} - - \begin{example} Recall that $S_1 = \{z \in \C : |z| = 1\}$. Let $X = S_1$, $T = S_1$ @@ -200,6 +201,7 @@ Recall: and $\alpha + \beta$ denotes the addition of \emph{angles}, i.e.~$\alpha \cdot \beta$ in complex numbers.} \end{example} +}{} \begin{definition} \label{def:isometricextension} @@ -237,6 +239,7 @@ Recall: An isometric extension of a distal flow is distal. \end{proposition} \begin{proof} + \gist{% Let $\pi\colon X\to Y$ be an isometric extension. Towards a contradiction, suppose that $x_1,x_2 \in X$ are proximal. @@ -253,6 +256,11 @@ Recall: we get $\rho(g_n x_1, g_n x_2) \to \rho(z,z) = 0$. Therefore $\rho(x_1,x_2) = 0$. Hence $x_1 = x_2$ $\lightning$. + }{Let $\pi\colon X \to Y$ isometric extension. + Suppose $x_1,x_2 \in X$ is proximal. + Then $\pi(x_1) = \pi(x_2)$. + But there exists a $T$-equivariant metric on the fibers. + } \end{proof} \begin{definition} diff --git a/inputs/lecture_18.tex b/inputs/lecture_18.tex index ab5a461..01b3c75 100644 --- a/inputs/lecture_18.tex +++ b/inputs/lecture_18.tex @@ -149,6 +149,7 @@ i.e.~show that if $(Z,T)$ is a proper factor of a minimal distal flow bet the quotient space. It is compact, second countable and Hausdorff. Let $\pi\colon X\to M$ denote the quotient map. +\gist{% \item $(Y,T) \mathbin{\text{\reflectbox{$\coloneqq$}}} (M,T)$ is an isometric flow: \begin{enumerate} @@ -204,6 +205,7 @@ i.e.~show that if $(Z,T)$ is a proper factor of a minimal distal flow (\yaref{thm:usccomeagercont}) this implies that $X \setminus \{x_2\}$ is meager. But then $X = \{\star\} \lightning$. +}{} \end{enumerate} \end{proof} diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index 6ad74b0..91b6a76 100644 --- a/inputs/lecture_19.tex +++ b/inputs/lecture_19.tex @@ -173,6 +173,7 @@ More generally we can show: then $(Y,T)$ is an isometric extension of $(Z_2, T)$. \end{lemma} \begin{proof} +\gist{% % TODO TODO TODO Think about this For $z_1,z_1' \in Z_1$ with $w_1(z_1) = w_1(z_1')$ let @@ -185,9 +186,16 @@ More generally we can show: on the fibers of $Y$ over $Z_2$ and invariant under $T$. - $\sigma$ is a metric, + $\sigma$ is a metric on fibers, since if $\pi_2(y) = \pi_2(y')$ and $\sigma(y,y') = 0$, then $\pi_1(y) = \pi_1(y')$ or $y = y'$. +}{% + \begin{itemize} + \item Let $\rho\colon Z_1 \times_W Z_1 \to \R$. + \item Consider $\sigma\colon Y \times_{Z_2} Y \to \R$ + given by $\sigma(y,y') \coloneqq \rho(\pi_1(y), \pi_1(y'))$. + \end{itemize} +} \end{proof} @@ -242,7 +250,7 @@ More generally we can show: $(X'_\xi, T) = \theta((X_\xi, T)$. Let $\pi_\xi$ and $\pi'_\xi$ denote the maps from $X$ to $X_\xi$ resp.~$X'_\xi$. Set - \[Y \coloneqq \{(\pi_\xi(x), \pi'_{\xi+1}(x)) \in X_\xi \times X'_{\xi+ 1}: x \in X\}.\] + \[Y \coloneqq \{(\pi_\xi(x), \pi'_{\xi+1}(x)) \in X_\xi \times X'_{\xi+ 1}: x \in X\} \subseteq X \times X'_{\xi+1}.\] Then % https://q.uiver.app/#q=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 diff --git a/inputs/lecture_20.tex b/inputs/lecture_20.tex index 8e0efde..48ffd08 100644 --- a/inputs/lecture_20.tex +++ b/inputs/lecture_20.tex @@ -46,9 +46,6 @@ Let $\pi_n\colon X \to (S^1)^n$ be the projection to the first $n$ coordinates. -% TODO ANKI-MARKER - - \begin{lemma} \label{lem:lec20:1} Let $x,x' \in X$ with $\pi_n(x) = \pi_n(x')$ @@ -66,7 +63,7 @@ coordinates. where $d$ is the metric on $X$, $d((x_i), (y_i)) = \max_n \frac{1}{2^n} | x_n - y_n|$.% TODO use multiplicative notation \end{lemma} -\begin{proof} +\begin{refproof}{lem:lec20:1} Let \begin{IEEEeqnarray*}{rCl} x &=& (\alpha^0_1, \alpha^0_2, \ldots, \alpha^0_{n-1}, \alpha_n, \alpha_{n+1}, \alpha_{n+2},\ldots)\\ @@ -74,7 +71,7 @@ coordinates. \end{IEEEeqnarray*} We will choose $x_k$ of the form \[ - (\alpha^0_1, \alpha^0_2, \ldots, \alpha^0_{n-1} \alpha_n e^{\i \beta_k}, \alpha_{n+1}, \alpha_{n+2}, \ldots), + (\alpha^0_1, \alpha^0_2, \ldots, \alpha^0_{n-1}, \alpha_n e^{\i \beta_k}, \alpha_{n+1}, \alpha_{n+2}, \ldots), \] where $\beta_k$ is such that $\frac{\beta_k}{\pi}$ is irrational and $|\beta_k| < 2^{-k}$. @@ -89,7 +86,6 @@ coordinates. $u' = (\xi'_n)_{n \in \N}$, let $\frac{u}{u'} = (\frac{\xi_n}{\xi'_n})_{n \in \N}$ ($X$ is a group). - We are interested in $F(x_k, x') = \inf_m d(\tau^m x_k, \tau^m x')$, but it is easier to consider the distance between their quotient and $1$. @@ -97,6 +93,7 @@ coordinates. \[ w_k \coloneqq \frac{x_k}{x'} = (\underbrace{1,\ldots,1}_{n-1}, e^{\i \beta_k}, \overbrace{\frac{\alpha_{n+1}}{\alpha'_{n+1}}, \frac{\alpha_{n+2}}{\alpha'_{n+2}}, \ldots}^{\mathclap{\text{not interesting}}}). \] +\gist{% \begin{claim} $F(x_k, x') = \inf_m d(\sigma^m(w_k), 1)$, where $\sigma(\xi_1, \xi_2, \ldots) = (\xi_1, \xi_1\xi_2, \xi_2\xi_3, \ldots)$. @@ -121,7 +118,8 @@ coordinates. &\le & 2^{-n} | e^{\i \beta_k}- 1|\\ &<& 2^{-n-k}. \end{IEEEeqnarray*} -\end{proof} +}{[some technical details omitted]} +\end{refproof} \begin{definition} diff --git a/inputs/lecture_22.tex b/inputs/lecture_22.tex index 2b99068..ae8fdba 100644 --- a/inputs/lecture_22.tex +++ b/inputs/lecture_22.tex @@ -1,6 +1,7 @@ \lecture{22}{2024-01-16}{} \begin{refproof}{thm:21:xnmaxiso} + % TODO TODO TODO We have the following situation: % https://q.uiver.app/#q=WzAsNCxbMCwwLCJYIl0sWzEsMSwiWF97bn0iXSxbMSwyLCJYX3tuLTF9Il0sWzIsMSwiWSJdLFswLDEsIlxccGlfe259Il0sWzEsMiwiXFx0ZXh0e2lzb21ldHJpY30iLDFdLFswLDIsIlxccGlfe24tMX0iLDIseyJjdXJ2ZSI6Mn1dLFswLDMsIlxccGknIiwwLHsiY3VydmUiOi0zfV0sWzMsMSwiaCIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFszLDIsIlxcb3ZlcmxpbmV7Z30sIFxcdGV4dHsgbWF4LiBpc29tLn0iLDAseyJjdXJ2ZSI6LTJ9XV0= \[\begin{tikzcd} @@ -17,7 +18,7 @@ We want to show that this tower is normal, i.e.~the isometric extensions are maximal isometric extension. - +\gist{% Let $Y$ be a maximal isometric extension of $X_{n-1}$ in $X$ and let $\overline{g} = \pi^n_{n-1} \circ h$. % factor map? We need to show that $h$ is an isomorphism. @@ -62,6 +63,25 @@ But $x$ and $x'$ don't depend on $k$, hence $R(x,x') = 0$. It follows that $\pi'(x) = \pi'(x')$ $\lightning$. +}{ + \begin{itemize} + \item $Y$ max.~isometric extension of $X_{n-1}$ in $X$ + and $\overline{g} = \pi^n_{n-1} \circ h$. + \item $h$ isomorphism. + Suppose not, then $\exists y_0,y_1 \in X.~\pi'(y_0) \neq \pi'(y_1), + \pi_n(y_0) = \pi_n(y_1) = t$. + + \item Apply \yaref{lem:lec20:1} $\leadsto$ sequence $(x_k)$ in $X$, + such that $\pi_{n-1}(x_k) = \pi_{n-1}(y_i)$, + $F(x_k,y_i) \to 0$. + \item $\rho\colon Y \times_{X_{n-1}} Y \to \R$ witnessing isometric. + \item $R(a,b) \coloneqq \rho(\pi'(a), \pi'(b))$ for $a,b \in X$ with + $\overline{g}(\pi'(a)) = \overline{g}(\pi'(b))$. + (defined for any two of $x_k$, $y_0$, $y_1$, $\tau$-equivariant) + \item $F(y_0,x_{k}) \to 0$, so $d(\tau^{m_k} y_0, \tau^{m_k} x_k) \to 0$. + \item $R(y_0,x_k) \to 0$, hence $\underbrace{R(y_0,y_1)}_{\text{no } k} \to 0$ $\lightning$. + \end{itemize} +} \end{refproof}