lecture 20

inputs/lecture_18.tex

@@ -15,6 +15,7 @@ sketch of \yaref{thm:l16:3} in the case of $|Z| = 1$.
 Let $(X,T)$ be a distal flow.
 Then $G \coloneqq  E(X,T)$ is a group.
 \begin{definition}
+    \label{def:F}
 For $x, x' \in X$ define
 \[
 F(x,x') \coloneqq \inf \{d(gx, gx') : g \in G\}.

inputs/lecture_19.tex

@@ -274,6 +274,7 @@ More generally we can show:
     $(X'_{\xi+1}, T)$ is a factor of $(X_{\xi+1}, T)$.
 \end{proof}
 \begin{example}[{\cite[p. 513]{Furstenberg}}]
+    \label{ex:19:inftorus}
     Let $X$ be the infinite torus
     \[
     X \coloneqq \{(\xi_1, \xi_2, \ldots) : \xi_i \in \C, |\xi_i| = 1\}.

inputs/lecture_20.tex

@@ -0,0 +1,196 @@
+\lecture{20}{2024-01-09}{The infinite Torus}
+
+\begin{example}
+    \footnote{This is the same as \yaref{ex:19:inftorus},
+        but with new notation.}
+    Let $X = (S^1)^{\N}$\footnote{We identify $S^1$ and $\faktor{\R}{\Z}$.}
+    and consider $\left(X, \Z \right)$
+    where the action is generated by
+    \[
+        \tau\colon (x_1,x_2,x_3,\ldots) \mapsto(x_1 + \alpha, x_1 + x_2, x_2 + x_3, \ldots)
+    \]
+    for some irrational $\alpha$.
+\end{example}
+\begin{remark}+
+    Note that we can identify $S^1$ with a subset of  $\C$ (and use multiplication)
+    or with  $\faktor{\R}{\Z}$ (and use addition).
+    In the lecture both notations were used.% to make things extra confusing.
+    Here I'll try to only use multiplicative notation.
+\end{remark}
+We will be studying projections to the first $d$ coordinates,
+i.e.
+\[
+    \tau_d \colon (x_1,\ldots,x_d) \mapsto (e^{\i \alpha} x_1, x_1x_2, \ldots, x_{d-1}x_d).
+\]
+$\tau_d$ is called the \vocab{$d$-skew shift}.
+For $d = 1$ we get the circle rotation $x \mapsto e^{\i \alpha} x$.
+\begin{fact}
+    \label{fact:tau1minimal}
+    The circle rotation $x \mapsto e^{\i \alpha} x$ is minimal.
+    In fact, every subgroup of $S^1$ is either dense in $S^1$
+    or it is of the form
+    \[
+    H_m \coloneqq  \{x \in S^1 : x^m = 0\}
+    \]
+    for some $m \in \Z$.
+\end{fact}
+\todo{Homework!}
+We will show that $\tau_d$ is minimal for all $d$,
+i.e.~every orbit is dense.
+From this it will follow that $\tau$ is minimal.
+
+Let $\pi_n\colon X \to  (S^1)^n$ be the projection to the first $n$
+coordinates.
+
+
+
+
+
+\begin{lemma}
+    Let $x,x' \in X$ with $\pi_n(x) = \pi_n(x')$
+    for some $n$.
+    Then there is a sequence of points $x_k$ with
+    \[\pi_{n-1}(x_k) = \pi_{n-1}(x) = \pi_{n-1}(x')\]
+    for all $k$
+    and
+    \[
+    F(x_k, x) \xrightarrow{k \to \infty} 0,
+    F(x_k, x') \xrightarrow{k \to  \infty} 0,
+    \]
+    where $F$ is as in \yaref{def:F},
+    i.e.~$F(a,b) = \inf_{n \in \Z} d(\tau^n a, \tau^n b)$,
+    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}
+    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)\\
+        x' &=& (\alpha^0_1, \alpha^0_2, \ldots, \alpha^0_{n-1}, \alpha_n, \alpha'_{n+1}, \alpha'_{n+2},\ldots).\\
+    \end{IEEEeqnarray*}
+    We will choose $x_k$ of the form
+    \[
+        (\alpha^0_1, \alpha^0_2, \ldots, \alpha^0_{n-1} \alpha_n e^{\i \beta_k}, \alpha_{n+1}, \alpha_{n+2}, \ldots),
+    \]
+    where $\beta_k$ is such that $\frac{\beta_k}{\pi}$ is irrational
+    and $|\beta_k| < 2^{-k}$.
+    Fix a sequence of such $\beta_k$.
+    Then
+    \[d(x_k,x) = 2^{-n} |e^{\i \beta_k} - 1| < 2^{-n-k} \xrightarrow{k\to \infty} 0.\]
+    In particular $F(x_k, x) \to 0$.
+
+    We want to show that $F(x_k, x') < 2^{-n-k}$.
+    For $u, u' \in X$,
+    $u = (\xi_n)_{n \in \N}$,
+    $u' = (\xi'_n)_{n \in \N}$,
+    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 $1$.
+    Consider
+    \[
+        w_k \coloneqq \frac{x_k}{x'} = (\underbrace{1,\ldots,1}_{n-1}, e^{\i \beta_k}, \overbrace{\frac{\alpha_{n+1}}{\alpha'_{n+1}}, \frac{\alpha_{n+2}}{\alpha'_{n+2}}, \ldots}^{\mathclap{\text{not interesting}}}).
+    \]
+    \begin{claim}
+        $F(x_k, x') = \inf_m d(\sigma^m(w_k), 1)$,
+        where  $\sigma(\xi_1, \xi_2, \ldots) = (\xi_1, \xi_1\xi_2, \xi_2\xi_3, \ldots)$.
+    \end{claim}
+    \begin{subproof}
+        We have
+        \begin{IEEEeqnarray*}{rCl}
+            F(u,u') &=& \inf_m d(\tau^m u, \tau^m u')\\
+                    &=& \inf_m d(\frac{\tau^m u}{\tau^m u'}, 1)\\
+                    &=& \inf_m d(\sigma^m\left( \frac{u}{u'} \right), 1).
+        \end{IEEEeqnarray*}
+    \end{subproof}
+
+    Fix $k$. Let $w^\ast = (1,\ldots,1, e^{\i \beta_k}, 1, \ldots)$.
+    By minimality of $(X,T)$ for any $\epsilon >0$,
+    there exists $m \in \Z$ such that 
+    $d(\sigma^m w_k, w^\ast) < \epsilon$.
+    % TODO Think about this
+    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}
+
+
+
+
+

inputs/tutorial_11.tex

@@ -0,0 +1,113 @@
+\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}

logic3.tex

@@ -44,10 +44,7 @@
 \input{inputs/lecture_17}
 \input{inputs/lecture_18}
 \input{inputs/lecture_19}
-
+\input{inputs/lecture_20}
-
-
-
 
 \cleardoublepage
 
@@ -64,6 +61,7 @@
 \input{inputs/tutorial_08}
 \input{inputs/tutorial_09}
 \input{inputs/tutorial_10}
+\input{inputs/tutorial_11}
 
 \section{Facts}
 \input{inputs/facts}