diff --git a/inputs/lecture_18.tex b/inputs/lecture_18.tex index 3ce94dc..3587f49 100644 --- a/inputs/lecture_18.tex +++ b/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\}. diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index d8fb2f7..8cecc60 100644 --- a/inputs/lecture_19.tex +++ b/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\}. diff --git a/inputs/lecture_20.tex b/inputs/lecture_20.tex new file mode 100644 index 0000000..f151903 --- /dev/null +++ b/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} + + + + + diff --git a/inputs/tutorial_11.tex b/inputs/tutorial_11.tex new file mode 100644 index 0000000..716e0c6 --- /dev/null +++ b/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} diff --git a/logic3.tex b/logic3.tex index 01247c9..24fbe79 100644 --- a/logic3.tex +++ b/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}