From 6eba6b70eb231f95728a8126ae73b81059d94b43 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Mon, 4 Dec 2023 20:40:52 +0100 Subject: [PATCH] updated typo analysy2 + dynamical systems --- .../Mathematical_analysis.tex | 14 ++-- .../Advanced_dynamical_systems.tex | 74 +++++++++++++------ preamble_formulas.sty | 2 +- 3 files changed, 59 insertions(+), 31 deletions(-) diff --git a/Mathematics/2nd/Mathematical_analysis/Mathematical_analysis.tex b/Mathematics/2nd/Mathematical_analysis/Mathematical_analysis.tex index 049f65c..67351e3 100644 --- a/Mathematics/2nd/Mathematical_analysis/Mathematical_analysis.tex +++ b/Mathematics/2nd/Mathematical_analysis/Mathematical_analysis.tex @@ -10,7 +10,7 @@ Let $(a_n)$ be a sequence of real numbers. A \emph{numeric series} is an expression of the form $$\sum_{n=1}^\infty a_n$$ We call $a_n$ \emph{general term of the series} and $\displaystyle S_N=\sum_{n=1}^N a_n$, for all $N\in\NN $, \emph{$N$-th partial sum of the series}\footnote{From now on we will write $\sum a_n$ to refer $\displaystyle\sum_{n=1}^\infty a_n$.}. \end{definition} \begin{definition} - We say the series $\sum a_n$ is \emph{convergent} if the sequence of partial sums is convergent, that is, if $\displaystyle S=\lim_{N\to\infty}S_N$ exists and it is finite. In that case, $S$ is called the \emph{sum of the series}. If the previous limit doesn't exists or it is infinite, we say the series is \emph{divergent}\footnote{We will use the notation $\sum a_n<\infty$ or $\sum a_n=+\infty$ to express that the series converges or diverges, respectively.}. + We say the series $\sum a_n$ is \emph{convergent} if the sequence of partial sums is convergent, that is, if $\displaystyle S=\lim_{N\to\infty}S_N$ exists and it is finite. In that case, $S$ is called the \emph{sum of the series}. If the previous limit doesn't exist or it is infinite, we say the series is \emph{divergent}\footnote{We will use the notation $\sum a_n<\infty$ or $\sum a_n=+\infty$ to express that the series converges or diverges, respectively.}. \end{definition} \begin{proposition} Let $(a_n)$ be a sequence such that $\sum a_n<\infty$. Then, $\forall\varepsilon>0$ $\exists n_0\in\NN $ such that $$\left|\sum_{n=1}^N a_n-\sum_{n=1}^\infty a_n\right|<\varepsilon$$ if $N\geq n_0$. @@ -431,7 +431,7 @@ \begin{enumerate} \item $\langle f,f\rangle\geq 0$. \item $\langle f+h,g\rangle=\langle f,g\rangle+\langle h,g\rangle$ and $\langle f,g+h\rangle=\langle f,g\rangle+\langle f,h\rangle$. - \item\label{MA:orto3} $\langle f,g\rangle=\overline{\langle g,f\rangle}$. + \item\label{MA:orto3} $\langle f,g\rangle=\overline{\langle g,f\rangle}$. \item $\langle \alpha f,g\rangle=\alpha\langle f,g\rangle$ and $\langle f,\alpha g\rangle=\overline{\alpha}\langle f,g\rangle$. \end{enumerate} \end{proposition} @@ -491,7 +491,7 @@ \begin{enumerate} \item For all $\lambda,\mu\in\CC $: $$\widehat{(\lambda f+\mu g)}(n)=\lambda\widehat{f}(n)+\mu\widehat{g}(n)$$ - \item\label{MA:fouriercoeffs2} Let $\tau\in\RR $. We define $f_\tau(x)=f(x-\tau)$. Then: $$\widehat{f_\tau}(n)=\exp{-\frac{2\pi\ii n\tau}{T}}\widehat{f}(n)$$ + \item\label{MA:fouriercoeffs2} Let $\tau\in\RR $. We define $f_\tau(x)=f(x-\tau)$. Then: $$\widehat{f_\tau}(n)=\exp{-\frac{2\pi\ii n\tau}{T}}\widehat{f}(n)$$ \item If $f$ is even, then $\widehat{f}(n)=\widehat{f}(-n)$, $\forall n\in\ZZ $.\newline If $f$ is odd, then $\widehat{f}(n)=-\widehat{f}(-n)$, $\forall n\in\ZZ $. \item If $f\in \mathcal{C}^k$ such that $f^{(r)}(-T/2)=f^{(r)}(T/2)$ $\forall r=0,\ldots,k-1$, then $$\widehat{f^{(k)}}(n)=\left(\frac{2\pi\ii n}{T}\right)^k\widehat{f}(n)$$ \item $\widehat{(f*g)}(n)=\widehat{f}(n)\widehat{g}(n)$. @@ -613,8 +613,8 @@ \begin{align*} S_Nf(x_0)-\ell & =\frac{1}{T}\int_0^{T/2}[f(x_0+t)+f(x_0-t) - 2\ell]D_N(t)\dd{t} \\ \begin{split} - &=\frac{1}{T}\int_{0}^{T/2}\frac{f(x_0+t)+f(x_0-t) - 2\ell}{t}\frac{t}{\sin\left(\frac{\pi t}{T}\right)}\cdot\\ - &\hspace{3.4cm}\cdot\sin\left(\frac{(2N+1)\pi t}{T}\right)\dd{t} + & =\frac{1}{T}\int_{0}^{T/2}\frac{f(x_0+t)+f(x_0-t) - 2\ell}{t}\frac{t}{\sin\left(\frac{\pi t}{T}\right)}\cdot \\ + & \hspace{3.4cm}\cdot\sin\left(\frac{(2N+1)\pi t}{T}\right)\dd{t} \end{split} \end{align*} Since the first terms form an integrable function, we can use now the \mnameref{MA:riemannlebesgue}. @@ -707,8 +707,8 @@ \begin{align*} \abs{\sigma_Nf(x)-f(x)}= & \abs{\int_0^{T/2}[f(x-t)-f(x)]F_N(t)\dd{t}} \\ \begin{split} - & \leq \int_0^{\delta}\abs{f(x-t)-f(x)}F_N(t)\dd{t}+\\ - &\hspace{1cm}+\int_\delta^{T/2}\abs{f(x-t)-f(x)}F_N(t)\dd{t} + & \leq \int_0^{\delta}\abs{f(x-t)-f(x)}F_N(t)\dd{t}+ \\ + & \hspace{1cm}+\int_\delta^{T/2}\abs{f(x-t)-f(x)}F_N(t)\dd{t} \end{split} \end{align*} To bound the first integral use the uniform continuity of $f$ in $[0,\delta]$ and for the second one use \mcref{MA:fejerprop}. diff --git a/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex b/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex index d265e0c..ca72b14 100644 --- a/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex +++ b/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex @@ -255,8 +255,7 @@ \begin{definition} We define the set: \begin{multline*} - \mathcal{D}^0(\TT^1):=\{f:\RR\to\RR:f\text{ increasing and}\\ - \text{ homeomorphism}, f(x+1)=f(x)+1\} + \mathcal{D}^0(\TT^1):=\{f\in \Homeo(\RR):f\text{ increasing and}\\ f(x+1)=f(x)+1\} \end{multline*} Note that we have the projection: $$ @@ -289,7 +288,7 @@ Note that ${f_{\alpha,\varepsilon}}'>0\iff \varepsilon<\frac{1}{2\pi}$. Thus, $f_{\alpha,\varepsilon}$ is strictly increasing, and therefore it is a homeomorphism. Moreover, $f_{\alpha,\varepsilon}(x+1)=f_{\alpha,\varepsilon}(x)+1$. \end{proof} \subsubsection{Rotation number}\label{ADS:rotation_number_section} - \begin{lemma} + \begin{lemma}\label{ADS:lema_sum} Recall that $f=\id+\varphi$ with $\varphi$ 1-periodic. And thus: $$ f^n=\id + \sum_{i=0}^{n-1} \varphi\circ f^i=: \id + \varphi_n @@ -322,10 +321,10 @@ Now take the supremum in $x$. The other inequality is analogous. \end{proof} \begin{lemma}\label{ADS:lema3} - Let $(u_n)\in\RR$ be a subadditive sequence. Then, $\displaystyle\lim_{n\to\infty}\frac{u_n}{n}$ exists and it is equal to $\inf_{n\in\NN}\frac{u_n}{n}$. Analogously, if $(u_n)$ is superadditive, then $\displaystyle\lim_{n\to\infty}\frac{u_n}{n}$ exists and it is equal to $\sup_{n\in\NN}\frac{u_n}{n}$. + Let $(u_n)\in\RR$ be a subadditive sequence. Then, $\displaystyle\lim_{n\to\infty}\frac{u_n}{n}$ exists, and it is equal to $\displaystyle\inf_{n\in\NN}\frac{u_n}{n}$. Analogously, if $(u_n)$ is superadditive, then $\displaystyle\lim_{n\to\infty}\frac{u_n}{n}$ exists, and it is equal to $\displaystyle\sup_{n\in\NN}\frac{u_n}{n}$. \end{lemma} \begin{proof} - Assume $(u_n)$ is positive and subadditive and fix $p\in\NN$. Let $n\geq p$ be such that $n=k_np+r_n$ with $r-1$ (using the same argument as before) and so $\norm{\psi_n}_{\mathcal{C}(\TT^1)}<1$. + Dividing by $n$ and taking limits, we have that $\rho(f)=\mu(\varphi)$. This also shows the second point. To prove the first one, note that $\min\psi_n\leq 0$ and so by \mcref{ADS:lema1} we have $\max\psi_n <1$. Moreover, $\min\psi_n =-\max(-\psi_n) >-1$ (using the same argument as before) and so $\norm{\psi_n}_{\mathcal{C}(\RR)}<1$ for all $n\in\NN$. \end{proof} \subsubsection{Rational rotation number} \begin{proposition}\label{ADS:characterisation_rot_number} - Let $f\in \mathcal{D}^0(\TT^1)$, $p\in\ZZ$ and $q\in\NN$. Then: + Let $f\in \mathcal{D}^0(\TT^1)$, $p\in\ZZ$ and $q\in\NN$ be such that the fraction $\frac{p}{q}$ is irreducible. Then: \begin{align*} \rho(f)=\frac{p}{q} & \iff \exists x\in\RR\text{ such that }f^q(x)=x+p \\ \rho(f)>\frac{p}{q} & \iff \forall x\in\RR\text{ we have }f^q(x)>x+p \\ @@ -484,7 +502,7 @@ Let $F\in \Homeoplus(\TT^1)$. Then, $R^\pm(F)$ are invariant non-closed sets. \end{proposition} \begin{definition} - Let $F\in \Homeoplus(\TT^1)$ and $x\in \TT^1$. We say that $x$ is a \emph{wandering point} if there exists a neighborhood $U$ of $x$ such that $\forall n\geq 1$ we have $F^n(U)\cap U=\varnothing$. The orbit $U$ is called a \emph{wandering domain}. We define the set: + Let $F\in \Homeoplus(\TT^1)$ and $x\in \TT^1$. We say that $x$ is a \emph{wandering point} if there exists a neighborhood $U$ of $x$ such that $\forall n\geq 1$ we have $F^n(U)\cap U=\varnothing$. The neighborhood $U$ is called a \emph{wandering domain}. We define the set: $$ \Omega(F):=\{ x\in\TT^1: x\text{ is not wandering}\} $$ @@ -495,18 +513,28 @@ \begin{proposition} Let $F\in \Homeoplus(\TT^1)$. Then, $\Omega(F)$ is an invariant closed set. \end{proposition} - \begin{remark} - Note that: + \begin{lemma} + Let $F \in \Homeoplus(\TT^1)$. Then: $$ \Fix(F)\subseteq \Per(F)\subseteq R^\pm(F)\subseteq \Omega(F)\subseteq \TT^1 $$ - \end{remark} + \end{lemma} + \begin{proof} + All the inclusions are clear except for maybe $R^\pm(F)\subseteq \Omega(F)$. Let $x\in R^\pm(F)$. Then, $\exists (n_k)\in\NN$ with $n_k \nearrow \infty$ such that $F^{n_k}(x)\to x$. Now, let $U$ be a neighborhood of $x$. Then, $x\in U$ and thus $F^{n_1}(U)\cap U\ne\varnothing$. So $x\in \Omega(F)$. + \end{proof} \begin{definition} - Let $F\in \Homeoplus(\TT^1)$ and $X\subseteq \TT^1$ be nonempty closed invariant set. We say that $X$ is \emph{minimal} if $\forall x\in X$, $\overline{\mathcal{O}(x)}=X$. If $X=\TT^1$, we say that $F$ is \emph{minimal}. + Let $F\in \Homeoplus(\TT^1)$ and $X\subseteq \TT^1$ be a non-empty closed invariant set. We say that $X$ is \emph{minimal} if $\forall x\in X$, $\overline{\mathcal{O}(x)}=X$. If $X=\TT^1$, we say that $F$ is \emph{minimal}. \end{definition} \begin{proposition} Let $F\in \Homeoplus(\TT^1)$ and $X\subseteq \TT^1$ be a closed and invariant. Then, $X$ is minimal $\iff$ $\forall Y\subseteq X$ closed, invariant and non-empty, $Y=X$. \end{proposition} + \begin{proof} + \begin{itemizeiff} + For some $y\in Y\subseteq X$ closed, invariant and non-empty, we have: + $$Y\subseteq X = \overline{\mathcal{O}(y)}\subseteq \overline{\mathcal{O}(Y)}\subseteq \overline{Y}=Y$$ + \item Let $x\in X$. Since $\overline{\mathcal{O}(x)}\subseteq X$ is closed, invariant and non-empty, we have that $\overline{\mathcal{O}(x)}=X$. + \end{itemizeiff} + \end{proof} \begin{theorem} Let $F\in \Homeoplus(\TT^1)$ with $\rho(F)=\frac{p}{q}\in \quot{\QQ}{\ZZ}$. Then: \begin{enumerate} @@ -515,7 +543,7 @@ \end{enumerate} \end{theorem} \begin{proof} - First we assume $q=1$ and $p=0$. Let $f\in \mathcal{D}^0(\TT^1)$ be a lift of $F$. By \mcref{ADS:characterisation_rot_number}, we have that $\exists x\in \RR$ with $f(x)=x$. So $\Fix(f)\ne \varnothing$, and it is closed and invariant by translations. Now we write $\RR\setminus\Fix(f)$ as union of open intervals. Let $(a,b)$ be one connected component. Inside it, we must have either $f(x)>x$ or $f(x)x$ or $f(x)0$, $(n_k)\in\NN$ with $n_k\nearrow +\infty$ and $(x_k)\in\TT^1$ such that $\forall k\geq 0$: diff --git a/preamble_formulas.sty b/preamble_formulas.sty index 210f855..4c6db19 100644 --- a/preamble_formulas.sty +++ b/preamble_formulas.sty @@ -304,7 +304,7 @@ \newcommand{\conn}{\mathrel{\#}} % connected sum. \mathrel gives the space of a relation (like +,-,...) while \mathbin gives the space of a binary operator (like =). \renewcommand{\S}{S} % S of the S ^ n (n-th dimensional sphere) \newcommand{\Homeo}{\mathrm{Homeo}} % set of homeomorphisms -\newcommand{\Homeoplus}{\mathrm{Homeo}^+} % set of orientation-preserving homeomorphisms +\newcommand{\Homeoplus}{\mathrm{Homeo}_+} % set of orientation-preserving homeomorphisms \newcommand{\Diff}{\mathrm{Diff}} % set of diffeomorphisms \newcommand{\Diffplus}{\mathrm{Diff}_+} % set of orientation-preserving diffeomorphisms