From 740a9c8f452e408a01bcd963737553295c46a4d2 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Tue, 28 Nov 2023 15:33:51 +0100 Subject: [PATCH] solve typo --- .../Advanced_dynamical_systems.tex | 48 +++++++++---------- 1 file changed, 24 insertions(+), 24 deletions(-) diff --git a/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex b/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex index 6e4b978..cc933f6 100644 --- a/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex +++ b/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex @@ -241,10 +241,10 @@ Recall that $f:\RR\to\RR$ is a homeomorphism if and only if $f$ is monotone. \end{remark} \begin{definition} - We say that a homeomorphism $F$ \emph{preserves orientation} if and only if $f$ is strictly increasing. We define the set of $\Homeo(\TT^1)$ as the set of homeomorphisms of $\TT^1$ that preserve orientation. + We say that a homeomorphism $F$ \emph{preserves orientation} if and only if $f$ is strictly increasing. We define the set of $\Homeoplus(\TT^1)$ as the set of homeomorphisms of $\TT^1$ that preserve orientation. \end{definition} \begin{proposition} - Let $F\in\Homeo(\TT^1)$. Then, $F$ admits a lift $f$ such that $f(x)=x+\varphi(x)$, where $\varphi:\RR\to\RR$ is a 1-periodic function. + Let $F\in\Homeoplus(\TT^1)$. Then, $F$ admits a lift $f$ such that $f(x)=x+\varphi(x)$, where $\varphi:\RR\to\RR$ is a 1-periodic function. \end{proposition} \begin{proof} We already now that $F$ admits a lift $f$. A straightforward calculation shows that $f_1:\RR\to\RR$ defined by $f_1(x)=f(x+1)$ is also a lift of $F$. Thus, $f_1-f=k\in \ZZ$. Now, since $f$ must be strictly increasing, we need $k\in \NN$. Moreover, since $F$ is injective, $f|_{[0,1)}$ is injective and its image cannot contain 2 points whose difference is an integer. Thus, $k=1$. Now, define $\varphi(x)=f(x)-x$, which is 1-periodic: @@ -260,7 +260,7 @@ \end{multline*} Note that we have the projection: $$ - \function{}{\mathcal{D}^0(\TT^1)}{\Homeo(\TT^1)}{f}{F} + \function{}{\mathcal{D}^0(\TT^1)}{\Homeoplus(\TT^1)}{f}{F} $$ We can define a distance in $\mathcal{D}^0(\TT^1)$ as: $$ @@ -347,16 +347,16 @@ \end{enumerate} \end{proposition} \begin{definition} - Let $F\in \Homeo(\TT^1)$ with lift $f$. We define the \emph{rotation number} of $F$ as $\rho(F):=[\rho(f)]\in \TT^1$. + Let $F\in \Homeoplus(\TT^1)$ with lift $f$. We define the \emph{rotation number} of $F$ as $\rho(F):=[\rho(f)]\in \TT^1$. \end{definition} \begin{definition} - Let $F,G\in\Homeo(\TT^1)$. We say that $G$ is \emph{semi-conjugate} to $F$ if there exists a continuous surjective map $H:\TT^1\to \TT^1$ such that $H\circ F=G\circ H$. We say that $G$ is \emph{conjugate} to $F$ if $H$ is a homeomorphism. + Let $F,G\in\Homeoplus(\TT^1)$. We say that $G$ is \emph{semi-conjugate} to $F$ if there exists a continuous surjective map $H:\TT^1\to \TT^1$ such that $H\circ F=G\circ H$. We say that $G$ is \emph{conjugate} to $F$ if $H$ is a homeomorphism. \end{definition} \begin{lemma} - Let $F,G\in\Homeo(\TT^1)$ be such that $G$ is semi-conjugate to $F$. Then, if $G$ has a periodic point, then $F$ has a periodic point. + Let $F,G\in\Homeoplus(\TT^1)$ be such that $G$ is semi-conjugate to $F$. Then, if $G$ has a periodic point, then $F$ has a periodic point. \end{lemma} \begin{theorem} - Let $F,G\in\Homeo(\TT^1)$ be conjugate by $H\in \Homeo(\TT^1)$. Then, $\rho(F)=\rho(G)$. + Let $F,G\in\Homeoplus(\TT^1)$ be conjugate by $H\in \Homeoplus(\TT^1)$. Then, $\rho(F)=\rho(G)$. \end{theorem} \begin{proof} Let $h$ and $f$ be lifts of $H$ and $F$ respectively. Then, an easy check shows that $g:=h\circ f\circ h^{-1}$ is a lift of $G$. It suffices to prove that $\rho(g)=\rho(f)$. Note that, by induction we have $h\circ f^n=g^n\circ h$ for all $n\in\NN$. Now write $h=\id + \varphi$ with $\varphi\in \mathcal{C}(\TT^1)$. Then: @@ -388,13 +388,13 @@ $$\text{Leb}(\varphi):=\int_{0}^1\varphi(x)\dd{x}$$ \end{remark} \begin{definition} - Let $F\in \Homeo(\TT^1)$ and $\mu\in \mathcal{M}(\TT^1)$. We define the \emph{pushforward measure} as $F_*\mu(\varphi):=\mu(\varphi\circ F)$. + Let $F\in \Homeoplus(\TT^1)$ and $\mu\in \mathcal{M}(\TT^1)$. We define the \emph{pushforward measure} as $F_*\mu(\varphi):=\mu(\varphi\circ F)$. \end{definition} \begin{definition} - We say that a measure $\mu\in\mathcal{M}(\TT^1)$ is \emph{invariant} by $F\in\Homeo(\TT^1)$ (or \emph{$F$-invariant}) if $F_*\mu=\mu$. We will denote by $\mathcal{M}_F(\TT^1)$ the set of $F$-invariant probability measures. + We say that a measure $\mu\in\mathcal{M}(\TT^1)$ is \emph{invariant} by $F\in\Homeoplus(\TT^1)$ (or \emph{$F$-invariant}) if $F_*\mu=\mu$. We will denote by $\mathcal{M}_F(\TT^1)$ the set of $F$-invariant probability measures. \end{definition} \begin{proposition} - Let $F\in \Homeo(\TT^1)$, $x\in\TT^1$ and $n\in\NN$. + Let $F\in \Homeoplus(\TT^1)$, $x\in\TT^1$ and $n\in\NN$. \begin{itemize} \item Note that $\text{Leb}$ is invariant under $R_\alpha$ $\forall \alpha\in\RR$. \item $\delta_x$ is $F$-invariant $\iff F(x)=x$ @@ -402,10 +402,10 @@ \end{itemize} \end{proposition} \begin{theorem} - Let $F\in\Homeo(\TT^1)$. Then, $\mathcal{M}_F(\TT^1)\ne\varnothing$. + Let $F\in\Homeoplus(\TT^1)$. Then, $\mathcal{M}_F(\TT^1)\ne\varnothing$. \end{theorem} \begin{proposition} - Let $F\in\Homeo(\TT^1)$ and $f=\id+\varphi$ be a lift of $F$, with $\varphi\in\mathcal{C}(\TT^1)$. Then, $\forall\mu\in\mathcal{M}_F(\TT^1)$, $\rho(f)=\mu(\varphi)$. Moreover: + Let $F\in\Homeoplus(\TT^1)$ and $f=\id+\varphi$ be a lift of $F$, with $\varphi\in\mathcal{C}(\TT^1)$. Then, $\forall\mu\in\mathcal{M}_F(\TT^1)$, $\rho(f)=\mu(\varphi)$. Moreover: \begin{enumerate} \item $\norm{f^n-\id -n\rho(f)}_{\mathcal{C}(\TT^1)}<1$ \item $\forall n\in\NN$, $\exists x_n\in\RR$ such that $f^n(x_n)-x_n=n\rho(f)$. @@ -444,7 +444,7 @@ \end{enumerate} \end{proof} \begin{definition} - Let $F\in\Homeo(\TT^1)$ and $x\in \TT^1$. We define the \emph{orbit} of $x$ as: + Let $F\in\Homeoplus(\TT^1)$ and $x\in \TT^1$. We define the \emph{orbit} of $x$ as: $$ \mathcal{O}_F(x):=\{F^n(x):n\in\ZZ\} $$ @@ -456,7 +456,7 @@ If the homeomorphism is not specified, we will omit the subscript. \end{definition} \begin{definition} - Let $F\in\Homeo(\TT^1)$ and $X\subset \TT^1$. We say that $X$ is \emph{positively invariant} if $F(X)\subseteq X$ and \emph{negatively invariant} if $F^{-1}(X)\subseteq X$. We say that $X$ is \emph{invariant} if $F(X)=X$. + Let $F\in\Homeoplus(\TT^1)$ and $X\subset \TT^1$. We say that $X$ is \emph{positively invariant} if $F(X)\subseteq X$ and \emph{negatively invariant} if $F^{-1}(X)\subseteq X$. We say that $X$ is \emph{invariant} if $F(X)=X$. \end{definition} \begin{proposition} Let $X\subset \TT^1$ and $x\in \TT^1$. Then: @@ -467,17 +467,17 @@ \end{enumerate} \end{proposition} \begin{definition} - Let $F\in \Homeo(\TT^1)$. We define the \emph{positively recurrent points} and \emph{negatively recurrent points} as: + Let $F\in \Homeoplus(\TT^1)$. We define the \emph{positively recurrent points} and \emph{negatively recurrent points} as: \begin{align*} R^+(F):=\{x\in\TT^1:x\in\omega(x)\} \\ R^-(F):=\{x\in\TT^1:x\in\alpha(x)\} \end{align*} \end{definition} \begin{proposition} - Let $F\in \Homeo(\TT^1)$. Then, $R^\pm(F)$ are invariant non-closed sets. + Let $F\in \Homeoplus(\TT^1)$. Then, $R^\pm(F)$ are invariant non-closed sets. \end{proposition} \begin{definition} - Let $F\in \Homeo(\TT^1)$ and $x\in \TT^1$. We say that $x$ is a \emph{wandering point} if there exists a neighborhood $U$ of $x$ such that $\forall n\geq 1$ we have $F^n(U)\cap U=\varnothing$. The orbit $U$ is called a \emph{wandering domain}. We define the set: + Let $F\in \Homeoplus(\TT^1)$ and $x\in \TT^1$. We say that $x$ is a \emph{wandering point} if there exists a neighborhood $U$ of $x$ such that $\forall n\geq 1$ we have $F^n(U)\cap U=\varnothing$. The orbit $U$ is called a \emph{wandering domain}. We define the set: $$ \Omega(F):=\{ x\in\TT^1: x\text{ is not wandering}\} $$ @@ -486,7 +486,7 @@ A point $x\in\TT^1$ is \emph{non-wandering} if it is not wandering, i.e.\ if $\forall U$ neighborhood of $x$ $\exists n\geq 1$ such that $F^n(U)\cap U\ne\varnothing$. \end{remark} \begin{proposition} - Let $F\in \Homeo(\TT^1)$. Then, $\Omega(F)$ is an invariant closed set. + Let $F\in \Homeoplus(\TT^1)$. Then, $\Omega(F)$ is an invariant closed set. \end{proposition} \begin{remark} Note that: @@ -495,13 +495,13 @@ $$ \end{remark} \begin{definition} - Let $F\in \Homeo(\TT^1)$ 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 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}. \end{definition} \begin{proposition} - Let $F\in \Homeo(\TT^1)$ and $X\subseteq \TT^1$ be a closed and invariant. Then, $X$ is minimal $\iff$ $\forall Y\subseteq X$ closed, invariant and non-empty, $Y=X$. + 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{theorem} - Let $F\in \Homeo(\TT^1)$ with $\rho(F)=\frac{p}{q}\in \quot{\QQ}{\ZZ}$. Then: + Let $F\in \Homeoplus(\TT^1)$ with $\rho(F)=\frac{p}{q}\in \quot{\QQ}{\ZZ}$. Then: \begin{enumerate} \item $F$ has periodic points of period $q$, and any periodic point of $F$ has minimal period $q$. \item For any $x\in \TT^1$, $\omega(x)$ and $\alpha(x)$ are periodic orbits. @@ -539,7 +539,7 @@ $\supp\text{Leb}=\TT^1$ and $\supp\delta_{x}=\{x\}$. \end{remark} \begin{proposition} - Let $\mu\in \mathcal{M}(\TT^1)$ and $F\in\Homeo(\TT^1)$. $\mu$ is invariant by $F$ if and only if $\forall A\subseteq \TT^1$ Borel set, $\mu(A)=\mu(F^{-1}(A))$. + Let $\mu\in \mathcal{M}(\TT^1)$ and $F\in\Homeoplus(\TT^1)$. $\mu$ is invariant by $F$ if and only if $\forall A\subseteq \TT^1$ Borel set, $\mu(A)=\mu(F^{-1}(A))$. \end{proposition} \begin{lemma} Let $\mu\in \mathcal{M}(\TT^1)$. We have a lift to a measure $\mu$ on $\RR$ invariant by integer translations: $\mu(A+k)=\mu(A)$ $\forall k\in\ZZ$ and $A\subseteq \mathcal{B}(\RR)$. @@ -557,7 +557,7 @@ A subset $C\subseteq \RR$ is a \emph{Cantor set} if it is closed, it has no isolated points and it has empty interior. \end{definition} \begin{theorem}\label{ADS:theorem_irrational_rotation_number} - Let $F\in\Homeo(\TT^1)$ with $\rho(F)\notin\quot{\QQ}{\ZZ}$. Then, there exists a surjective continuous map $H:\TT^1\to \TT^1$ such that $H\circ F=R_{\rho(F)}\circ H$. Moreover, we have exactly one of the following two properties: + Let $F\in\Homeoplus(\TT^1)$ with $\rho(F)\notin\quot{\QQ}{\ZZ}$. Then, there exists a surjective continuous map $H:\TT^1\to \TT^1$ such that $H\circ F=R_{\rho(F)}\circ H$. Moreover, we have exactly one of the following two properties: \begin{enumerate} \item $F$ is conjugated to $R_{\rho(F)}$ and in that case $F$ is minimal. \item $\exists X\subsetneq \TT^1$ minimal which is a Cantor set and $X=\Omega(F)$. @@ -597,7 +597,7 @@ where the equality is due to the invariance of $\mu$. So, we also have $\int_{\TT^1}P_n(x)\dd{\mu}=a_0$. Now consider the Féjer trigonometric polynomial that converge uniformly to $\varphi$ and use the \mnameref{RFA:domianted}. \end{proof} \begin{proposition} - Let $F\in\Homeo(\TT^1)$ with $\rho(F)\notin\quot{\QQ}{\ZZ}$. Then, $F$ is uniquely ergodic. + Let $F\in\Homeoplus(\TT^1)$ with $\rho(F)\notin\quot{\QQ}{\ZZ}$. Then, $F$ is uniquely ergodic. \end{proposition} \begin{proof} Let $H$ be such that $H\circ F=R_\rho\circ H$ (by \mcref{ADS:theorem_irrational_rotation_number}) and so $F^{-1}(H^{-1}(A))=H^{-1}({R_\rho}^{-1}(A))$. Define: