From 6dbd814b905e3cd0e29a545b8e113ddf355d7cde Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Tue, 14 Nov 2023 12:33:47 +0100 Subject: [PATCH] updated advanced dyn systems + fixed points --- .../2nd/Linear_geometry/Linear_geometry.tex | 10 +- .../Advanced_dynamical_systems.tex | 114 +++++++++++++++--- preamble_formulas.sty | 3 + 3 files changed, 106 insertions(+), 21 deletions(-) diff --git a/Mathematics/2nd/Linear_geometry/Linear_geometry.tex b/Mathematics/2nd/Linear_geometry/Linear_geometry.tex index 24a1def..b42e94c 100644 --- a/Mathematics/2nd/Linear_geometry/Linear_geometry.tex +++ b/Mathematics/2nd/Linear_geometry/Linear_geometry.tex @@ -732,7 +732,7 @@ \end{enumerate} \end{proposition} \begin{proposition} - If the set of fixed points of an affinity $f$, $\text{Fix}(f)$, is non-empty, then $\text{Fix}(f)$ is a subvariety. + If the set of fixed points of an affinity $f$, $\Fix(f)$, is non-empty, then $\Fix(f)$ is a subvariety. \end{proposition} \begin{definition} Let $f$ be an affinity. We define the \emph{invariance level} of $f$, $\rho(f)$, as: $$\rho(f)=\min\{\dim L:f(L)\subset L\subset\mathbb{A}\}\in\{0,\ldots,\dim\mathbb{A}\}$$ @@ -743,7 +743,7 @@ \begin{proposition}[Properties of translations] Let $T_{\vf{v}}$ be a translation. Then: \begin{enumerate} - \item $\text{Fix}(T_{\vf{v}})=\varnothing$. + \item $\Fix(T_{\vf{v}})=\varnothing$. \item Invariant lines are those with director subspace $\langle\vf{v}\rangle$. \item If $\mathcal{R}=\{P;(\vf{v}_1,\ldots,\vf{v}_n)\}$ is an affine frame, then: $$\vf{M}_\mathcal{R}(T_{\vf{v}})=\left(\begin{array}{cccc|c} 1 & 0 & \cdots & 0 & 1 \\ @@ -762,7 +762,7 @@ \begin{proposition}[Properties of reflections] Let $f$ be a reflection with root $\vf{v}$ and mirror $H=P+E$. Then: \begin{enumerate} - \item $\text{Fix}(f)=H$. + \item $\Fix(f)=H$. \item Invariant lines are those contained on $H$ and those with director subspace $\langle\vf{v}\rangle$. \item If $\mathcal{R}=\{P;(\vf{v}_1,\ldots,\vf{v}_{n-1},\vf{v})\}$ is an affine frame such that $P\in H$ and $\vf{v}_1,\ldots,\vf{v}_{n-1}\in E$, then $$\vf{M}_\mathcal{R}(f)=\left(\begin{array}{cccc|c} 1 & 0 & \cdots & 0 & 0 \\ @@ -781,7 +781,7 @@ \begin{proposition}[Properties of projections] Let $f$ be a projection over $H=P+E$ in the direction of $\vf{v}$. Then: \begin{enumerate} - \item $\text{Fix}(f)=H$. + \item $\Fix(f)=H$. \item Invariant lines are those contained on $H$. \item If $\mathcal{R}=\{P;(\vf{v}_1,\ldots,\vf{v}_{n-1},\vf{v})\}$ is an affine frame such that $P\in H$ and $\vf{v}_1,\ldots,\vf{v}_{n-1}\in E$, then $$\vf{M}_\mathcal{R}(f)=\left(\begin{array}{cccc|c} 1 & 0 & \cdots & 0 & 0 \\ @@ -801,7 +801,7 @@ Let $f$ be a homothety of similitude ratio $\lambda$. Then: \begin{enumerate} \item $f$ has a unique fixed. - \item If $\mathcal{R}=\{P;\mathcal{B}\}$ is an affine frame with $P\in\text{Fix}(f)$ and $\mathcal{B}$ an arbitrary basis, then $$\vf{M}_\mathcal{R}(f)=\left(\begin{array}{ccc|c} + \item If $\mathcal{R}=\{P;\mathcal{B}\}$ is an affine frame with $P\in\Fix(f)$ and $\mathcal{B}$ an arbitrary basis, then $$\vf{M}_\mathcal{R}(f)=\left(\begin{array}{ccc|c} & & & 0 \\ & \lambda\vf{I} & & \vdots \\ & & & 0 \\ diff --git a/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex b/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex index 053a10f..c2d3b6a 100644 --- a/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex +++ b/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex @@ -255,25 +255,25 @@ \begin{definition} We define the set: \begin{multline*} - D^0(\TT^1):\{f:\RR\to\RR:f\text{ increasing and}\\ + \mathcal{D}^0(\TT^1):\{f:\RR\to\RR:f\text{ increasing and}\\ \text{ homeomorphism}, f(x+1)=f(x)+1\} \end{multline*} Note that we have the projection: $$ - \function{}{D^0(\TT^1)}{\Homeo(\TT^1)}{f}{F} + \function{}{\mathcal{D}^0(\TT^1)}{\Homeo(\TT^1)}{f}{F} $$ - We can define a distance in $D^0(\TT^1)$ as: + We can define a distance in $\mathcal{D}^0(\TT^1)$ as: $$ d(f,g)=\max\{ \sup_{x\in\RR}\abs{f(x)-g(x)},\sup_{x\in\RR}\abs{f^{-1}(x)-g^{-1}(x)}\} $$ \end{definition} \begin{lemma} - $D^0(\TT^1)$ is a complete metric space. Moreover: + $\mathcal{D}^0(\TT^1)$ is a complete metric space. Moreover: \begin{enumerate} - \item $f\to f^{-1}$ is continuous, $f\in D^0(\TT^1)$. - \item $(f,g)\to f\circ g$ is continuous, $(f,g)\in D^0(\TT^1)\times D^0(\TT^1)$. + \item $f\to f^{-1}$ is continuous, $f\in \mathcal{D}^0(\TT^1)$. + \item $(f,g)\to f\circ g$ is continuous, $(f,g)\in \mathcal{D}^0(\TT^1)\times \mathcal{D}^0(\TT^1)$. \end{enumerate} - Thus, $D^0(\TT^1)$ is a topological group with the composition. + Thus, $\mathcal{D}^0(\TT^1)$ is a topological group with the composition. \end{lemma} \begin{definition} Let $\varepsilon\geq 0$ and $\alpha\in\RR$. We define the \emph{Arnold family} as: @@ -282,7 +282,7 @@ $$ \end{definition} \begin{lemma} - If $0\leq \varepsilon<\frac{1}{2\pi}$, then $f_{\alpha,\varepsilon}\in D^0(\TT^1)$. + If $0\leq \varepsilon<\frac{1}{2\pi}$, then $f_{\alpha,\varepsilon}\in \mathcal{D}^0(\TT^1)$. \end{lemma} \begin{proof} 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. @@ -296,7 +296,7 @@ with $\varphi_n$ 1-periodic. \end{remark} \begin{lemma}\label{ADS:lema1} - Let $f\in D^0(\TT^1)$ be such that $f=\id +\varphi$, with $\varphi$ 1-periodic. Let $m:=\min_{x\in\RR}\varphi$ and $M:=\max_{x\in\RR}\varphi$. Then, we have $m\leq M< m+1$. + Let $f\in \mathcal{D}^0(\TT^1)$ be such that $f=\id +\varphi$, with $\varphi$ 1-periodic. Let $m:=\min_{x\in\RR}\varphi$ and $M:=\max_{x\in\RR}\varphi$. Then, we have $m\leq M< m+1$. \end{lemma} \begin{proof} Let $0\leq x\leq y<1\leq x+1$. Then, $f(y)-1$ (using the same argument as before) and so $\norm{\psi_n}_{\mathcal{C}(\TT^1)}<1$. \end{proof} - \begin{proposition} - Let $f\in D^0(\TT^1)$, $p\in\ZZ$ and $q\in\NN$. Then: + \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: \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 \\ \rho(f)<\frac{p}{q} & \iff \forall x\in\RR\text{ we have }f^q(x)x$ and write $f=\id+\varphi$ with $\varphi\in\mathcal{C}(\TT^1)$ and $\varphi>0$. Since, $\TT^1$ is compact, we have in fact that $\varphi\geq \min\varphi=:\varepsilon>0$. Now: + $$ + f^n-\id = \sum_{i=0}^{n-1}\varphi\circ f^i\geq n\varepsilon + $$ + And so $\rho(f)\geq \varepsilon>0$. + \item Proceed as in the previous case. + \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: + $$ + \mathcal{O}_F(x):=\{F^n(x):n\in\ZZ\} + $$ + We also define the \emph{positive orbit} of $x$ and the \emph{negative orbit} of $x$ as: + \begin{align*} + \mathcal{O}_F^+(x) & :=\{F^n(x):n\in\ZZ_{\geq 0}\} \\ + \mathcal{O}_F^-(x) & :=\{F^n(x):n\in\ZZ_{\leq 0}\} + \end{align*} + 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 $F(X)=X$. + \end{definition} + \begin{proposition} + Let $X\subset \TT^1$ and $x\in \TT^1$. Then: + \begin{enumerate} + \item $X$ is invariant $\iff \forall x\in X$, $\mathcal{O}(x)\subseteq X\iff X$ is a union of orbits. + \item $\mathcal{O}(x)$ is finite $\iff x$ is periodic. + \item The omega limit $\omega(x)$ and the alpha limit $\alpha(x)$ are non-empty compact invariant sets. + \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: + \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. + \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: + $$ + \Omega(F):=\{ x\in\TT^1: x\text{ is not wandering}\} + $$ + \end{definition} + \begin{remark} + 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. + \end{proposition} + \begin{remark} + Note that: + $$ + \Fix(F)\subseteq \Per(F)\subseteq R^\pm(F)\subseteq \Omega(F)\subseteq \TT^1 + $$ + \end{remark} + \begin{definition} + Let $F\in \Homeo(\TT^1)$ and $X\subseteq \TT^1$ be closed and invariant. 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, $Y=X$. + \end{proposition} + \begin{theorem} + Let $F\in \Homeo(\TT^1)$ 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. + \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)