solve typo

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:
     $$
@@ -347,16 +347,16 @@
     \end{enumerate}
   \end{proposition}
   \begin{definition}
-    Let $F\in \Homeo(\TT^1)$ with lift $f$. We define the \emph{rotation number} of $F$ as $\rho(F):=[\rho(f)]\in \TT^1$.
+    Let $F\in \Homeoplus(\TT^1)$ with lift $f$. We define the \emph{rotation number} of $F$ as $\rho(F):=[\rho(f)]\in \TT^1$.
   \end{definition}
   \begin{definition}
-    Let $F,G\in\Homeo(\TT^1)$. We say that $G$ is \emph{semi-conjugate} to $F$ if there exists a continuous surjective map $H:\TT^1\to \TT^1$ such that $H\circ F=G\circ H$. We say that $G$ is \emph{conjugate} to $F$ if $H$ is a homeomorphism.
+    Let $F,G\in\Homeoplus(\TT^1)$. We say that $G$ is \emph{semi-conjugate} to $F$ if there exists a continuous surjective map $H:\TT^1\to \TT^1$ such that $H\circ F=G\circ H$. We say that $G$ is \emph{conjugate} to $F$ if $H$ is a homeomorphism.
   \end{definition}
   \begin{lemma}
-    Let $F,G\in\Homeo(\TT^1)$ be such that $G$ is semi-conjugate to $F$. Then, if $G$ has a periodic point, then $F$ has a periodic point.
+    Let $F,G\in\Homeoplus(\TT^1)$ be such that $G$ is semi-conjugate to $F$. Then, if $G$ has a periodic point, then $F$ has a periodic point.
   \end{lemma}
   \begin{theorem}
-    Let $F,G\in\Homeo(\TT^1)$ be conjugate by $H\in \Homeo(\TT^1)$. Then, $\rho(F)=\rho(G)$.
+    Let $F,G\in\Homeoplus(\TT^1)$ be conjugate by $H\in \Homeoplus(\TT^1)$. Then, $\rho(F)=\rho(G)$.
   \end{theorem}
   \begin{proof}
     Let $h$ and $f$ be lifts of $H$ and $F$ respectively. Then, an easy check shows that $g:=h\circ f\circ h^{-1}$ is a lift of $G$. It suffices to prove that $\rho(g)=\rho(f)$. Note that, by induction we have $h\circ f^n=g^n\circ h$ for all $n\in\NN$. Now write $h=\id + \varphi$ with $\varphi\in \mathcal{C}(\TT^1)$. Then:
@@ -388,24 +388,24 @@
     $$\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$
       \item $\displaystyle\frac{\delta_x+\cdots+\delta_{F^{n-1}(x)}}{n}$ is $F$-invariant $\iff F^n(x)=x$
     \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: