From a79b7ba55782f229f75bf0b39847fab07f408cd4 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Tue, 28 Nov 2023 15:36:53 +0100 Subject: [PATCH] solve overfulls --- .../Advanced_dynamical_systems.tex | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex b/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex index cc933f6..78452b0 100644 --- a/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex +++ b/Mathematics/5th/Advanced_dynamical_systems/Advanced_dynamical_systems.tex @@ -633,17 +633,17 @@ \end{proof} \begin{definition} For $k\in\NN\cup\{0\}$ we define the set $\mathcal{D}^k(\TT^1)$ as: - $$ - \mathcal{D}^k(\TT^1):=\{f:\RR\to\RR \text{ $\mathcal{C}^k$-diffeomorphism such that }f(x+1)=f(x)+1\} - $$ + \begin{multline*} + \mathcal{D}^k(\TT^1):=\{f:\RR\to\RR \text{ $\mathcal{C}^k$-diffeomorphism such that}\\f(x+1)=f(x)+1\} + \end{multline*} Note that $f\in \mathcal{D}^k(\TT^1)$ if and only if $f=\id +\varphi$, with $\varphi\in\mathcal{C}^k(\TT^1)$. We also define the set $\Diffplus^k(\TT^1)$ as: - $$ - \Diffplus^k(\TT^1):=\{F:\TT^1\to\TT^1 \text{ $\mathcal{C}^k$-diffeomorphism with orientation preserving}\} - $$ + \begin{multline*} + \Diffplus^k(\TT^1):=\{F:\TT^1\to\TT^1 \text{ $\mathcal{C}^k$-diffeomorphism with}\\\text{orientation preserving}\} + \end{multline*} \end{definition} \begin{proposition} - Let $F\in\Diffplus^1(\TT^1)$ with $\rho(F)\notin\quot{\QQ}{\ZZ}$, $\mu$ be the unique invariant probability measure of $F$ and $f\in\mathcal{D}^1(\TT^1)$ be a lift of $F$. Then, $\displaystyle \lim_{n\to \infty}\frac{1}{n}\log Df^n(x)=\int_{\TT^1}\log(Df)\dd{\mu}=0$ + Let $F\in\Diffplus^1(\TT^1)$ with $\rho(F)\notin\quot{\QQ}{\ZZ}$, $\mu$ be the unique invariant probability measure of $F$ and $f\in\mathcal{D}^1(\TT^1)$ be a lift of $F$. Then, $\displaystyle \lim_{n\to \infty}\frac{1}{n}\log Df^n(x)=\int_{\TT^1}\log(Df)\dd{\mu}=0$. \end{proposition} \end{multicols} \end{document} \ No newline at end of file