Please use this identifier to cite or link to this item: http://hdl.handle.net/2445/164371
 Title: Herman rings and Arnold disks Author: Buff, XavierFagella Rabionet, NúriaGeyer, LukasHenriksen, Christian Keywords: Sistemes dinàmics complexosFuncions de variables complexesComplex dynamical systemsFunctions of complex variables Issue Date: 2005 Publisher: London Mathematical Society Abstract: For $(\l,a)\in \C^*\times \C$, let $f_{\l,a}$ be the rational map defined by $$f_{\l,a}(z) = \l z^2 \frac{az+1}{z+a}.$$ If $\a\in \R/\Z$ is a Bruno number, we let ${\cal D}_\a$ be the set of parameters $(\l,a)$ such that $f_{\l,a}$ has a fixed Herman ring with rotation number $\a$ (we consider that $(\ex^{2i\pi\a},0)\in {\cal D}_\a$). The results obtained in \cite{mcs} imply that for any $g\in {\cal D}_\a$ the connected component of ${\cal D}_\a\cap (\C^*\times(\C\setminus \{0,1\}))$ which contains $g$ is isomorphic to a punctured disk. In this article, we show that there is an isomorphism $\F_\a:\D\to {\cal D}_\a$ such that $$\F_\a(0) = (\ex^{2i\pi \a},0)\quad{\rm and}\quad \F_\a'(0)=(0,r_\a),$$ where $r_\a$ is the conformal radius at $0$ of the Siegel disk of the quadratic polynomial $z\mapsto \ex^{2i\pi \a}z(1+z)$. In particular, ${\cal D}_\a$ is a Riemann surface isomorphic to the unit disk. As a consequence, we show that for $a\in (0,1/3)$, if $f_{\l,a}$ has a fixed Herman ring with rotation number $\a$ and if $m_a$ is the modulus of the Herman ring, then, as $a\to 0$, we have $\ex^{\pi m_a} = \frac{r_\a}{a} + {\cal O}(a).$ We finally explain how to adapt the results to the complex standard family $z\mapsto \l z \ex^{\frac{a}{2}(z-1/z)}$. Note: Versió postprint del document publicat a: https://doi.org/10.1112/S0024610705007015 It is part of: Journal of the London Mathematical Society-Second Series, 2005, vol. 72, num. 3, p. 689-716 URI: http://hdl.handle.net/2445/164371 Related resource: https://doi.org/10.1112/S0024610705007015 ISSN: 0024-6107 Appears in Collections: Articles publicats en revistes (Matemàtiques i Informàtica)

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