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http://hdl.handle.net/2445/164371
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DC Field | Value | Language |
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dc.contributor.author | Buff, Xavier | - |
dc.contributor.author | Fagella Rabionet, Núria | - |
dc.contributor.author | Geyer, Lukas | - |
dc.contributor.author | Henriksen, Christian | - |
dc.date.accessioned | 2020-06-05T06:45:28Z | - |
dc.date.available | 2020-06-05T06:45:28Z | - |
dc.date.issued | 2005 | - |
dc.identifier.issn | 0024-6107 | - |
dc.identifier.uri | http://hdl.handle.net/2445/164371 | - |
dc.description.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 $z\mapsto \l z \ex^{\frac{a}{2}(z-1/z)}$. | - |
dc.format.extent | 28 p. | - |
dc.format.mimetype | application/pdf | - |
dc.language.iso | eng | - |
dc.publisher | London Mathematical Society | - |
dc.relation.isformatof | Versió postprint del document publicat a: https://doi.org/10.1112/S0024610705007015 | - |
dc.relation.ispartof | Journal of the London Mathematical Society-Second Series, 2005, vol. 72, num. 3, p. 689-716 | - |
dc.relation.uri | https://doi.org/10.1112/S0024610705007015 | - |
dc.rights | (c) London Mathematical Society, 2005 | - |
dc.source | Articles publicats en revistes (Matemàtiques i Informàtica) | - |
dc.subject.classification | Sistemes dinàmics complexos | - |
dc.subject.classification | Funcions de variables complexes | - |
dc.subject.other | Complex dynamical systems | - |
dc.subject.other | Functions of complex variables | - |
dc.title | Herman rings and Arnold disks | - |
dc.type | info:eu-repo/semantics/article | - |
dc.type | info:eu-repo/semantics/acceptedVersion | - |
dc.identifier.idgrec | 550480 | - |
dc.date.updated | 2020-06-05T06:45:29Z | - |
dc.rights.accessRights | info:eu-repo/semantics/openAccess | - |
Appears in Collections: | Articles publicats en revistes (Matemàtiques i Informàtica) |
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File | Description | Size | Format | |
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550480.pdf | 638.69 kB | Adobe PDF | View/Open |
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