Please use this identifier to cite or link to this item: http://hdl.handle.net/2445/164370
 Title: Asymptotic size of Herman rings of the complex standard family by quantitative quasiconformal surgery Author: Fagella Rabionet, NúriaMartínez-Seara Alonso, M. TeresaVillanueva Castelltort, Jordi Keywords: Sistemes dinàmics complexosTeoria geomètrica de funcionsSuperfícies de RiemannComplex dynamical systemsGeometric function theoryRiemann surfaces Issue Date: May-2004 Publisher: Cambridge University Press Abstract: In this paper we consider the complexification of the Arnold standard family of circle maps given by $\widetilde F_{\alpha,\epsilon}(u)=ue^{i\alpha} e^{({\epsilon}/{2}) (u-{1}/{u})}$, with $\alpha=\alpha(\epsilon)$ chosen so that $\widetilde F_{\alpha(\epsilon),\epsilon}$ restricted to the unit circle has a prefixed rotation number $\theta$ belonging to the set of Brjuno numbers. In this case, it is known that $\widetilde F_{\alpha(\epsilon),\epsilon}$ is analytically linearizable if $\epsilon$ is small enough and so it has a Herman ring $\widetilde U_{\epsilon}$ around the unit circle. Using Yoccoz's estimates, one has that the size$\widetilde R_\epsilon$ of $\widetilde U_{\epsilon}$ (so that $\widetilde U_{\epsilon}$ is conformally equivalent to $\{u\in{\mathbb C}: 1/\widetilde R_\epsilon < |u| < \widetilde R_\epsilon\}$) goes to infinity as $\epsilon\to 0$, but one may ask for its asymptotic behavior. We prove that $\widetilde R_\epsilon=({2}/{\epsilon})(R_0+\mathcal{O}(\epsilon\log\epsilon))$, where R0 is the conformal radius of the Siegel disk of the complex semistandard map $G(z)=ze^{i\omega}e^z$, where $\omega= 2\pi\theta$. In the proof we use a very explicit quasiconformal surgery construction to relate $\widetilde F_{\alpha(\epsilon),\epsilon}$ and G, and hyperbolic geometry to obtain the quantitative result. Note: Reproducció del document publicat a: https://doi.org/10.1017/S0143385704000045 It is part of: Ergodic Theory and Dynamical Systems, 2004, vol. 24, num. 3, p. 735-766 URI: http://hdl.handle.net/2445/164370 Related resource: https://doi.org/10.1017/S0143385704000045 ISSN: 0143-3857 Appears in Collections: Articles publicats en revistes (Matemàtiques i Informàtica)

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