03-256 N ria Fagella, Tere M. Seara, Jordi Villanueva

Asymptotic Size of Herman Rings of the Complex Standard Family by Quantitative Quasiconformal Surgery (733K, Postscript) Jun 3, 03

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Abstract. In this paper we consider the complexification of the Arnold standard family of circle maps given by $\widetilde F_{\alpha,\ep}(u)=u{\rm e}^{{\rm i}\alpha} {\rm e}^{\frac{\ep}{2}(u-\frac{1}{u})}$, with $\alpha=\alpha(\ep)$ chosen so that $\widetilde F_{\alpha(\ep),\ep}$ 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 $\widetilde F_{\alpha(\ep),\ep}$ is analytically linearizable if $\ep$ is small enough, and so, it has a Herman ring $\widetilde U_{\ep}$ around the unit circle. Using Yoccoz's estimates, one has that \emph{the size} $\widetilde R_\ep$ of $\widetilde U_{\ep}$ (so that $\widetilde U_{\ep}$ is conformally equivalent to $\{u\in\bc:\mbox{ } 1/\widetilde R_\ep < |u| < \widetilde R_\ep\}$) goes to infinity as $\ep\to 0$, but one may ask for its asymptotic behavior. We prove that $\widetilde R_\ep=\frac{2}{\ep}(R_0+{\cal O}(\ep\log\ep))$, where $R_0$ is the conformal radius of the Siegel disk of the complex semistandard map $G(z)=z{\rm e}^{{\rm i}\omega}{\rm e}^z$, where $\omega= 2\pi\theta$. In the proof we use a very explicit quasiconformal surgery construction to relate $\widetilde F_{\alpha(\ep),\ep}$ and $G$, and hyperbolic geometry to obtain the quantitative result.

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