Asymptotic size of Herman rings of the complex standard family by quantitative quasiconformal surgery

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.