N ria Fagella, Tere M. Seara, Jordi Villanueva
Asymptotic Size of Herman Rings of the Complex Standard Family by Quantitative Quasiconformal Surgery
(733K, Postscript)

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.