 03256 N ria Fagella, Tere M. Seara, Jordi Villanueva
 Asymptotic Size of Herman Rings of the Complex Standard Family by Quantitative Quasiconformal Surgery
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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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