Asymptotic size of Herman Rings using quasiconformal surgery

Abstract

This thesis gives an introduction to the theory of quasiconformal surgery. To that end, we consider the complexification of the Arnol'd standard family of circle maps. These functions are analytically linearisable under certain conditions, and therefore have a Herman ring $\wtilde U_\ve$ (where $\ve$ is a parameter of the map) around the unit circle, whose size $\wtilde R_\ve$ tends to infinity as $\ve$ tends to zero. We study the asymptotic size of these Herman rings and check that \wtilde R_\ve=\frac{2}{\ve}(R_0+\cO(\ve\log\ve)), where $R_0$ is the conformal radius of the Siegel disc of the complex semistandard map. In order to achieve this, we perform a quasiconformal surgery construction to relate $\wtilde F_{\alpha(\ve),\ve}$ and $G$, and hyperbolic geometry to obtain the quantitative result.