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 F̃α,ε(u) = ue iαe((ε/2(u-1/u with α = α(ε) chosen so that F̃α(ε),ε restricted to the unit circle has a prefixed rotation number θ belonging to the set of Brjuno numbers. In this case, it is known that F̃α(ε), ε analytically linearizable if ε is small enough and so it has a Herman ring Ũε around the unit circle. Using Yoccoz's estimates, one has that the size R̃ε of Ũ ε (so that Uε is conformally equivalent to {u ∈ ℂ : 1/R̃ε < |u|