HERMAN RINGS AND ARNOLD DISKS

Abstract

For (λ,a)∈ C* × C, let fλ,a be the rational map defined by fλ,a(z) = λ z2 (az+1)/(z+a). If α∈ R/Z is a Brjuno number, we let Dα be the set of parameters (λ,a) such that fλ,a has a fixed Herman ring with rotation number α (we consider that (e2iπα,0)∈ Dα). Results obtained by McMullen and Sullivan imply that, for any g∈ Dα, the connected component of Dα(C* × (C/{0,1})) that contains g is isomorphic to a punctured disk. We show that there is a holomorphic injection Fα:D→Dα such that Fα(0) = (e2iπ α,0) and F ' ( 0 ) = ( 0 , r α ) , where rα is the conformal radius at 0 of the Siegel disk of the quadratic polynomial z↦ e2iπ αz(1+z). As a consequence, we show that for a∈ (0,1/3), if fl,a has a fixed Herman ring with rotation number α and if ma is the modulus of the Herman ring, then, as a→0, we have eπ ma=(rα/a) + O(a). We finally explain how to adapt the results to the complex standard family z↦ λ\se(a/2)(z-1/z).