HERMAN RINGS AND ARNOLD DISKS

For (λ, a) ∈ C ∗ × C, let fλ ,a be the rational map defined by fλ ,a (z)=λz2(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 (e2iπ α ,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 α (0) = (0, rα ), where rα is the conformal radius at 0 of the Siegel disk of the quadratic polynomial z −→ e2iπ α z(1 + z). As a consequence, we show that for a ∈ (0, 1/3), if fλ ,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 eπma = (rα /a) + O(a). We finally explain how to adapt the results to the complex standard family z −→ λze(a/2)(z−1/z ).