Herman Rings and Arnold Disks

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