Surgery on Herman rings of the complex standard family

Abstract

We consider the standard family (or Arnold family) of circle maps given by fα,β(x) = x + α + β sin(x) (mod 2π), for x, α ∈ [0, 2π), β ∈ (0, 1) and its complexification Fα,β(z) = zeiα exp[1/2β(z - 1/z)]. If fαβ is analytically linearizable, there is a Herman ring around the unit circle in the dynamical plane of Fαβ. Given an irrational rotation number θ, the parameters (α, β such that fα,β has rotation number θ form a curve Tθ in the parameter plane. Using quasi-conformal surgery of the simplest type, we show that if θ is a Brjuno number, the curve Tθ can be parametrized real-analytically by the modulus of the Herman ring, from β = 0 up to a point (α0, β0) with β0 ≤ 1, for which the Herman ring collapses. Using a result of Herman and a construction in I. N. Baker and P. Domínguez (Complex Variables 37 (1998), 67-98) we show that for a certain set of angles θ ∈ B \ H, the point β0 is strictly less than 1 and, moreover, the boundary of the Herman rings with the corresponding rotation number have two connected components which are quasi-circles, and do not contain any critical point. For rotation numbers of constant type, the boundary consists of two quasi-circles, each containing one of the two critical points of Fα,β.