Abstract

The standard map, $F$_t,a: C → C, z → x + t + a/2π sin(2π z), restricts to an analytic circle diffeomorphism on $T\; =\; R\; /Z$ when and $t\; \in \; R\; /\; Z$. It can therefore be assigned a rotation number ρ, a number that, in a certain sense, describes how much $F$ rotates a point on the circle on the average. When ρ is irrational, there exists a homeomorphism φ : T → T which conjugates the rigid rotation $z\; \to \; z\; +\; \rho$ to $F$_{t, a}. We are interested in the case where this conjugacy is real analytic; then it extends analytically to a maximal neighborhood of the circle whose image is called a Herman ring. Allowing the parameters $t$, a to be complex numbers we can still define the notation of a fixed Herman ring and we study the set where $F$_t,a possesses such a ring with a given fixed rotation number. It turns out that this set is a holomorphic injection of the unit disk in $C$. Finally we link the parameters tocertain geometric properties of the Herman ring for small values of |a|.