Asymptotic behaviour of the domain of analyticity of invariant curves of the standard map

Abstract

In this paper we consider the standard map, and we study the invariant curve obtained by analytical continuation, with respect to the perturbative parameter E, of the invariant circle of rotation number the golden mean corresponding to the case E=0. We show that, if we consider the parameterization that conjugates the dynamics of this curve to an irrational rotation, the domain of definition of this conjugation has an asymptotic boundary of analyticity when E->0 (in the sense of the singular perturbation theory). This boundary is obtained studying the conjugation problem for the so-called semi-standard map. To prove this result we have used KAM-like methods adapted to the framework of singular perturbation theory, as well as matching techniques to join di erent pieces of the conjugation, obtained in different parts of its domain of analyticity.