This text deals with an area preserving diffeomorphism of the cylinder into itself. The variables are of action \((p)\)-angle \((q)\) type, the diffeomorphism being defined by the relations: \(p'= p+ e \sin q\), \(q'= p+ q+e \sin q\), \(e\) being a small parameter. So one has a particular case, with the nonlinear part in the relations defining the action and the angle only depending on the angle \(q\). This map is called “standard map”. When the parameter \(e\) is equal to zero (integrable twist map), any initial action \(p(0)\) gives rise to an invariant circle, the dynamics on the variable \(q\) being a rotation of angle \(p(0)\).
The paper’s purpose is a study of the invariant closed curve (on the cylinder), obtained by analytical continuation (with respect to \(e\)) of the invariant circle with the irrational rotation number equal to the golden mean. Considering a parametrization which conjugates the dynamics on the invariant closed curve to the irrational rotation number, it is shown that the definition domain of this conjugation has an asymptotic boundary of analyticity, when \(e\) tends toward zero (in the sense of the singular perturbation theory). This is made from a change of variables leading to the so-called “semi-standard map”. The result is proved using KAM-like methods adapted to the singular perturbation theory, as well as matching techniques to join different pieces of conjugation, obtained in different parts of its domain of analyticity. This gives rise to long and delicate proofs of the convergence in the theorems.