Abstract :
In this paper we describe the moduli space of germs of generic analytic families of complex 1-dimensional resonant analytic diffeomorphisms of codimension 1. In Rousseau and Christopher (2007) [11], it was shown that the Ecalle modulus can be unfolded to give a complete modulus for such germs. As function of the canonical parameter, the modulus is defined on two sectors giving a covering of a neighborhood of the origin in the parameter space. As in the case of the Ecalle modulus, the modulus is defined up to a linear scaling depending only on the parameter.
The compatibility condition is obtained by considering the region of intersection of the two sectors in parameter space, which we call the Glutsyuk sectors. There, both the fixed point and the periodic orbit are hyperbolic and they are connected by the orbits of the diffeomorphism. This yields an alternate description of the equivalence class by the Glutsyuk modulus: near each of the fixed point and of the periodic orbit we construct a change of coordinate to the normal form. The Glutsyuk modulus measures the obstruction of having one being the analytic extension of the other. In the intersection of the two sectors, we have two representatives of the modulus which describe the same dynamics. A necessary compatibility condition is that they have the same Glutsyuk modulus. This necessary condition becomes sufficient for realizability.
The compatibility condition implies the existence of a linear scaling for which the modulus is 1-summable in , whose directions of non-summability coincide with the direction of real multipliers at the fixed point and periodic orbit. Conversely, we show that the compatibility condition (which implies the summability property) is sufficient to realize the modulus as coming from an analytic unfolding, thus giving a complete description of the space of moduli.
