The Moduli Space of Germs of Generic Families of Analytic Diffeomorphisms Unfolding a Parabolic Fixed Point

In this paper, we describe the moduli space of germs of generic families of analytic diffeomorphisms which unfold a parabolic fixed point of codimension 1. In [11] (and also [14]), it was shown that the Ecalle–Voronin modulus can be unfolded to give a complete modulus for such germs. The modulus is defined on a ramified sector in the canonical perturbation parameter ϵ. As in the case of the Ecalle–Voronin modulus, the modulus is defined up to a linear scaling depending only on ϵ. Here, we characterize the moduli space for such unfoldings by finding the compatibility conditions on the modulus which are necessary and sufficient for realization as the modulus of an unfolding. The compatibility condition is obtained by considering the region of sectorial overlap in ϵ-space. This lies in the Glutsyuk sector where the two fixed points are hyperbolic and connected by the orbits of the diffeomorphism. In this region, we have two representatives of the modulus which describe the same dynamics. We identify the necessary compatibility condition between these two representatives by comparing them both with their common Glutsyuk modulus. The compatibility condition implies the existence of a linear scaling for which the modulus is -summable in ϵ, whose direction of nonsummability coincides with the direction of real multipliers at the fixed points. 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. The terminology “space” of moduli is justified by the fact that the moduli depend analytically on extra parameters.