# Moscow Mathematical Journal

Volume 23, Issue 2, April–June 2023 pp. 169–203.

Analytic Classification of Generic Unfoldings of Antiholomorphic Parabolic Fixed Points of Codimension 1

**Authors**:
Jonathan Godin (1) and Christiane Rousseau (1)

**Author institution:**(1) Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal, Qc, H3C 3J7, Canada

**Summary: **

We classify generic unfoldings of germs of antiholomorphic
diffeomorphisms with a parabolic point of codimension 1 (i.e., a double fixed point) under conjugacy. These generic unfoldings depend on one real parameter.
The classification is done by assigning to each such germ a weak and a strong modulus, which are unfoldings of the modulus assigned to the antiholomorphic parabolic point. The weak and the strong moduli are unfoldings of the Écalle–Voronin modulus of the second iterate of the germ which is a real unfolding of a holomorphic parabolic point. A preparation of the unfolding allows to identify one real analytic *canonical parameter* and any conjugacy between two prepared generic unfoldings preserves the canonical
parameter. We also solve the realisation problem by giving necessary and sufficient conditions for a strong modulus to be realized. This is done simultaneously with solving the problem of the existence of an antiholomorphic square root to a germ of generic analytic unfolding of a holomorphic parabolic germ. As a second application we establish the condition for the existence of a real analytic invariant curve.

2020 Math. Subj. Class. 37F46, 32H50, 37F34, 37F44.

**Keywords:**Discrete dynamical systems, antiholomorphic dynamics, parabolic fixed point, classification, unfoldings, modulus of analytic classification.

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