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

TL;DR

This paper classifies generic unfoldings of antiholomorphic parabolic fixed points of codimension 1 using moduli, providing a comprehensive framework for understanding their conjugacy classes and invariants.

Contribution

It introduces a classification scheme based on weak and strong moduli for antiholomorphic unfoldings, including canonical parameters and realization conditions.

Findings

Established a classification using weak and strong moduli.

Identified a real analytic canonical parameter for unfoldings.

Provided necessary and sufficient conditions for realizing a strong modulus.

Abstract

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 unfolding 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 \'Ecalle-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 probem 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.