Generic unfolding of an antiholomorphic parabolic point of codimension k

2023/01/27 by Christiane Rousseau, Rousseau, Christiane
Mathematics · #32H50 #37F34 #37F44 #37F46 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2301.11684

openalex publication_date 2023/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We classify generic unfoldings of germs of antiholomorphic diffeomorphisms with a parabolic point of codimension~k (i.e.~a fixed point of multiplicity k+1) under conjugacy. Such generic unfoldings depend real analytically on k real parameters. A preparation of the unfolding allows to identify real analytic canonical parameters, which are preserved by any conjugacy between two prepared generic unfoldings. A modulus of analytic classification is defined, which is an unfolding of the modulus assigned to the antiholomorphic parabolic point. Since the second iterate of such a germ is a real unfolding of a holomorphic parabolic point, the modulus is a special form of an unfolding of the Écalle-Voronin modulus of the second iterate of the antiholomorphic parabolic germ. We also solve the problem of the existence of an antiholomorphic square root to a germ of generic analytic unfolding of a holomorphic parabolic germ.

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