Generic unfolding of an antiholomorphic parabolic point of codimension $k$

TL;DR

This paper classifies and constructs canonical parameters for generic unfoldings of antiholomorphic parabolic points of codimension k, providing a detailed analytic classification and addressing the existence of antiholomorphic square roots.

Contribution

It introduces a classification of generic unfoldings of antiholomorphic parabolic points, defines canonical parameters and a modulus of classification, and solves the antiholomorphic square root problem.

Findings

Canonical parameters are preserved under conjugacy.

A modulus of classification is constructed as an unfolding of the Ecalle-Voronin modulus.

Existence of antiholomorphic square roots for certain unfoldings is established.

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 \emph{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 \'Ecalle-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.