Abstract

AbstractWe investigate the local dynamics of antiholomorphic diffeomorphisms around a parabolic fixed point. We first give a normal form. Then we give a complete classification including a modulus space for antiholomorphic germs with a parabolic fixed point under analytic conjugacy. We then study some geometric applications: existence of real analytic invariant curves, existence of holomorphic and antiholomorphic roots of holomorphic and antiholomorphic parabolic germs, and commuting holomorphic and antiholomorphic parabolic germs.

Highlights

  • We are interested in the local dynamics of antiholomorphic diffeomorphisms with a parabolic fixed point, that is, a fixed point of multiplicity k + 1

  • We study the classification under conjugacy by analytic changes of coordinate of a germ of an antiholomorphic diffeomorphism f with a parabolic fixed point

  • Is well known. (See e.g. [5] or [6].) The dynamics of g is determined by a topological invariant, the integer k, a formal invariant, the complex number b, and an analytic invariant given by an equivalence class of 2k germs of diffeomorphisms which are the transition functions on the space of orbits of g

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Summary

Introduction

We are interested in the local dynamics of antiholomorphic diffeomorphisms with a parabolic fixed point, that is, a fixed point of multiplicity k + 1 (that is, of codimension k). When studying the space of orbits of an antiholomorphic germ of a parabolic diffeomorphism of any codimension, we see that the Écalle height has a meaning only on the Écalle cylinder of the petals containing the formal symmetry axis of f.

Antiholomorphic parabolic fixed points
Properties of the formal normal form
Fatou coordinates
Modulus of analytic classification
Space of orbits and classification under analytic conjugacy
Applications of the classification theorem
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