Periodic complex map germs and foliations.

Leonardo M Câmara,Bruno A Scárdua

doi:10.1590/0001-3765201720170233

Abstract

In this paper we study topological and analytical conditions on the orbits of a germ of diffeomorphism in the complex plane in order to obtain periodicity. In particular, we give a simple proof of a finiteness criteria for groups of analytic diffeomorphisms, stated in Brochero Martínez 2003. As an application, we derive some consequences about the integrability of complex vector fields in dimension three in a neighborhood of a singular point.

Highlights

The relationship between periodic subgroups of Diff(C, 0) and the integrability of germs of vector fields at (C2, 0) was established in Mattei and Moussu 1980
There the authors show that the topological condition of finiteness of the orbits is sufficient to ensure the periodicity of a finitely generated sugbroup at Diff(C, 0)
They obtain that the topological condition of closeness of the orbits of a germ of vector field X is equivalent to the existence of a first integral, i.e., a germ of map f : (C2, 0) −→ (C, 0) whose level sets contain the orbits of X

Summary

- INTRODUCTION

The relationship between periodic subgroups of Diff(C, 0) and the integrability of germs of vector fields at (C2, 0) was established in Mattei and Moussu 1980. Let G ∈ Diff(C2, 0), the group generated by G is finite if and only if there exists a neighborhood V of the origin such that |OV (G, x)| < ∞ for all x ∈ V and G preserves infinitely many analytic invariant curves at 0. Let f ∈ Diff(C2, 0) and {Sj}∞ j=1 be distinct periodic separatrices for f , after a finite number of blowing-ups on S there appears a local map-germ f ∈ Diff(C2, 0) admitting an infinite set of irreducible periodic separatrices in general position. Let f, g ∈ O2 be generically transverse germs and G ∈ Diff(C2, 0) be a complex map germ having finite orbits and preserving the level sets of both f and g.

Then h

Let now X

- REFERENCES