On fixed points of conformal pseudogroups

Abstract

We show in this paper Theorem 2 that if (H, H1) is a pseudogroup generated by a finite numberH1 of germs of conformal diffeomorphisms of ℂ defined on a sufficiently small discD, which is not linearizable and such that the linear group (L,H1)={g′(0)/g∈(H,H1)}⊂ℂ* is dense in ℂ*, then the set of fixed points of the pseudogroup (H, H1) is dense inD. This implies the abundance of distinct homotopy classes of loops in leaves of foliations defined in ℂ2 by generic polynomial vector fields as well as for germs of holomorphic vector fields in ℂ2 beginning with generic jets, both of degree at least 2. These homotopy classes may be realized arbitrarily close to the line at infinity or to 0, respectively. This shows the genericity of polynomial vector fields with infinite Petrovsky-Landis genus ([5]).