On fixed points of conformal pseudogroups

Abstract

We show in this paper Theorem 2 that if (H, H ₁) is a pseudogroup generated by a finite numberH ₁ of germs of conformal diffeomorphisms of ℂ defined on a sufficiently small discD, which is not linearizable and such that the linear group (L,H ₁)={g′(0)/g∈(H,H ₁)}⊂ℂ* is dense in ℂ*, then the set of fixed points of the pseudogroup (H, H ₁) 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]).

The idea of the proof is very simple. Ifg is a non-linear conformal diffeomorphism with multiplier λ=g'(0), then the map obtained by the composition ofg and the linear map with multiplier λ⁻¹ will have at 0 a fixed point of multiplicity at least 2. Since we may approximate λ⁻¹ by elementsh in the pseudogroup and the multiplicity of fixed points satisfy a law of conservation of number, we obtain thath o g has fixed points close to 0. These fixed points appear as a by product of the relative nonlinearity of the generators of the pseudogroup, since linearizable pseudogroups have 0 as an isolated fixed point. The fixed points obtained are not conjugate since they have distinct multipliers.

The main technical tool is the angular derivative introduced in [8]. It allows one to split the search for fixed points into two parts: One is to obtain a contraction and the other is to return arbitrarily close to the starting point without modifying the property of contraction. This is carried out since the angular derivative is multiplicative for compositions and is identically 1 for linear maps.