Sur certains pseudogroupes de biholomorphismes locaux de $(\mathbb{C}^n,0)$

Abstract

Let Γ be a pseudogroup of local holomorphic transformations of ℂ n fixing zero. We study the dynamics of Γ. We show that if Γ contains two elements whose 2-jets are in “general position” and sufficiently near the identity, then: 1) Γ acts minimally on the bundle of infinite-order jets on some pointed neighborhood ℬ of 0 (that is to say: for any z 0 ,z 1 ∈ℬ and any germ φ:z 0 →z 1 of biholomorphism, there exists a sequence γ n ∈Γ which converges to φ uniformly on some neighborhood of z 0 ). 2) Γ preserves no geometric structure near 0 (this is a trivial consequence of 1). 3) For any holomorphic pseudogroup topologically conjugate to Γ, the germ of conjugacy at 0 is either holomorphic or antiholomorphic. The main feature of the proof is to attach to any pseudogroup Γ a sheaf 𝔤 Γ of Lie algebrae on ℂ n such that Γ is “dense” in 𝔤 Γ in a natural sense. Then we prove that under some natural assumption on Γ, 𝔤 Γ (U) must be the sheaf of all holomorphic vector fields for any U open in ℬ, where ℬ is the (open) complementary of 0 in its basin of attraction.