On the dynamics of some diffeomorphisms of ℂ² near parabolic fixed points

In this paper we consider diffeomorphisms of ℂ² of the special form F(z,w) = (w,-z + 2G(w)). For such maps the origin is a parabolic fixed point. Under certain hypotheses on G we prove the existence of a domain Ω ⊂ ℂ with 0 ∈ ∂Ω and of invariant complex curves w = f(z) and w = g(z), z ∈ Ω, for F^-1 and F, such that F^-n(z,f(z)) → 0 and Fⁿ(z,g(z)) → 0 as n → ∞.