Stable Manifolds of Two-dimensional Biholomorphisms Asymptotic to Formal Curves

Abstract

Let F ∈ Diff (C2, 0) be a germ of a holomorphic diffeomorphism and let G be an invariant formal curve of F. Assume that the restricted diffeomorphism F|G is either hyperbolic attracting or rationally neutral non-periodic (these are the conditions that the diffeomorphism F|G should satisfy, if G were convergent, in order to have orbits converging to the origin). Then we prove that F has finitely many stable manifolds, either open domains or parabolic curves, consisting of and containing all converging orbits asymptotic to G. Our results generalize to the case where G is a formal periodic curve of F.