Stable manifolds for holomorphic automorphisms

We give a sucient condition for the abstract basin of attraction of a sequence of holomorphic self-maps of balls in C d to be biholomorphic to C d . As a consequence, we get a sucient condition for the stable manifold of a point in a compact hyperbolic invariant subset of a complex manifold to be biholomorphic to a complex Euclidean space. Our result immediately implies previous theorems obtained by Jonsson-Varolin and by Peters; in particular, we prove (without using Oseledec’s theory) that the stable manifold of any point where the negative Lyapunov exponents are well-dened is biholomorphic to a complex Euclidean space. Our approach is based on the solution of a linear control problem in spaces of subexponential sequences, and on careful estimates of the norm of the conjugacy operator by a lower triangular matrix on the space of k-homogeneous polynomial endomorphisms of C d . Mathematics Subject Classication 2010. Primary: 37F99. Secondary: 32H50, 37D25, 37H99.