  We give a sufficient condition for the abstract basin of attraction of a
sequence of holomorphic self-maps of balls in \mathbb{C}^{d} to be
biholomorphic to \mathbb{C}^{d}. As a consequence, we get a sufficient
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-defined 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 hte conjugacy operator by a
lower triangular matrix on the space of \textit{k}-homogeneous polynomial
endomorphisms of \mathbb{C}^{d}.
