We give a sufficient condition for the abstract basin of attraction of a sequence of holomorphic self-maps of balls in ℂd to be biholomorphic to ℂd. 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 the conjugacy operator by a lower triangular matrix on the space of k-homogeneous polynomial endomorphisms of ℂd.
|Autori:||Abate, Marco; Abbondandolo, Alberto; Majer, Pietro|
|Titolo:||Stable manifolds for holomorphic automorphisms|
|Anno del prodotto:||2014|
|Digital Object Identifier (DOI):||10.1515/crelle-2012-0069|
|Appare nelle tipologie:||1.1 Articolo in rivista|