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.

Stable manifolds for holomorphic automorphisms

ABATE, MARCO;ABBONDANDOLO, ALBERTO;MAJER, PIETRO
2014-01-01

Abstract

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.
2014
Abate, Marco; Abbondandolo, Alberto; Majer, Pietro
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/157937
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