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arXiv · 1104.4561

Stable manifolds for holomorphic automorphisms

Marco Abate·Alberto Abbondandolo·Pietro Majer
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Abstract

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}.

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Marco Abate, Alberto Abbondandolo, Pietro Majer. 2011-04-23. Stable manifolds for holomorphic automorphisms. https://arxiv.org/abs/1104.4561

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