Stable manifolds for holomorphic automorphisms
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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 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.
The authors would like to thank Eric Bedford and Jasmin Raissy for several useful conversations. The support of the INdAM grant Local discrete dynamics in one, several and infinitely many variables during the initial stages of this work is gratefully acknowledged. Part of this work was done while the last two authors were visiting the University of Leipzig and the Max Planck Institut für Mathematik in den Naturwissenschaften. We would like to thank both institutions for their hospitality and the Humboldt Foundation and the Ateneo Italo Tedesco for financial support in the form of a Humboldt Fellowship and of a Vigoni Project.
© 2014 by Walter de Gruyter Berlin/Boston
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Articles in the same Issue
- Frontmatter
- UC hierarchy and monodromy preserving deformation
- Vector-valued p-adic Siegel modular forms
- K-theoretic Schubert calculus for OG(n,2n+1) and jeu de taquin for shifted increasing tableaux
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Strichartz estimates for partially periodic solutions to Schrödinger equations in
- Inverse tunneling estimates and applications to the study of spectral statistics of random operators on the real line
- Mean curvature flow of entire graphs in a half-space with a free boundary
- Ricci flow and the holonomy group
- Uniqueness of compact tangent flows in Mean Curvature Flow
- Even unimodular Lorentzian lattices and hyperbolic volume
- Algebraic points on Shimura curves of Γ0(p)-type
- Errata to Algebraic points on Shimura curves of Γ0(p)-type (J. reine angew. Math., DOI 10.1515/crelle-2012-0068)
- Property A and the operator norm localization property for discrete metric spaces
- Stable manifolds for holomorphic automorphisms
