Complex Hénon mappings in \({\mathbb{C}}^ 2\) and Fatou-Bieberbach domains. (English) Zbl 0761.32015

Fornæss, John Erik; Sibony, Nessim

Duke Math. J. 65, No. 2, 345-380 (1992).

The authors study iteration theory for certain polynomial automorphisms of \(\mathbb{C}^ 2\), including the Hénon maps \(g(z,w)=(z^ 2+c+aw,z)\). The following sets, pluriharmonic Green’s functions, and currents have been used in recent work in this area: \[ \begin{aligned} K^ \pm &= \{p\in\mathbb{C}^ 2\mid\;g^{\pm n}(p) \text{ is a bounded sequence}\},\\ K &= K^ +\cap K^ -, \qquad J^ \pm = \partial K^ \pm,\\ G^ \pm(z,w) &= \lim_{n\to\infty} {1\over 2^ n}\log^ + \| g^ n(z,w)\|,\\ \mu^ \pm &= dd^ c G^ \pm.\end{aligned} \] The authors show that \(G^ \pm\) are Hölder continuous and estimate the Hausdorff dimension of \(K^ \pm\). They generalize a theorem on convergence of currents due to Bedford and Smillie. They also study iterates of the family \(g(z,w)=(z^ 2+c+aw,az)\). The behaviour is related to that of the iterates of the one- variable map \(P_ c\) defined by \(P_ c(z)=z^ 2+c\). Properties of \(K^ \pm\) are obtained, and \(g\) is shown to be hyperbolic on \(K\), in case \(P_ c^ n(0)\) is unbounded and \(a\) is sufficiently small.
Some interesting phenomena for Fatou-Bieberbach domains are exhibited. If \(P_ c\) has in \(\mathbb{C}\) attractive cycle of order \(k\), then if \(a\) is small enough, \(g\) has such attractive cycle in \(\mathbb{C}^ n\). The interior of \(K^ +\) has \(k\) connected components with \(J^ +\) as their common boundary. \(J^ +\) is therefore the boundary of \(k\) distinct Fatou- Bieberbach regions, so the boundary of the Fatou-Bieberbach region need not be a topological manifold.
In addition the authors prove a result about the domain of linearization of a polynomial automorphism of \(\mathbb{C}^ 2\) which fixes 0, with one eigenvalue of modulus less than 1 and the other of the form \(e^{i\theta}\) where \(\theta\) satisfies a diophantine condition. The domain of linearization must be either \(\mathbb{C}^ 2\) or else biholomorphic to \(\Delta\times\mathbb{C}\).

Reviewer: I.Graham (Toronto)

MSC:

32H50	Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32C30	Integration on analytic sets and spaces, currents
37B99	Topological dynamics
28A78	Hausdorff and packing measures

Keywords:

iteration; polynomial automorphisms; Hénon maps; convergence of currents; Fatou-Bieberbach domains; attractive cycle

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