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Iteration theory, compactly divergent sequences and commuting holomorphic maps. (English) Zbl 0760.32014

The well-known Wolff-Denjoy theorem describes the behaviour of iterations of holomorphic automorphism of the unit disk of the complex plane. This theorem was generalized to holomorphic endomorphisms of hyperbolic Riemann surfaces by M. Heins and D. Sullivan (see an excellent exposition by J. Milnor, Dynamics in one complex variable: Introductory Lectures, SUNY Stony-Brook, Inst. Math. Sci., Preprint # 1990/5).
The author gives its generalizations to a large class of bounded domains in \(\mathbb{C}^ n\).
Let \(X\) and \(Y\) be two complex manifolds; denote by \(\text{Hol}(X,Y)\) the set of holomorphic maps from \(X\) into \(Y\), endowed with the compact-open topology. A sequence \(\{f_ \nu\}\subset\text{Hol}(X,Y)\) is compactly divergent if, for any pair of compact subsets \(H\subset X\) and \(K\subset Y\) we have \(f_ \nu(H)\cap K=\emptyset\) eventually. A family \({\mathcal F}\subset\text{Hol}(X,Y)\) is normal if every sequence \(\{f_ \nu\}\subset{\mathcal F}\) has a subsequence which is either converging in \(\text{Hol}(X,Y)\) or compactly divergent. A complex manifold \(X\) is taut if \(\text{Hol}(\Delta,X)\) is a normal family; here \(\Delta\) is the unit disk. Denote by \(\Gamma(f)\) the set of all limit points of iterations \(\{f^ k\}\), here \(f\in\text{Hol}(X,X)\).
Theorem 1. Let \(X\) be a taut manifold. If \(f\in\text{Hol}(X,X)\) is such that \(\{f^ k\}\) is not compactly divergent then \(\Gamma(f)\) is isomorphic to a compact abelian group of the form \(\mathbb{Z}_ q\times\mathbb{T}^ r\), where \(\mathbb{Z}_ q\) is the cyclic group of order \(q\), and \(\mathbb{T}^ r\) is the real torus of group of rank \(r\).
Theorem 2. Let \({\mathcal D}\Subset\mathbb{C}^ n\) be a strongly pseudoconvex domain, and \(f\in\text{Hol}({\mathcal D},{\mathcal D})\). If \(\{f^ k\}\) is compactly divergent then \(\{f^ k\}\) converges, uniformly on compact sets, to a point \(z_ 0\in\partial{\mathcal D}\).

MSC:

32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32A30 Other generalizations of function theory of one complex variable
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
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References:

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