Complex dynamics in higher dimension. I. (English) Zbl 0813.58030

Fornaess, John Erik; Sibony, Nessim

Camacho, C. (ed.) et al., Complex analytic methods in dynamical systems. Proceedings of the congress held at Instituto de Matemática Pura e Aplicada, IMPA, Rio de Janeiro, Brazil, January 1992. Paris: Société Mathématique de France, Astérisque. 222, 201-231 (1994).

Approximate solutions of any polynomial equation with one variable can be obtained by Newton’s iteration method. This can be extended to the case of complex value equations by the study of iteration of rational functions. The extension was made also to the case of two variables.
In this paper the authors discuss global questions of iteration of rational maps in higher dimension. The analogue of Montel’s theorem comes from the Kobayashi hyperbolicity of the complement of certain complex hypersurfaces in the complex projective space of higher dimension. The authors discuss holomorphic maps, count the number of periodic points, describe the family of exceptional maps and classify critically finite maps of dimension two.
For the entire collection see [Zbl 0797.00019].

Reviewer: Y.Kozai (Tokyo)

MSC:

37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50	Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables

Keywords:

complex dynamics; polynomial equation; iteration of rational maps; holomorphic maps; number of periodic points; exceptional maps; finite maps