Dynamics of rational maps on \(\mathbb P^k\). (Dynamique des applications rationnelles de \(\mathbb P^k\).) (French) Zbl 1020.37026

Sibony, Nessim

Cerveau, Dominique et al., Dynamique et géométrie complexes. Paris: Société Mathématique de France. Panor. Synth. 8, 97-185 (1999).

The paper surveys the theory of iterations of rational maps of the complex projective space \(\mathbb{P}^k\). It is divided into three parts. The first part is on the dynamics of very general rational mappings of \(\mathbb{P}^k\) and develops the basic notions (periodic points, critical set, Fatou and Julia sets). The invariant Green current \(T\) and its basic properties are given. Part two studies regular polynomial automorphisms of \(\mathbb{C}^k\) (generalized Hénon mappings). The last chapter treats holomorphic mappings of \(\mathbb{C}^k\) and the author surveys results by himself, Fomaes, Ueda, Duval, Brient, etc. The paper concludes with a consistent appendix with basic facts from pluripotential theory.
For the entire collection see [Zbl 1010.00008].

Reviewer: Viorel Vâjâitu (Bucureşti)

MSC:

37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F50	Small divisors, rotation domains and linearization in holomorphic dynamics
32H50	Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32U15	General pluripotential theory

Keywords:

holomorphic dynamics; Julia sets in \(\mathbb{P}^k\); invariant currents; pluripotential theory; Fatou-Bieberbach domains; dynamic hyperbolicity