On complex dynamics in high dimensions

Question

I am a fresh Ph.D student and I'm interested in complex dynamics in high dimensions. I have the following questions.

1. What research directions are there in several complex dynamics and what problems are in these directions?
2. In complex dynamics in high dimension, who are famous or active mathematicians should I follow?
3. If I want to go into this field, what are the necessary background knowledge (e.g complex geometry)? Are there some good survey papers or books to help me?

Thank you!  

Answer 1

1. Main areas are dynamics of automorphisms (for example, Henon maps), dynamics of endomorphisms, dynamics of foliations, and local dynamics.

2. Eric Bedford, Tien-Cuong Dinh, John Fornaess, Misha Lyubich, Nessim Sibony, John Smiley, and their students and collaborators.

3. One-dimensional holomorphic dynamics, functions of several complex variables (especially holomorphic maps and currents), basic complex algebraic geometry, and basics of the general theory of dynamical systems.

Some books and collections of surveys,

Abate, Marco; Bedford, Eric; Brunella, Marco; Dinh, Tien-Cuong; Schleicher, Dierk; Sibony, Nessim Holomorphic dynamical systems. Lectures given at the C.I.M.E. Summer School held in Cetraro, July 7–12, 2008. Edited by Graziano Gentili, Jacques Guenot and Giorgio Patrizio. Lecture Notes in Mathematics, 1998. Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence, 2010.

Fornæss, John Erik; Sibony, Nessim Complex dynamics in higher dimension. Several complex variables (Berkeley, CA, 1995–1996), 273–296, Math. Sci. Res. Inst. Publ., 37, Cambridge Univ. Press, Cambridge, 1999.

Fornæss, John Erik; Sibony, Nessim Complex dynamics in higher dimension. I. Complex analytic methods in dynamical systems (Rio de Janeiro, 1992). Astérisque No. 222 (1994), 5, 201–231.