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Dear mathematicians, I want to know how much advance there has been in complex dynamics of several variables. I am at present reading Carleson's book on Complex Dynamics on one variables.Curious to know about several variables case.t Specifically,what are the best papers in this area.

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Papers of Bedford and Smillie in the early-mid 1990's are a good start. – Lee Mosher Apr 11 '13 at 17:14
This question is way too unfocused; I voted to close. – Misha Apr 11 '13 at 17:39
up vote 3 down vote accepted

There has been indeed much development in the dynamics of several complex variables in the last 20 years or so. The directions which the subject took focus on different aspects. E.g., is one interested in maps in the affine space, projective space or more general complex manifolds? Are the maps holomorphic diffeomorphisms, just holomorphic endomorphisms or birational maps? Polynomial, rational or transcendental (where this distinction makes sense)? What are invariant objects associated with the maps of interest: attracting/repelling sets, measures, currents etc.? The papers studying such problems are numerous (and many of them carry 32H50 as their primary MSC classification). In addition to sources already mentioned, let me recommend somewhat more up-to-date accounts of these subjects:

MR2572393 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. xiv+342 pp. ISBN: 978-3-642-13170-7

This book contains several tutorials by people involved in the development of the subject, including:

Marco Abate, Discrete holomorphic local dynamical systems (1–55) MR2648687; Eric Bedford, Dynamics of rational surface automorphisms (57–104) MR2648688; Marco Brunella, Uniformisation of foliations by curves (105–163) MR2648689; Tien-Cuong Dinh and Nessim Sibony, Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings (165–294) MR2648690.

If some particular subject catches your attention, look up the bibliography on it given in these tutorials. It is a matter of taste to decide which papers are "the best", although those mentioned above (in Alex Eremenko's and in my answer) are very well written. It helps to read French, as many papers in dynamics of several complex variables are written in this language.

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There is a book by Erik Fornaess and Nessim Sibony, MR1363948, survey papers of the same authors, MR1810536, MR1748606, MR1285389 and on various specific questions I also recommend papers of Misha Lyubich with various co-authors, and Eric Bedford and John Smillie, especially "Polynomial diffeomorphisms" in 8 parts.

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