# Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mapping - Mathematics > Dynamical Systems

Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mapping - Mathematics > Dynamical Systems - Download this document for free, or read online. Document in PDF available to download.

Abstract: The emphasis of this course is on pluripotential methods in complex dynamicsin higher dimension. They are based on the compactness properties ofplurisubharmonic functions and on the theory of positive closed currents.Applications of these methods are not limited to the dynamical systems that weconsider here. We choose to show their effectiveness and to describe the theoryfor two large families of maps. The first chapter deals with holomorphicendomorphisms of the projective space P^k. We establish the first propertiesand give several constructions for the Green currents and the equilibriummeasure \mu. The emphasis is on quantitative properties and speed ofconvergence. We then treat equidistribution problems and establish ergodicproperties of \mu: K-mixing, exponential decay of correlations for variousclasses of observables, central limit theorem and large deviations theorem.Finally, we study the entropy, the Lyapounov exponents and the dimension of\mu. The second chapter develops the theory of polynomial-like maps in higherdimension. We introduce the dynamical degrees and construct the equilibriummeasure \mu of maximal entropy. Then, under a natural assumption, we proveequidistribution properties of points and various statistical properties of themeasure \mu. The assumption is stable under small pertubations on the map. Wealso study the dimension of \mu, the Lyapounov exponents and their variation.Our aim is to get a self-contained text that requires only a minimalbackground. In order to help the reader, an appendix gives the basics on p.s.h.functions, positive closed currents and super-potentials on projective spaces.Some exercises are proposed and an extensive bibliography is given.

Author: ** Tien-Cuong Dinh, Nessim Sibony**

Source: https://arxiv.org/