Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mapping

TL;DR

This paper explores complex dynamics in higher dimensions using pluripotential theory, focusing on holomorphic endomorphisms of projective spaces and polynomial-like maps, establishing properties of equilibrium measures and their statistical behavior.

Contribution

It develops a comprehensive theory for higher-dimensional complex dynamics, including Green currents, equilibrium measures, and statistical properties for both endomorphisms and polynomial-like maps.

Findings

Construction of Green currents and equilibrium measures for projective space endomorphisms

Establishment of ergodic properties like mixing and decay of correlations

Analysis of entropy, Lyapunov exponents, and measure dimension

Abstract

The emphasis of this course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. We choose to show their effectiveness and to describe the theory for two large families of maps. The first chapter deals with holomorphic endomorphisms of the projective space P^k. We establish the first properties and give several constructions for the Green currents and the equilibrium measure \mu. The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems and establish ergodic properties of \mu: K-mixing, exponential decay of correlations for various classes 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 higher dimension. We introduce the dynamical degrees and construct the equilibrium measure \mu of maximal entropy. Then, under a natural assumption, we prove equidistribution properties of points and various statistical properties of the measure \mu. The assumption is stable under small pertubations on the map. We also study the dimension of \mu, the Lyapounov exponents and their variation. Our aim is to get a self-contained text that requires only a minimal background. 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.