Dynamics in Several Complex Variables

Abstract

The emphasis of this introductory course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic (p.s.h.) functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. Nervertheless, we choose to show their effectiveness and to describe the theory for two large families of maps: the endomorphisms of projective spaces and the polynomial-like mappings.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 T p and the equilibrium measure µ = T k . The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems. We show the existence of a proper algebraic set E , totally invariant, i.e. f −1 (E ) = f (E ) = E , such that when a ∈ E , the probability measures, equidistributed on the fibers f −n (a), converge towards the equilibrium measure µ, as n goes to infinity. A similar result holds for the restriction of f to invariant subvarieties. We survey the equidistribution problem when points are replaced by varieties of arbitrary dimension, and discuss the equidistribution of periodic points. We then establish ergodic properties of µ: Kmixing, exponential decay of correlations for various classes of observables, central limit theorem and large deviations theorem. We heavily use the compactness of the space DSH(P k ) of differences of quasi-p.s.h. functions. In particular, we show that the measure µ is moderate, i.e. µ, e α|ϕ| ≤ c, on bounded sets of ϕ in DSH(P k ), for suitable positive constants α, c. Finally, we study the entropy, the Lyapounov exponents and the dimension of µ.The second chapter develops the theory of polynomial-like maps, i.e. proper holomorphic maps f : U → V where U, V are open subsets of C k with V convex and U ⋐ V . We introduce the dynamical degrees for such maps and construct the equilibrium measure µ of maximal entropy. Then, under a natural assumption on the dynamical degrees, we prove equidistribution properties of points and various statistical properties of the measure µ. The assumption is stable under small pertubations on the map. We also study the dimension of µ, 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.