Nessim Sibony

Complex dynamics in higher dimensions

The field of complex dynamics in higher dimension was initiated in the 1920’s by Fa-tou. It was motivated by studies in Newton’s method, celestial mechanics and functional equations. Recently, new methods from pluripotential theory have been introduced to the subject. These techniques have produced many new interesting results. We give an introduction to this subject and a summary of the most relevant developments.

Dynamique des applications d'allure polynomiale

We study the dynamics of polynomial-like mappings in several variables. A special case of our results is the following theorem: Let f:U→V be a proper holomorphic map from an open set U⋐V onto a Stein manifold V. Assume f is of topological degree dt⩾2. Then there is a probability measure μ supported on K:=⋂n⩾0f−n(V) satisfying the following properties: (1) The measure μ is invariant, K-mixing, of maximal entropy logdt. (2) If J is the Jacobian of f with respect to a volume form Ω then ∫logJdμ⩾logdt. (3) For every probability measure ν on V with no mass on pluripolar sets dt−n(fn)∗ν⇀μ. (4) If the p.s.h. functions on V are μ-integrables (μ is PLB), then (a) The Lyapounov exponents for μ are strictly positive; (b) μ is exponentially mixing; (c) There is a proper analytic subset E0 of V such that f−1(E0)⊂E0 and for z∉E, μzn:=dt−n(fn)∗δz⇀μ where E=⋃n⩾0fn(E0); (d) The measure μ is a limit of Dirac masses on the repelling periodic points. The condition μ is PLB is stable under small pertubation of f. This gives large families where it is satisfied.

Random iterations of rational functions

We study the asymptotic behavior of iterates of rational functions with small perturbations. In presence of attractive cycles we show that almost surely, in the parameter space, the iterates converge to a given neighborhood of the attractive cycles. When there is no attractive cycle, we prove an ergodic theorem with respect to Lebesgue measure.

Dynamics in Several Complex Variables: Endomorphisms of Projective Spaces and Polynomial-like Mappings

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 section deals with holomorphic endomorphisms of the projective space \({\mathbb{P}}^{k}\). 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. \({f}^{-1}(\mathit{E}) = f(\mathit{E}) = \mathit{E}\), 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 with varieties of arbitrary dimension, and discuss the equidistribution of periodic points. We then establish ergodic properties of μ: K-mixing, exponential decay of correlations for various classes of observables, central limit theorem and large deviations theorem. We heavily use the compactness of the space \(\mathrm{DSH}({\mathbb{P}}^{k})\) 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 \(\mathrm{DSH}({\mathbb{P}}^{k})\), for suitable positive constants α, c. Finally, we study the entropy, the Lyapounov exponents and the dimension of μ. The second section 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.

Green currents for holomorphic automorphisms of compact Kahler manifolds

and some additional regularity properties. Very likely, the currents T will describe the distribution of invariant manifolds of codimension s corresponding to the small est Lyapounov exponents. Let dp denote the dynamical degree of order p of /. It describes the growth under iteration of the volume of p-dimensional manifolds. When d\ > 1, it is natural to introduce first a positive closed (1, l)-current as ifnYoj 7\ := lim ^?-.

Subharmonicity for Uniform Algebras

Let A be a uniform algebra with maximal ideal space MA. A notion of subharmonicity is defined for functions on MA. Under certain hypotheses of continuity, it is proved that the notion of subharmonicity is local. A consequence is that the notion of Jensen boundary point is local. The solutions to an abstract Dirichlet problem are studied in the context of uniform algebras. The methods are applied to algebras of analytic functions, and in particular a version of the extended maximum principle is obtained for analytic functions of several complex variables.

Dynamics of polynomial automorphisms of Ck

We study the dynamics of polynomial automorphisms ofCk. To an algebraically stable automorphism we associate positive closed currents which are invariant underf, consideringf as a rational map onPk. These currents give information on the dynamics and allow us to construct a canonical invariant measure which is shown to be mixing.

Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms

We introduce a notion of super-potential (canonical function) associated to positive closed (p, p)-currents on compact Kahler manifolds and we develop a calculus on such currents. One of the key points in our study is the use of deformations in the space of currents. As an application, we obtain several results on the dynamics of holomorphic automorphisms: regularity and uniqueness of the Green currents. We also get the regularity, the entropy, the ergodicity and the hyperbolicity of the equilibrium measures. AMS classification : 37F, 32H50, 32U40. Key-words : super-potential, structural variety of currents, moderate measure, Green current, equilibrium measure.

Some open problems in higher dimensional complex analysis and complex dynamics

We present a collection of problems in complex analysis and complex dynamics in several variables.

Spectre de A(\̄gW) pour des domaines bornés faiblement pseudoconvexes réguliers

Abstract If Ω is a weakly pseudoconvex domain in a Stein manifold, then the spectrum of the Frechet Algebra A ∞ ( \ gW) is isomorphic to \ gW, A ∞ ( \ gW) denotes the space of holomorphic functions smooth up to the boundary. The spectrum of the uniform algebra A( \ gW) is also isomorphic to -Ω. As a corollary we prove an approximation theorem for plurisubharmonic functions in Ω continuous in -Ω.