Tien-Cuong Dinh

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.

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 ^?-.

Comparison of dynamical degrees for semi-conjugate meromorphic maps

Let f be a dominant meromorphic self-map on a projective manifold X which preserves a meromorphic fibration pi: X --> Y of X over a projective manifold Y. We establish formulas relating the dynamical degrees of f, the dynamical degrees of f relative to the fibration and the dynamical degrees of the self-map g on Y induced by f. Applications are given.

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.

Suites d’Applications Méromorphes Multivaluées et Courants Laminaires

Let Fn: X1 → X2 be a sequence of (multivalued) meromorphic maps between compact Kähler manifolds. We study the asymptotic distribution of preimages of points by Fn and, for multivalued self-maps of a compact Riemann surface, the asymptotic distribution of repelling fixed points.Let (Zn) be a sequence of holomorphic images of ℙs in a projective manifold. We prove that the currents, defined by integration on Zn, properly normalized, converge to currents which satisfy some laminarity property. We also show this laminarity property for the Green currents, of suitable bidimensions, associated to a regular polynomial automorphism of ℂk or an automorphism of a projective manifold.

Dimension de la mesure d’équilibre d’applications méromorphes

Let ƒ be a dominating meromorphic self-map of a compact Kähler manifold. Assume that the topological degree of ƒ is larger than the other dynamical degrees. We give estimates of the dimension of the equilibrium measure of ƒ, which involve the Lyapounov exponents.

Attracting current and equilibrium measure for attractors on ℙ k

Let f be a holomorphic endomorphism of ℙk having an attracting setA. We construct an attracting current and an equilibrium measure associated toA. The attracting current is weakly laminar and extremal in the cone of invariant currents. The equilibrium measure is mixing and has maximal entropy onA.

Holomorphic Dynamical Systems

This chapter is a survey on local dynamics of holomorphic maps in one and several complex variables, discussing in particular normal forms and the structure of local stable sets in the non-hyperbolic case, and including several proofs and a large bibliography.

EQUIDISTRIBUTION OF ZEROS OF HOLOMORPHIC SECTIONS IN THE NON COMPACT SETTING

We consider tensor powers LN of a positive Hermitian line bundle (L,hL) over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed as N→∞ with respect to the natural measure coming from the curvature of L. Under certain boundedness assumptions on the curvature of the canonical line bundle of X and on the Chern form of L we prove a non-compact version of this result. We give various applications, including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of SL2(ℤ) and to the hyperbolic measure, the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity. We also give estimates for the speed of convergence of the currents of integration on the zero-divisors.