arXiv subjects
Publications and source records attributed to Tien-Cuong Dinh.
Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d>2 odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a. We show for generic potentials V that the number of resonances of -Delta+V with modulus less than r is approximatively equal to a constant times a^dr^d.
We show that the complex Monge-Ampere equation on a compact Kaehler manifold (X,ω) of dimension n admits a Holder continuous omega-psh solution if and only if its right-hand side is a positive measure with Holder continuous super-potential. This property is true in particular when the measure has locally Holder continuous potentials or when it belongs to the Sobolev space W^{2n/p-2+epsilon,p}(X) or to the Besov space B^{epsilon-2}_{\infty,\infty}(X) for some epsilon>0 and p>1.
Consider a Brody hyperbolic foliation by Riemann surfaces with linearizable isolated singularities on a compact complex surface. We show that its hyperbolic entropy is finite. We also estimate the modulus of continuity of the Poincare metric on leaves. The estimate holds for foliations on manifolds of higher dimension.
Let f be a non-invertible holomorphic endomorphism of the complex projective space P^k and f^n its iterate of order n. Let V be an algebraic subvariety of P^k which is generic in the Zariski sense. We give here a survey on the asymptotic equidistribution of the sequence $f^{-n}(V)$ when n goes to infinity.
Let f be a dominant meromorphic self-map on a compact Kaehler manifold X which preserves a fibration given by a meromorphic map from X to a compact Kaehler manifold Y. We compute the dynamical degrees of f in term of its dynamical degrees relative to the fibration and the dynamical degrees of the map g on Y which is induced by f. We derive from this result new properties of some fibrations intrinsically associated to X when this manifold admits an interesting dynamical system.
We develop a notion of entropy, using hyperbolic time, for laminations by hyperbolic Riemann surfaces. When the lamination is compact and transversally smooth, we show that the entropy is finite and the Poincare metric on leaves is transversally Holder continuous. A notion of metric entropy is also introduced for harmonic measures.
Let T be an exterior modular correspondence on an irreducible locally symmetric space X. In this note, we show that the isolated fixed points of the power T^n are equidistributed with respect to the invariant measure on X as n tends to infinity. A similar statement is given for general sequences of modular correspondences.
We study holomorphic automorphisms on compact Kähler manifolds having simple actions on the Hodge cohomology ring. We show for such automorphisms that the main dynamical Green currents admit complex laminar structures (woven currents) and the Green measure is the unique invariant probability measure of maximal entropy.
We give an abstract version of the hard Lefschetz theorem, the Lefschetz decomposition and the Hodge-Riemann theorem for compact Kaehler manifolds.
We introduce the heat equation relative to a positive dd-bar-closed current and apply it to the invariant currents associated with Riemann surface laminations possibly with singularities. The main examples are holomorphic foliations by Riemann surfaces in projective spaces. We prove two kinds of ergodic theorems for such currents: one associated to the heat diffusion and one close to Birkhoff's averaging on orbits of a dynamical system. The heat diffusion theorem with respect to a harmonic measure is also developed for real laminations.
Let f be a holomorphic automorphism of positive entropy on a compact Kaehler surface. We show that the equilibrium measure of f is exponentially mixing. The proof uses some recent development on the pluripotential theory. The result also holds for automorphisms on compact Kaehler manifolds of higher dimension under a natural condition on their dynamical degrees.
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.
Let f be a non-invertible holomorphic endomorphism of the complex projective space P^k, f^n its iterate of order n and μthe equilibrium measure of f. We estimate the speed of convergence in the following known result. If a is a Zariski generic point in P^k, the probability measures, equidistributed on the preimages of a under f^n, converge to μas n goes to infinity.
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 μ. The emphasis is on quantitative properties and speed of convergence. We then treat equidistribution problems and establish ergodic properties of μ: 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 μ. The second chapter develops the theory of polynomial-like maps in higher dimension. We introduce the dynamical degrees and construct the equilibrium measure μof maximal entropy. Then, under a natural assumption, 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.
We introduce a notion of super-potential for positive closed currents of bidegree (p,p) on projective spaces. This gives a calculus on positive closed currents of arbitrary bidegree. We define in particular the intersection of such currents and the pull-back operator by meromorphic maps. One of the main tools is the introduction of structural discs in the space of positive closed currents which gives a "geometry" on that space. We apply the theory of super-potentials to construct Green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms.
We prove some properties of analytic multiplicative and sub-multiplicative cocycles. The results allow to construct natural invariant analytic sets associated to complex dynamical systems.
We introduce a notion of super-potential (canonical function) associated to positive closed (p,p)-currents on compact Kaehler 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.
We prove exponential estimates for plurisubharmonic functions with respect to Monge-Ampere measures with Holder continuous potential. As an application, we obtain several stochastic properties for the equilibrium measures associated to holomorphic maps on projective spaces. More precisely, we prove the exponential decay of correlations, the central limit theorem for general d.s.h. observables, and the large deviations theorem for bounded d.s.h. observables and Holder continuous observables.