arXiv subjects
Publications and source records attributed to Tien-Cuong Dinh.
Let f be a non-invertible holomorphic endomorphism of a projective space and f^n its iterate of order n. We prove that the pull-back by f^n of a generic (in the Zariski sense) hypersurface, properly normalized, converge to the Green current associated to f when n tends to infinity. We also give an analogous result for the pull-back of positive closed (1,1)-currents.
We define the pull-back operator, associated to a meromorphic transform, on several types of currents. We also give a simple proof to a version of a classical theorem on the extension of currents.
Let f be a holomorphic endomorphism of P^k having an attracting set A. We construct an attracting current and an equilibrium measure associated to A. The attracting current is weakly laminar and extremal in the cone of invariant currents. The equilibrium measure is mixing and has maximal entropy.
Let f be a meromorphic correspondence on a compact Kahler manifold. We show that the topological entropy of f is bounded from above by the logarithm of its maximal dynamical degree. An analogous estimate for the entropy on subvarieties is given. We also discuss a notion of Julia and Fatou sets.
We show that there is a spectral gap for the transfer operator associated to a rational map f on the Riemann sphere. Using this and the method of pertubed operators we establish the (Local) Central Limit Theorem for the measure of maximal entropy of f, with an estimate on the speed of convergence.
We introduce a geometry on the cone of positive closed currents of bidegree (p,p) and apply it to define the intersection of such currents. We also construct and study the Green currents and the equilibrium measure for horizontal-like mappings. The Green currents satisfy some extremality properties. The equilibrium measure is invariant, mixing and has maximal entropy. It is equal to the intersection of the Green currents associated to the horizontal-like map and to its inverse.
We show that the pseudoconcave holes of some naturally arising class of manifolds, called hyperconcave ends, can be filled in, including the case of complex dimension 2 . As a consequence we obtain a stronger version of the compactification theorem of Siu-Yau and extend Nadel's theorems to dimension 2.
We prove the Hodge-Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.
Let f be a holomorphic automorphism of a compact Kahler manifold (X,ω) of dimension k>1. We study the convex cones of positive closed (p,p)-currents T_p, which satisfy a functional relation $f^*(T_p)=λT_p, λ>1,$ and some regularity condition (PB). Under appropriate assumptions on dynamical degrees we introduce closed finite dimensional cones, not reduced to zero, of such currents. In particular, when the topological entropy h(f) of f is positive, then for some m, there is a positive closed (m,m)-current T_m which satisfies the relation $f^* T_m=\exp(h(f)) T_m$. Moreover,every quasi-p.s.h. function is integrable with respect to the trace measure of T_m. When the dynamical degrees of f are all distinct, we construct an invariant measure $μ$ as an intersection of closed currents. We show that this measure is mixing and gives no mass to pluripolar sets and to sets of small Hausdorff dimension.
Let $f$ be a dominating meromorphic self-map of a compact Kähler manifold. Assume that the topological degree of $f$ is larger than the other dynamical degrees. We give estimates of the dimension of the equilibrium measure of $f$, which involve the Lyapounov exponents.
We construct an equilibrium measure $μ$ for a polynomial correspondence F of Lojasiewicz exponent l>1. We then show that $μ$ can be built as the distribution of preimages of a generic point and that the expansive periodic points are equidistributed on the support of $μ$. Using this results, we will give a characterization of infinite unique sets for polynomials.
We develop the study of some spaces of currents of bidegree (p,p). As an application we construct the equilibrium measure for a large class of birational maps of P^k as intersection of Green currents. We show that these currents are extremal and that the corresponding measure is mixing.
Let T be a positive closed (p,p)-current of mass 1 on a compact Kahler manifold X. Then, there exist a constant c, independent of T, and smooth positive closed (p,p)-currents Tn and Sn of mass c such that Tn-Sn converge weakly to T. We also extend this result to positive pluriharmonic currents. Then we study the wedge product of positive closed (1,1)-currents having continuous potential with positive pluriharmonic currents. As an application, we give an estimate of the topological entropy of meromorphic maps on compact Kahler manifolds.
Let f be a dominating meromorphic self-map of large topological degree on a compact Kaehler manifold. We give a new construction of the equilibrium measure of f and prove that it is exponentially mixing. Then, we deduce the central limit theorem for Lipschitzian observables.
We show, for a class of automorphisms of C^k, that their equilibrium measures are exponentially mixing. In particular, this holds for (generalized) Henon maps.
We construct the Green current for a random iteration of "horizontal-like" mappings in two complex dimensions. This is applied to the study of a polynomial map $f:\mathbb{C}^2\to\mathbb{C}^2$ with the following properties: 1. infinity is $f$-attracting, 2. $f$ contracts the line at infinity to a point not in the indeterminacy set. Then the Green current of $f$ can be decomposed into pieces associated with an itinerary determined by the indeterminacy points. We also study the set of escape rates near infinity, i.e. the possible values of the function $\limsup \frac{1}{n}\log^+\log^+ \norm{f^n}$. We exhibit examples for which this set contains an interval.
Let (F_n) be a sequence of (multivalued) meromorphic maps between compact Kaehler manifolds X1 and X2. We study the asymptotic distribution of preimages of points by F_n and the asymptotic distribution of fixed points for multivalued self-maps of a compact Riemann surface. Let (Z_n) be a sequence of holomorphic images of the projective space P^s in a projective manifold. We prove that the currents, defined by integration on Z_n, properly normalized, converge to weakly laminar currents. We also show that the Green currents, of suitable bidimensions, associated to a regular polynomial automorphism, are (weakly) laminar.
Let f be a polynomial endomorphism of degree d>1 of C^k (k>1) which extends to a holomorphic endomorphism of P^k. Assume that the maximal order Julia set of f is laminated by real hypersurfaces in some open set. We show that f is homogenous and is a polynomial lift of a Lattes endomorphism of P^{k-1}.