Suites d’Applications Méromorphes Multivaluées et Courants Laminaires

Let F_n: X₁ → X₂ be a sequence of (multivalued) meromorphic maps between compact Kähler manifolds. We study the asymptotic distribution of preimages of points by F_n and, for multivalued self-maps of a compact Riemann surface, the asymptotic distribution of repelling fixed points. Let (Z_n) be a sequence of holomorphic images of ?^s in a projective manifold. We prove that the currents, defined by integration on Z_n, 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 ?^k or an automorphism of a projective manifold.     

Attracting current and equilibrium measure for attractors on ℙ<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>

Let f be a holomorphic endomorphism of ℙ ^k 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.     

Non-Autonomous basins of attraction and their boundaries

We study whether the basin of attraction of a sequence of automorphisms of ℂ ^{k is biholomorphic to} ℂ ^{k. In particular, we show that given any sequence of automorphisms with the same attracting fixed point, the basin is biholomorphic to} ℂ ^{k, if every map is iterated sufficiently many times. We also construct Fatou-Bieberbach domains in} ℂ² whose boundaries are four-dimensional.     

A local extension theorem for proper holomorphic mappings in ℂ<Superscript>2</Superscript>

Let ƒ: D → D′ be a proper holomorphic mapping between bounded domains D, D′ in ℂ².Let M, M′ be open pieces on δD, δD′, respectively that are smooth, real analytic and of finite type. Suppose that the cluster set of M under ƒ is contained in M′. It is shown that ƒ extends holomorphically across M. This can be viewed as a local version of the Diederich-Pinchuk extension result for proper mappings in ℂ².     

On the nonextendability of germs of holomorphic functions

We construct explicit supporting manifolds and local holomorphic peak functions as obstructions to the extendability of holomorphic functions on a class of domains not necessarily pseudoconvex in C^N, N >2.     

Reflection Ideals and mappings between generic submanifolds in complex space

Results on finite determination and convergence of formal mappings between smooth generic submanifolds in ℂ ^N are established in this article. The finite determination result gibes sufficient conditions to guarantee that a formal map is uniquely determined by its jet, of a preassigned order, at a point. Convergence of formal mappings for real-analytic generic submanifolds under appropriate assumptions is proved, and natural geometric conditions are given to assure that if two germs of such submanifolds are formally equivalent, then, they are necessarily biholomorphically equivalent. It is also shown that if two real-algebraic hypersurfaces in ℂ ^N are biholomorphically equivalent, then, they are algebraically equivalent. All the results are first proved in the more general context of “reflection ideals” associated to formal mappings between formal as well as real-analytic and real-algebraic manifolds.     

Structure properties of laminar currents on ℙ2

We study the structure of a class of laminar closed positive currents on ℂℙ², naturally appearing in birational dynamics. We prove such a current admits natural non intersecting leaves, that are closed under analytic continuation. As a consequence it can be seen as a foliation cycle a weak lamination.     

A reflection principle for real—Analytic submanifolds of complex spaces

We give an invariant nondegeneracy condition for CR-maps between generic submanifolds in different dimensions and use it to prove a reftection principle for these maps.     

Polynomial diffeomorphisms of C2: V. critical points and Lyapunov exponents

There is an invariant measure μ, which is the pluri-complex version of the harmonic measure of the Julia set for polynomial maps of C.In this paper we give an integral formula for the Lyapunov exponents of a polynomial automorphism with respect to μ, analogous to the Brolin-Manning formula polynomial maps of C.Our formula relates the Lyapunov exponents to the value of a Green function at a type of critical point which we define in this paper. We show that these the critical points have a natural dynamical interpretation.     

Pull-back of currents by holomorphic maps

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.     

Attracting basins in ℙ2

We prove the existence of infinitely many attracting basins for some holomorphic mappings in ℙ². We also show that if a family of mappings has a complex generic homoclinic tangency, then some of the mappings in the family have an attractive periodic fixed point.     

Analytic sets extending the graphs of holomorphic mappings

Let D, D′ ⊂ ℂⁿ be bounded domains with smooth real analytic boundaries and ƒ: D → D′ be a proper holomorphic map. Our main result implies that if the graph of ƒ extends as an analytic set to a neighborhood of a poìnt (a, a′) ∈ ∂D × 3D′ with a′ ∈ cl_ƒ(a), then ƒ extends holomorphically to a neighborhood of a.     

Nowhere minimal CR submanifolds and Levi-flat hypersurfaces

A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is studied and characterized. This question is completely resolved for algebraic submanifolds of codimension 2 and a sufficient condition for noncontainment is given for non algebraic submanifolds. As a consequence, an example of a submanifold of codimension 2, not biholomorphically equivalent to an algebraic one, is given. We also investigate the structure of singularities of Levi-flat hypersurfaces.     

Principe de bloch et estimations de la métrique de Kobayashi des domaines de<Emphasis Type="Bold">C</Emphasis><Superscript>2</Superscript>

Nous ramenons l'existence d'estimations optimales pour la métrique de Kobayashi dans les domaines pseudoconvexes de type fini deC ² à un principe de Bloch asymptotique. Nous établissons ce principe en combinant la méthode de renormalisation utilisée par Gromov dans le contexte des applications harmoniques aux techniques de dilatation des coordonnées. Cecifournit une preuve totalement élémentaire d'un résultat de Catlin particulièrement utile dans l'étude des questions de prolongement et de rigidité d'applications holomorphes.

     

One-parameter families of operators in ℂ

We introduce classes of one-parameter families (OPF) of operators on C _c ^t8 (ℂ) which characterize the behavior of operators associated to the problem in the weighted space L² (ℂ, e^−2p) where p is a subharmonic, nonharmonic polynomial. We prove that an order 0 OPF operator extends to a bounded operator from L^q (ℂ) to itself, 1 < q < ∞, with a bound that depends on q and the degree of p but not on the parameter τ or the coefficients of p. Last, we show that there is a one-to-one correspondence given by the partial Fourier transform in τ between OPF operators of order m ≤ 2 and nonisotropic smoothing (NIS) operators of order m ≤ 2 on polynomial models in ℂ².     

On estimates of biharmonic functions on Lipschitz and convex domains

Using Maz ’ya type integral identities with power weights, we obtain new boundary estimates for biharmonic functions on Lipschitz and convex domains in ℝⁿ. Forn ≥ 8, combinedwitharesultin[18], these estimates lead to the solvability of the L^p Dirichlet problem for the biharmonic equation on Lipschitz domains for a new range of p. In the case of convex domains, the estimates allow us to show that the L^p Dirichlet problem is uniquely solvable for any 2 − ε < p < ∞ and n ≥ 4.     

Algebraicity of real analytic hypersurfaces and blowing-down

We show that a germ (M, 0) of real analytic strongly pseudoconvex hypersurface is holomorphically equivalent to a germ of real algebraic hypersurface if and only if there is a blowing-down from (M, 0) to a germ of real algebraic hypersurface.     

The Balian-Low theorem and regularity of Gabor systems

For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L²(ℝ), such that the generating function g∈L²(ℝ) satisfies the condition:∫_ℝ|g(x)|²(1+|x| ^A )/log ^d (2+|x|)dx < ∞ and .