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Documents  Berteloot, François | enregistrements trouvés : 7

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How to study the dynamics of a holomorphic polynomial vector field in $\mathbb{C}^{2}$? What is the replacement of invariant measure? I will survey some surprising rigidity results concerning the behavior of these dynamical system. It is helpful to consider the extension of this dynamical system to the projective plane.
Consider a foliation in the projective plane admitting a unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. With T.-C. Dinh, we showed that there is a unique positive $dd^{c}$-closed (1, 1)-current of mass 1 which is directed by the foliation. This is the current of integration on the invariant curve. A unique ergodicity theorem for the distribution of leaves follows: for any leaf $L$, appropriate averages on $L$ converge to the current of integration on the invariant curve (although generically the leaves are dense). The result uses our theory of densities for currents. It extends to Foliations on Kähler surfaces.
I will describe a recent result, with T.-C. Dinh and V.-A. Nguyen, dealing with foliations on compact Kähler surfaces. If the foliation, has only hyperbolic singularities and does not admit a transverse measure, in particular no invariant compact curve, then there exists a unique positive $dd^{c}$-closed (1, 1)-current of mass 1 which is directed by the foliation( it’s like uniqueness of invariant measure for discrete dynamical systems). This improves on previous results, with J.-E. Fornæss, for foliations (without invariant algebraic curves) on the projective plane. The proof uses a theory of densities for positive $dd^{c}$-closed currents (an intersection theory).
How to study the dynamics of a holomorphic polynomial vector field in $\mathbb{C}^{2}$? What is the replacement of invariant measure? I will survey some surprising rigidity results concerning the behavior of these dynamical system. It is helpful to consider the extension of this dynamical system to the projective plane.
Consider a foliation in the projective plane admitting a unique invariant algebraic curve. Assume that the foliation is ...

37F75 ; 37Axx

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I will talk about recent joint work with Dusty Grundmeier and Lars Simon. We have proved sup-norm estimates for dbar on a wide class of pseudoconvex domains in $\mathbb{C}^{3}$, including all known examples of bounded, pseudoconvex domains with real-analytic boundary of finite D’Angelo type.

32A26 ; 32F18 ; 32T25

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I will report on a work in progress with Federico Lo Bianco, Erwan Rousseau, and Frédéric Touzet about the structure of codimension one foliations having an infinite group of birational symmetries.

37F75 ; 32S65 ; 14E05

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Algebraic and Complex Geometry

We show, among other things, one way to construct a smooth complex projective rational variety of any dimension n ≥ 3, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic real forms. This is a joint work in progress with Professors Tien-Cuong Dinh and Xun Yu.

14J50

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For a complex Lie group $G$ with a real form $G_{0}\subset G$, we prove that any Hamiltionian automorphism $\phi$ of a coadjoint orbit $\mathcal{O}_{0}$ of $G_{0}$ whose connected components are simply connected, may be approximated by holomorphic $O_{0}$-invariant symplectic automorphism of the corresponding coadjoint orbit of $G$ in the sense of Carleman, provided that $\mathcal{O}$ is closed. In the course of the proof, we establish the Hamiltonian density property for closed coadjoint orbits of all complex Lie groups. For a complex Lie group $G$ with a real form $G_{0}\subset G$, we prove that any Hamiltionian automorphism $\phi$ of a coadjoint orbit $\mathcal{O}_{0}$ of $G_{0}$ whose connected components are simply connected, may be approximated by holomorphic $O_{0}$-invariant symplectic automorphism of the corresponding coadjoint orbit of $G$ in the sense of Carleman, provided that $\mathcal{O}$ is closed. In the course of the proof, we establish the ...

32E30 ; 32H04 ; 32M05

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- 160 p.
ISBN 978-2-86883-521-5

Cours spécialisés , 0007

Localisation : Collection 1er étage;Disparu

dimension de Hausdorff # ensemble de Fatou # ensemble de Julia # famille holomorphe # hyperbolicité # mesure de Green # méthode quasi-conforme # théorème de non-errance de Sullivan

22Exx ; 30C62 ; 31A05 ; 37A25 ; 37F15 ; 37F45 ; 37F50

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Publication de l'Institut de Recherche Mathématique Avancée

Localisation : Publication 1er étage

théorème ergodique # spectre

47A35 ; 47A10

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