CIRM - Videos & books Library - Unique ergodicity for foliations on compact Kähler surfaces

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Auteurs : Sibony, Nessim (Auteur de la Conférence)
CIRM (Editeur )

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ergodic theory foliations by Riemann surfaces characteristic function Ahlfors current ddc-closed currents foliations by Riemann surfaces hyperbolic singularities characteristic function unique ergodicity in case of invariant curves generic unique ergodicity rigidity for Hénon maps density for currents

Résumé : 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).

Keywords : foliations by Reimann surfaces; currents; density

Codes MSC :
37Axx - Ergodic theory
37F75 - Holomorphic foliations and vector fields

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https://videos.cirm-math.fr/2020-01-27_Sibony.mp4

Informations sur la Rencontre

Nom de la rencontre : Complex Dynamics / Dynamique complexe
Organisateurs de la rencontre : Berteloot, François ; Nguyên, Viêt Anh
Dates : 27/01/2020 - 31/01/2020
Année de la rencontre : 2020
URL Congrès : https://conferences.cirm-math.fr/2148.html

Données de citation

DOI : 10.24350/CIRM.V.19602503
Citer cette vidéo: Sibony, Nessim (2020). Unique ergodicity for foliations on compact Kähler surfaces. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19602503
URI : http://dx.doi.org/10.24350/CIRM.V.19602503

Bibliographie

DINH, Tien-Cuong et SIBONY, Nessim. Unique ergodicity for foliations in $\mathbb {P}^ 2$ with an invariant curve. Inventiones mathematicae, 2018, vol. 211, no 1, p. 1-38. - https://arxiv.org/abs/1509.07711

DINH, Tien-Cuong et SIBONY, Nessim. Rigidity of Julia sets for Hénon type maps. arXiv preprint arXiv:1301.3917, 2013. - https://arxiv.org/abs/1301.3917

DINH, Tien-Cuong et SIBONY, Nessim. Rigidity of Julia sets for Hénon type maps. Journal of Modern Dynamics, 2014, vol. 8, no 3&4, p.499-548. - https://arxiv.org/abs/1301.3917

FORNÆSS, John Erik et SIBONY, Nessim. Harmonic currents of finite energy and laminations. Geometric & Functional Analysis GAFA, 2005, vol. 15, no 5, p. 962-1003. - https://arxiv.org/abs/math/0402432

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