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Unique ergodicity for foliations on compact Kähler surfaces

De Nessim Sibony

Apparaît dans la collection : Complex Dynamics / Dynamique complexe

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).

Informations sur la vidéo

Données de citation

  • DOI 10.24350/CIRM.V.19602503
  • Citer cette vidéo Sibony, Nessim (27/01/2020). Unique ergodicity for foliations on compact Kähler surfaces. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19602503
  • URL https://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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