Français
Unique ergodicity for foliations on compact Kähler surfaces
Post-edited
Authors : Sibony, Nessim (Author of the conference)
CIRM (Publisher )
37Axx - Ergodic theory
37F75 - Holomorphic foliations and vector fields
CIRM (Publisher )
Abstract :
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
MSC Codes : 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
37Axx - Ergodic theory
37F75 - Holomorphic foliations and vector fields
|
Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 18/02/2020 Conference Date : 27/01/2020 Subseries : Research talks arXiv category : Complex Variables ; Dynamical Systems Mathematical Area(s) : Dynamical Systems & ODE ; Algebraic & Complex Geometry Format : MP4 (.mp4) - HD Video Time : 00:51:18 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2020-01-27_Sibony.mp4 
|
Information on the Event
Event Title : Complex Dynamics / Dynamique complexeEvent Organizers : Berteloot, François ; Nguyên, Viêt Anh Dates : 27/01/2020 - 31/01/2020 Event Year : 2020 Event URL : https://conferences.cirm-math.fr/2148.html
Citation Data
DOI : 10.24350/CIRM.V.19602503Cite this video as: 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 |
See Also
- [Multi angle] Sup-norm estimates for $ \overline{\partial}$ in $\mathbb{C}^{3}$ / Author of the conference Stensønes, Berit.
- [Multi angle] Birational transformations preserving codimension one foliations / Author of the conference Pereira, Jorge Vitório.
- [Multi angle] Projective rational manifolds with non-finitely generated discrete automorphism group and infinitely many real forms / Author of the conference Oguiso, Keiji.
- [Multi angle] Symplectic Carleman approximation on co-adjoint orbits / Author of the conference Wold, Erlend Fornæss.
Bibliography
- 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
