Dynamique Complexe à plusieurs variables – Dynacomplexe

We want study the dynamics of holomorphic and meromorphic maps in several complex variables. The study of these maps started with E. Bedford, J.E. Fornaess, M. Lyubich, N. Sibony and J. Smillie's fundamental works. Actually, this subject is very important in mathematics and receives more attention. For these dynamical systems, the goal is to construct dynamical currents, the equilibrium measure and to study them (for example Lyapounov exponents, laminar structure for the current, Hausdorff dimension for the measure, entropy…). The topological aspect of the dynamics is poorly understood. Until very recently, the results used only currents of bidimension (1,1) or currents of bicodimension (1,1) and consequently the class of examples was small. Very recently, thanks to the contribution of the members of the project and N. Sibony, geometric methods using pluripotential theory and real dynamics have been develop to solve the previous fundamental problem. If f is a meromorphic map or a correspondence on a compact Kähler manifold, then we can construct in many situations the dynamical currents and the equilibrium measure. Moreover, we can compute the entropy of the map and give sharp estimates for the Lyapounov exponents. We want now to develop our methods and to use other theories in order to solve completely the previous problem. The members of the project, in order to realize their objectives, have to develop their knowledge in Real Dynamics, Ergodic Theory, Foliations and Kähler Geometry. So, we would like: 1)To develop our workgroup: Since several years, the members of the project are in a workgroup 'Complex Dynamics and Geometry' at the university Paris 11. We would like to develop this workgroup and to make it financially independent from Harmonic Analysis team in which H. de Thelin and C. Dupont are. This workgroup will meet every week and we will invite french and foreigner researchers. The subjects of this workgroup will be Real Dynamics, Ergodic Theory, Foliations and Kähler Geometry which are the fields useful in order to solve our problem. 2)Organization of a short course every years: We would like to organize a course every year on one of the previous subject. This course will be accessible for students in Master 2. 3)Travels in other laboratories: We would like to meet researchers in other laboratories in order to develop our knowledge in the previous subjects. Connections with F. Ledrappier (University of Notre Dame), Y. T. Siu (Harvard University) or M. Brunella (University of Dijon) will be useful.