course: Dynamics of foliations by Riemann Surfaces
speaker: Nessim Sibony (Université Paris-Sud)abstract: Consider the polynomial differential equation in $C2$ $frac{dz}{dt}=P(z,w), frac{dw}{dt}=Q(z,w).$ The polynomials $P$ and $Q$ are holomorphic, the time is complex. We want to study the global behavior of the solutions. It is convenient to consider the extension as a foliation in the projective plane $P2$. There are however singular points. When the line at infinity is invariant, Il'yashenko has shown that generically leaves are dense and that the foliation is ergodic. This follows from the study of the holonomy on the invariant line. But generically on the vector field, there is no invariant line and even no invariant algebraic surface as shown by Jouanolou. This example is a special case of a lamination (with singularities) by Riemann surfaces. In particular, one can consider similar questions in any number of dimensions. Other laminations appear in the study of polynomial automorphisms of $Ck$. I will discuss some ergodicity results for such foliations. The analysis on $partial overline{partial}$-closed current is a crucial tool. The talk is based on joint works with J.E Fornaess and with T.C Dinh and V.A.Nguyen.
timetable:
Wed 16 Apr, 14:30 - 15:30, Aula Bianchi Scienze
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