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Mathematics Colloquium


Date:  Tuesday, September 28, 2010

Title:  Analysis on laminations by Riemann Surfaces

Abstract:  Consider the polynomial differential equation in $ \C^2$ \begin{equation*} \frac{dz}{dt}=P(z,w),\qquad \frac{dw}{dt}=Q(z,w). \end{equation*} The polynomials $ P$ and $Q $ are holomorphic, the time is complex. In order to study the global behavior of the solutions. It is convenient to consider the extension as a foliation in the projective plane $P^2$. 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. In order to understand their dynamics we need some analysis on such objects. We will discuss harmonic currents directed by the lamination. The heat equation with respect to a harmonic measure and geometric ergodic theorems for laminations with singularities. They are the analogues in this context of the classical Birkhoff's ergodic theorem.

Speaker:  Nessim Sibony
Institution:  Université Paris-Sud


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