Heat equation and ergodic theorems for Riemann surface laminations [Equation de la chaleur et théorèmes ergodiques pour les laminations par surfaces de Riemann]

摘要

We study the dynamics of possibly singular foliations by Riemann surfaces. The main examples are holomorphic foliations by Riemann surfaces in projective varieties. We introduce the heat equation relative to a positive ??? -closed current and apply it to the directed currents associated with Riemann surface laminations possibly with singularities. This permits to construct the heat diffusion with respect to various Laplacians that could be defined almost everywhere with respect to the ???-closed current. We prove two kinds of ergodic theorems for such currents: one associated to the heat diffusion and one of geometric nature close to Birkhoff's averaging on orbits of a dynamical system. Here the averaging is on hyperbolic leaves and the time is the hyperbolic time. The heat diffusion theorem with respect to a harmonic measure is also developed for real laminations.