DMITRY DOLGOPYAT  AND  CARLANGELO LIVERANI                        
                                                  
                                                  
                                     Energy transfer in a fast-slow Hamiltonian                           
                                                    Abstract

We consider a finite region of a lattice of weakly interacting geodesic 
flows on manifolds of negative curvature and we show that, when rescaling 
the interactions and the time appropriately, the energies of the flows 
evolve according to a non linear diffusion equation. This is a first step 
toward the derivation of macroscopic equations from a Hamiltonian 
microscopic dynamics in the case of weakly coupled systems.