Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems

We establish the existence of weak limits of solutions (in the class $L_p$, $p\ge1$) of the Liouville equation for nondegenerate quasihomogeneous Hamilton equations. We find the limit probability distributions in the configuration space. We give conditions for a uniform distribution of Gibbs ensembles for geodesic flows on compact manifolds.