  We examine high energy eigenfunctions for the Dirichlet Laplacian on domains
where the billiard flow exhibits mixed dynamical behavior. (More generally, we
consider semiclassical Schrodinger operators with mixed assumptions on the
Hamiltonian flow.) Specificially, we assume that the billiard flow has an
invariant ergodic component, U, and study defect measures, mu, of positivie
density subsequences of eigenfunctions (and, more generally, of almost
orthogonal quasimodes). We show that any defect measure associated to such a
subsequence, when restricted to U, satisfies mu = c mu_L where mu_L is the
Liouville measure. This proves part of a conjecture of Percival.
