We examine high energy eigenfunctions for the Dirichlet Laplacian on domains where the billiard flow exhibits mixed dynamical behavior. (More generally, we consider semiclassical Schrödinger operators with mixed assumptions on the Hamiltonian flow.) Specifically, we assume that the billiard flow has an invariant ergodic component, , and study defect measures, , of positive density subsequences of eigenfunctions (and, more generally, of almost orthogonal quasimodes). We show that any defect measure associated to such a subsequence satisfies , where is the Liouville measure. This proves part of a conjecture of Percival .
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Jeffrey Galkowski, Quantum ergodicity for a class of mixed systems. J. Spectr. Theory 4 (2014), no. 1, pp. 65–85DOI 10.4171/JST/62