Percival’s conjecture for the Bunimovich mushroom billiard

Abstract

The Laplace–Beltrami eigenfunctions on a compact Riemannian manifold M whose geodesic billiard flow has mixed character have been conjectured by Percival to split into two complementary families, with all semiclassical mass supported in the completely integrable and ergodic regions of phase space respectively. In this paper, we consider the Dirichlet Laplacian on a family of mushroom billiards Mt parametrised by the length of their rectangular component. We prove that there exist eigenfunction subsequences of Mt with full upper density that split as conjectured by Percival for almost all , providing the first example of a billiard known to satisfy this weak form of Percival’s conjecture.