Percival's conjecture for mixed dynamical systems

Let $M$ be a compact Riemannian manifold with flow domain $\mathcal{D}$, and suppose that $\mathcal{D}$ is the disjoint union of two invariant subsets $U$ and $\mathcal{D}\setminus U$, where $U$ is ergodic and $\mathcal{D}\setminus U$ is integrable. Let $(u_k)_{k\in\mathbb{N}}$ be a complete system of eigenfunctions of the Laplace--Beltrami operator. Percival's conjecture. There exist subsets $A,B\subset\mathbb{N}$ such that

$$A\cup B\text{ has density }1,$$

$(u_k)_{k\in A}$ equidistributes in the ergodic region $U$, every semiclassical measure associated to the subset $B$ is supported in the completely integrable region $\mathcal{D}\setminus U$, and

$$\operatorname{dens}(A)=\mu_L(U).$$

This conjecture predicts a spectral decomposition reflecting the ergodic and integrable components of a mixed dynamical system. In particular, the natural densities of the two subsets of eigenfunctions should be proportional to the Liouville measures of the corresponding invariant regions; the paper presents the Bunimovich mushroom billiard as a model case where a satisfactory general analogue of quantum ergodicity is not yet known.