A cornerstone result in the field of quantum chaos is the quantum ergodicity theorem, which asserts that the Laplacian eigenfunctions of a compact manifold with ergodic Hamiltonian flow satisfy a certain notion of equidistribution in the high energy limit.
In this talk I will discuss the conjectured generalisation of this result to systems with multiple invariant subsets of positive measure, and in particular my recent work which establishes this conjecture for a family of mushroom billiards.