In joint work with Semyon Dyatlov and Nicholas Miller, we study semiclassical measures for Laplacian eigenfunctions on compact complex hyperbolic quotients. Geodesic flows on these quotients are a model case of hyperbolic dynamical systems with different expansion/contraction rates in different directions. We show that the support of any semiclassical measure is either equal to the entire cosphere bundle or contains the cosphere bundle of a compact immersed totally geodesic complex submanifold. The proof uses the one-dimensional fractal uncertainty principle of Bourgain-Dyatlov along the fast expanding/contracting directions, in a way similar to the work of Dyatlov-Jézéquel on quantum cat maps, together with a description of the closures of fast unstable/stable trajectories relying on Ratner theory.
