Talk by Semyon Dyatlov

Semiclassical measures are a standard object studied in quantum chaos, capturing macroscopic behavior of sequences of eigenfunctions in the high energy limit. In previous work with Jin and Nonnenmacher we showed that for Laplacian eigenfunctions on negatively curved surfaces, semiclassical measures have full support. This result is not available in higher dimensions because the key new ingredient, the fractal uncertainty principle (proved by Bourgain and the speaker), is only known for subsets of the real line. In this talk I will present joint work with Jézéquel which focuses on eigenfunctions of quantum cat maps in higher dimensions. We show that, if the quantized matrix has a unique largest eigenvalue and its characteristic polynomial is irreducible over the rationals, then all semiclassical measures have full support. The proof uses the one-dimensional fractal uncertainty principle and a careful choice of propagation time.