The APDE seminar on Monday, 4/11, will be given by Malo Jézéquel (MIT) both in-person (891 Evans) and online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: Semiclassical measures for higher dimensional quantum cat maps.

Abstract: Quantum chaos is the study of quantum systems whose
associated classical dynamics is chaotic. For instance, a central
question concerns the high frequencies behavior of the eigenstates of
the Laplace-Beltrami operator on a negatively curved compact
Riemannian manifold M. In that case, the associated classical dynamics
is the geodesic flow on the unit tangent bundle of M, which is
hyperbolic and hence chaotic. Quantum cat maps are a popular toy model
for this problem, in which the geodesic flow is replaced by a cat map,
i.e. the action on the torus of a matrix with integer coefficients. In
this talk, I will introduce quantum cat maps, and then discuss a
result on delocalization for the associated eigenstates. It is deduced
from a \emph{fractal uncertainty principle}. Similar statements have
been obtained in the context of negatively curved surfaces by
Dyatlov-Jin and Dyatlov-Jin-Nonnenmacher, and the case of
two-dimensional cat maps have been dealt with by Schwartz. The novelty
of our result is that we are sometimes able to bypass the restriction
to low dimensions. This is a joint work with Semyon Dyatlov.