Eigenmode delocalization on Anosov surfaces
(joint with Semyon Dyatlov and Long Jin)
The eigenmodes of the Laplacian on a smooth compact Riemannian
manifold (with no boundary) $(M,g)$ can exhibit various localization properties in the
high frequency limit, which depend on the dynamical properties of the geodesic flow.
I will focus on a "quantum chaotic" situation, namely assume that the
geodesic flow is strongly chaotic (Anosov);
this is the case if the sectional curvature of $(M,g)$ is strictly negative.
The Quantum Ergodicity theorem then states that almost all the eigenmodes become
equidistributed on $M$ in the the high frequency limit. The Quantum Unique Ergodicity
conjecture states that this behaviour admits no exception, namely
all eigenstates should equidistribute in this limit.
This conjecture remaining inaccessible, a less ambitious goal is to constrain the possible
localization behaviours of the eigenmodes. I will discuss a recent progress in the case of Anosov surfaces:
we show that all the eigenmodes must remain delocalized across all of $M$ in the high frequency limit.
More precisely, any semiclassical measure (flow-invariant probability measure on $S^*M$,
embodying the limiting localization properties of a subsequence of eigenmodes) has full support.
The proof, which generalizes a work by Dyatlov-Jin in the constant curvature
case, uses dynamical properties of Anosov flows (the foliation of $S^*M$ into stable and
unstable manifolds), methods from semiclassical analysis, and a recent
Fractal Uncertainty Principle due to Bourgain-Dyatlov.