Abstract:

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.