Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold

Abstract

We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized.Comment: We added the proof of the Entropic Uncertainty Principle. 45 pages, 2 EPS figure