 
Summary: HALFDELOCALIZATION OF EIGENFUNCTIONS FOR THE
LAPLACIAN ON AN ANOSOV MANIFOLD
NALINI ANANTHARAMAN AND STÉPHANE NONNENMACHER
Abstract. We study the highenergy eigenfunctions of the Laplacian on a compact Rie
mannian manifold with Anosov geodesic flow. The localization of a semiclassical measure
associated with a sequence of eigenfunctions is characterized by the KolmogorovSinai
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, highenergy eigenfunctions are at least halfdelocalized.
The theory of quantum chaos tries to understand how the chaotic behaviour of a clas
sical Hamiltonian system is reflected in its quantum version. For instance, let M be a
compact Riemannian C # manifold, such that the geodesic flow has the Anosov property
 the ideal chaotic behaviour. The corresponding quantum dynamics is the unitary flow
generated by the LaplaceBeltrami operator on L 2 (M). One expects that the chaotic
properties of the geodesic flow influence the spectral theory of the Laplacian. The Random
Matrix conjecture [6] asserts that the highlying eigenvalues should, after proper renormal
ization, statistically resemble those of a large random matrix, at least for a generic Anosov
metric. The Quantum Unique Ergodicity conjecture [27] (see also [5, 30]) deals with the
corresponding eigenfunctions #: it claims that the probability density #(x) 2 dx should
approach (in a weak sense) the Riemannian volume, when the eigenvalue corresponding
