 
Summary: ENTROPY AND THE LOCALIZATION OF EIGENFUNCTIONS.
NALINI ANANTHARAMAN
Abstract. We study the large eigenvalue limit for the eigenfunctions of the
laplacian, on a compact manifold of negative curvature in fact, we only
assume that the geodesic flow has the Anosov property. In the semiclassical
limit, we prove that the Wigner measures associated to eigenfunctions have
positive metric entropy. In particular, they cannot concentrate entirely on
closed geodesics.
1. Introduction, statement of results
We consider a compact Riemannian manifold M of dimension d 2, and assume
that the geodesic flow (gt
)tR, acting on the unit tangent bundle of M, has a
"chaotic" behaviour. This refers to the asymptotic properties of the flow when
time t tends to infinity: ergodicity, mixing, hyperbolicity...: we assume here that
the geodesic flow has the Anosov property, the main example being the case of
negatively curved manifolds. The words "quantum chaos" express the intuitive
idea that the chaotic features of the geodesic flow should imply certain special
features for the corresponding quantum dynamical system: that is, according to
Schr¨odinger, the unitary flow exp(i t
2 ) tR
