Entropy and the localization of eigenfunctions

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 semi-classical limit, we prove that the Wigner measures associated to eigenfunctions have positive metric entropy. In particular, they cannot concentrate entirely on closed geodesies. 1. Introduction, statement of results We consider a compact Riemannian manifold M of dimension d > 2, and assume that the geodesic flow (#*)teR> acting on the unit tangent bundle of M, has a 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 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 Schrodinger, the unitary flow (exp(iht^))t£R acting on the Hilbert space L2(M), where A stands for the Laplacian on M and h is proportional to the Planck constant. Recall that the quantum flow converges, in a sense, to the classical flow (g*) in the so-called semi-classical limit h - > 0; one can imagine that for small values ofh the quantum system will inherit certain qualitative properties of the classical flow. One expects, for instance, a very different behaviour of eigenfunctions of the Laplacian, or the distribution of its eigenvalues, if the geodesic flow is Anosov or, in the other extreme, completely integrable (see [Sa95]). The convergence of the quantum flow to the classical flow is stated in the Egorov theorem. Consider one of the usual quantization procedures Op^, which associates an operator Oph(a) acting on L2(M) to every smooth compactly supported function a e C%°(T*M) on the cotangent bundle T*M. According to the Egorov theorem, we have for any fixed t