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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 geodesics. 1. Introduction, statement of results We consider a compact Riemannian manifold M of dimension d ≥ 2, and assume that the geodesic flow (g t )t∈R , acting on the unit tangent bundle of M , has a “chaotic”. | Annals of Mathematics Entropy and the localization of eigenfunctions By Nalini Anantharaman Annals of Mathematics 168 2008 435 475 Entropy and the localization of eigenfunctions By 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 semi-classical 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 t2R 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 Schrodinger the unitary flow exp i ty R acting on the Hilbert space L2 M where A stands for the Laplacian on M and is proportional to the Planck constant. Recall that the quantum flow converges in a sense to the classical flow gt in the so-called semi-classical limit 0 one can imagine that for small values of 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 .
