Entropy of eigenfunctions of the Laplacian in dimension 2

We study asymptotic properties of eigenfunctions of the Laplacian on compact Riemannian surfaces of Anosov type (for instance negatively curved surfaces). More precisely, we give an answer to a question of Anantharaman and Nonnenmacher [4] by proving that the Kolmogorov-Sinai entropy of a semiclassical measure μ for the geodesic flow gt is bounded from below by half of the Ruelle upper bound. (This text has been written for the proceedings of the 37èmes Journées EDP (Port d’Albret-June, 7-11 2010)) 1. Motivations and results Consider a smooth, compact, connected and Riemannian manifold M which has no boundary and which is of finite dimension d. In this talk, the main result will give an information on the asymptotic behavior of eigenfunctions of the Laplace Beltrami operator ∆ on M in the case of a chaotic geodesic flow. The geodesic flow g on the cotangent bundle T ∗M is defined as the Hamiltonian flow corresponding toH(x, ξ) := ‖ξ‖ 2 x 2 , where ‖.‖x is the norm on T ∗ xM induced by the metric on M . Using pseudodifferential calculus with a small parameter ~ > 0 [12], the quantum operator corresponding to H is −~2∆. A way to look at eigenfunctions of ∆ in the large eigenvalue limit is to understand the eigenfunctions ψ~ of −~ 2∆ 2 associated to the eigenvalue1 1 in the semiclassical limit ~ → 0, i.e. look at the solutions of −~∆ψ~ = ψ~. Using again ~-pseudodifferential calculus, one can associate to every observable a in a good class of symbols an operator Op~(a) acting on L2(M). Using these operators, one can define a distribution μ~ on T ∗M : ∀a ∈ C∞ o (T ∗M), μ~(a) = ∫ T ∗M a(x, ξ)dμ~(x, ξ) := 〈ψ~,Op~(a)ψ~〉L2(M). 1As M is compact, a sequence of such semiclassical parameters ~ is a discrete subsequence that tends to 0.