### Nalini Anantharaman (Ecole Polytechnique): Entropy and localization of eigenfunctions

Abstract. On a compact negatively curved manifold, we study the asymptotic behaviour of the eigenfunctions $(\phi_n)$ of the laplacian, when the eigenvalue $\lambda_n$ goes to infinity. The Quantum Unique Ergodicity conjecture says that the probability measures $|\phi_n(x)|^2dx$ should converge weakly to the riemannian volume (the uniform measure). We prove a result going in this direction, saying that the `dynamical' entropy of these measures is asymptotically positive. This is a follow-up talk to the colloquium, and will give more details about the proof and recent developments.