L∞ norms of chaotic eigenfunctions: probabilistic and deterministic results

When studying Laplace eigenfunctions on compact manifolds, their localisation or delocalisation properties at large eigenvalues are strongly related to the dynamics of the geodesic flow. In this talk, I will be interested in delocalisation phenomena, through the study of L∞ norms of eigenfunctions, on manifolds of negative curvature. After recalling the existing results and conjectures, I will show how these results can be improved by adding small random perturbations to the Laplacian. I will also present some deterministic improvements, in the case of manifolds of constant curvature. These are joint works with Martin Vogel, and with Yann Chaubet.