Delocalization of slowly damped eigenmodes on Anosov manifolds

Gabriel Riviere (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

We look at the properties of high frequency eigenmodes for the damped wave equation on a compact manifold with an Anosov geodesic flow. We study eigenmodes with spectral parameters which are asymptotically close enough to the real axis. We prove that such modes cannot be completely localized on subsets satisfying a condition of negative topological pressure. As an application, one can deduce the existence of a ''strip'' of logarithmic size without eigenvalues below the real axis under this dynamical assumption on the set of undamped trajectories.

Mots clés: damped wave equation; semiclassical measures; Anosov flows; topological pressure; Kolmogorov-Sinai entropy

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Systèmes dynamiques [math.DS]

Gabriel Riviere. Delocalization of slowly damped eigenmodes on Anosov manifolds. Communications in Mathematical Physics, 2012, 316 (2), pp.555-593. ⟨10.1007/s00220-012-1588-7⟩. ⟨hal-00620636v2⟩