Equidistribution of Eisenstein series on convex co-compact hyperbolic manifolds

Colin Guillarmou (1) , Frederic Naud (2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 LANLG - Laboratoire d'Analyse non linéaire et Géométrie

For convex co-compact hyperbolic manifolds $\Gamma\backslash \mathbb{H}^{n+1}$ for which the dimension of the limit set satisfies $\delta_\Gamma< n/2$, we show that the high-frequency Eisenstein series associated to a point $\xi$ ''at infinity'' concentrate microlocally on a measure supported by (the closure of) the set of points in the unit cotangent bundle corresponding to geodesics ending at $\xi$. The average in $\xi$ of these limit measures equidistributes towards the Liouville measure.

Domaines: Théorie spectrale [math.SP]; Géométrie différentielle [math.DG]

Colin Guillarmou, Frederic Naud. Equidistribution of Eisenstein series on convex co-compact hyperbolic manifolds. American Journal of Mathematics, 2014, 136 (2), pp.445-479. ⟨10.1353/ajm.2014.0015⟩. ⟨hal-00608699⟩