Equidistribution of Eisenstein series on convex co-compact hyperbolic manifolds

Abstract

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.