Equidistribution of Eisenstein series on convex co-compact hyperbolic manifolds

TL;DR

This paper proves that high-frequency Eisenstein series on certain hyperbolic manifolds concentrate along geodesics and, on average, become uniformly distributed according to the Liouville measure.

Contribution

It establishes microlocal concentration of Eisenstein series and their average equidistribution on convex co-compact hyperbolic manifolds with small limit set dimension.

Findings

Eisenstein series concentrate microlocally on geodesic sets

Average limit measures equidistribute to Liouville measure

Results apply to manifolds with limit set dimension less than n/2

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.