Spherical means on compact locally symmetric spaces of non-positive curvature.

Abstract

We consider spherical means of continuous functions on the unit tangent bundle of a compact, non-positively curved locally symmetric space and study their behavior as the radius tends to infinity. In dimension greater or equal to 2, we prove that spherical means converge to a probability measure of maximal entropy. This limit measure has an easy characterization in both geometric and algebraic terms. On our way we also derive a convergence result for horospherical means on compact locally symmetric spaces of noncompact type.