The behavior under projection of dilating sets in a covering space

Let $M$ be a compact Riemannian manifold with covering space $S$, and suppose $d{\mu_r}\;(0 < r < \infty )$ is a family of Borel probability measures on $S$, all of which arise from some fixed measure by $r$-homotheties of $S$ about some point, followed by renormalization of the resulting measure. In this paper we study the ergodic properties, as a function of $r$, of the corresponding family of projected measures on $M$ in the Euclidean and hyperbolic cases. A typical example arises by considering the behavior of a dilating family of spheres under projection.