The behavior under projection of dilating sets in a covering space

Randol, Burton

doi:10.1090/S0002-9947-1984-0752507-0

The behavior under projection of dilating sets in a covering space
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Trans. Amer. Math. Soc. 285 (1984), 855-859 Request permission

Abstract:

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.

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