Amount of failure of upper-semicontinuity of entropy in noncompact rank one situations, and hausdorff dimension

Abstract

Recently, Einsiedler and the authors provided a bound in terms of escape of mass for the amount by which upper-semicontinuity for metric entropy fails for diagonal ows on homogeneous spaces 􀀀nG, where G is any connected semisimple Lie group of real rank 1 with nite center, and 􀀀 is any nonuniform lattice in G. We show that this bound is sharp, and apply the methods used to establish bounds for the Hausdorff dimension of the set of points which diverge on average.