Amount of failure of upper-semicontinuity of entropy in noncompact rank   one situations, and Hausdorff dimension
  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 flows on homogeneous spaces $\Gamma\backslash G$, where $G$ is any connected semisimple Lie group of real rank 1 with finite center and $\Gamma$ 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. 
Shirali Kadyrov, Anke D. Pohl
math.DS