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

TL;DR

This paper investigates the failure of upper-semicontinuity of entropy in noncompact rank-one homogeneous spaces, establishing sharp bounds and exploring the Hausdorff dimension of diverging points.

Contribution

It proves the sharpness of existing bounds on entropy failure and applies these methods to estimate the Hausdorff dimension of points diverging on average.

Findings

Bound on entropy failure is sharp.

Established bounds for Hausdorff dimension of diverging points.

Extended understanding of entropy behavior in noncompact rank-one spaces.

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 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.