On the Hausdorff dimension of geodesics that diverge on average

TL;DR

This paper establishes a relationship between the Hausdorff dimension of certain geodesic directions and the entropy at infinity of the geodesic flow in negatively curved manifolds, also relating measure entropy bounds.

Contribution

It proves that the Hausdorff dimension of diverging geodesic directions equals the entropy at infinity, and bounds measure entropy by the same entropy at infinity.

Findings

Hausdorff dimension matches entropy at infinity for diverging geodesics

Measure entropy is bounded above by entropy at infinity

Results apply to complete, pinched negatively curved manifolds

Abstract

In this article we prove that the Hausdorff dimension of geodesic directions that are recurrent and diverge on average coincides with the entropy at infinity of the geodesic flow for any complete, pinched negatively curved Riemannian manifold. Furthermore, we prove that the entropy of a $\sigma$-finite, infinite, ergodic and conservative invariant measure is bounded from above by the entropy at infinity of the geodesic flow.