Variations on a Theorem of Birman and Series

Abstract

Suppose that $\Sigma$ is a hyperbolic surface and $f:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ a monotonic function. We study the closure in the projective tangent bundle $PT\Sigma$ of the set of all geodesics $\gamma$ satisfying $I(\gamma ,\gamma )\le f({\ell }_{\Sigma }(\gamma ))$. For instance we prove that if $f$ is unbounded and sublinear then this set has Hausdorff dimension strictly bounded between 1 and 3.

21 pages, 3 figures

References in corpus (1)

• The Cheeger Constant, Isoperimetric Problems, and Hyperbolic Surfaces