Variations on a Theorem of Birman and Series

TL;DR

This paper investigates the geometric and fractal properties of geodesics on hyperbolic surfaces constrained by a function relating their intersection number and length, revealing dimension bounds for certain geodesic sets.

Contribution

It introduces a new analysis of geodesic closures constrained by intersection-length functions, establishing bounds on their Hausdorff dimension.

Findings

Set of geodesics has Hausdorff dimension between 1 and 3 when f is unbounded and sublinear.

Provides new bounds on the complexity of geodesic sets under intersection constraints.

Extends understanding of geodesic behavior in hyperbolic geometry.

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 $I(\gamma,\gamma)\leq f(\ell_\Sigma(\gamma))$. For instance we prove that if $f$ is unbounded and sublinear then this set has Hausdorff dimension strictly bounded between 1 and 3.