Geodesic excursions into cusps in finite-volume hyperbolic manifolds
The main goal of the paper is to prove that, for a given non-compact hyperbolic \(n\)-manifold \(M\) of finite volume, \(p\in M\), and a number \(\alpha\), \(0\leq\alpha \leq 1\), the Hausdorff dimension of the set \(\{v\in T_ p^ 1(M)\): \(\lim_{t\to\infty} \sup (\text{dist} (\gamma_ v(t),p)/t)\geq \alpha\}\) is equal to \(n(1-\alpha)\), where \(\gamma_ v(t)\) is the geodesic in \(M\) emanating from \(p\) in the direction of \(v\). This generalize a result of \textit{D. Sullivan} [Acta Math. 149, 215-237 (1982; Zbl 0517.58028)] that, for almost every direction \(v\), such a limit is \(1/n\), and it is one for just a countable set of directions \(v\). However we remark that one has to restrict this claim to the class of hyperbolic manifolds with only Abelian parabolic cusps because the authors assume in fact such property for all considered manifolds \(M\).
scientific article; zbMATH DE number 488357