Geodesic excursions into cusps in finite-volume hyperbolic manifolds
type: Item
instance of: scholarly article
title: Geodesic excursions into cusps in finite-volume hyperbolic manifolds
zbMATH Open document ID: 0793.53052
author: M. V. Melián
author: D. Pestana
published in: Michigan Mathematical Journal
publication date: 1994-01-19T00:00:00Z
review text: 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 \(M\).
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 488357
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: geodesic
zbMATH Keywords: hyperbolic manifolds
zbMATH Keywords: parabolic cusps
reviewed by: Boris N. Apanasov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004675
OpenAlex ID: W2050941284
DOI: 10.1307/MMJ/1029004675
description: scientific article; zbMATH DE number 488357