Publication: Geodesic excursions into cusps in finite-volume hyperbolic manifolds
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1993
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University of Michigan, Department of Mathematics
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Abstract
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 $\{v\in T\sb p\sp 1(M)$: $\lim\sb{t\to\infty} \sup (\text{dist} (\gamma\sb v(t),p)/t)\geq \alpha\}$ is equal to $n(1-\alpha)$, where $\gamma\sb v(t)$ is the geodesic in $M$ emanating from $p$ in the direction of $v$. This generalize a result of [Acta Math. 149, 215-237 (1982)] that, for almost every direction $v$, such a limit is $1/n$, and it is one for just a countable set of directions $v$.\par 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 $M$ [source: Zentralblatt MATH].
Description
18 pages, no figures.-- MSC1991 codes: Primary: 53C22; Secondary: 30F40, 58F17.
MR#: MR1214056 (94d:53067)
Zbl#: Zbl 0793.53052
MR#: MR1214056 (94d:53067)
Zbl#: Zbl 0793.53052
Keywords
Hausdorff dimension, Geodesic, Hyperbolic manifolds, Parabolic cusps
Bibliographic citation
Michigan Mathematical Journal, 1993, vol. 40, n. 1, p. 77-93