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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Dimension of escaping geodesics


Author: Zsuzsanna Gönye
Journal: Trans. Amer. Math. Soc. 360 (2008), 5589-5602
MSC (2000): Primary 30F40, 28A78; Secondary 30F35
Published electronically: May 22, 2008
MathSciNet review: 2415087
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Abstract: Suppose $ M=\mathbb{B}/G$ is a hyperbolic manifold. Consider the set of escaping geodesic rays $ \gamma (t)$ originating at a fixed point $ p$ of the manifold $ M$, i.e.  $ \operatorname{dist}(\gamma(t),p)\to \infty$. We investigate those escaping geodesics which escape at the fastest possible rate, and find the Hausdorff dimension of the corresponding terminal points on the boundary of $ \mathbb{B}$.

In dimension $ 2$, for a geometrically infinite Fuchsian group, if the injectivity radius of $ M=\mathbb{B}/G$ is bounded above and away from zero, then these points have full dimension. In dimension $ 3$, when $ G$ is a geometrically infinite and topologically tame Kleinian group, if the injectivity radius of $ M=\mathbb{B}/G$ is bounded away from zero, the dimension of these points is $ 2$, which is again maximal.


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Additional Information

Zsuzsanna Gönye
Affiliation: Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794
Address at time of publication: Department of Mathematics, University of West Hungary, Szombathely, H-9700, Hungary
Email: zgonye@ttmk.nyme.hu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04513-3
Keywords: Fuchsian groups, Kleinian groups, escaping geodesics, deep points, Hausdorff dimension
Received by editor(s): November 29, 2005
Received by editor(s) in revised form: March 9, 2007
Published electronically: May 22, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.