Dimension of escaping geodesics
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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
- Received by editor(s): November 29, 2005
- Received by editor(s) in revised form: March 9, 2007
- Published electronically: May 22, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 5589-5602
- MSC (2000): Primary 30F40, 28A78; Secondary 30F35
- DOI: https://doi.org/10.1090/S0002-9947-08-04513-3
- MathSciNet review: 2415087