Dimension of escaping geodesics

Author:
Zsuzsanna Gönye

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

Published electronically:
May 22, 2008

MathSciNet review:
2415087

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Abstract | References | Similar Articles | Additional Information

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

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.