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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Dimension of escaping geodesics
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by Zsuzsanna Gönye PDF
Trans. Amer. Math. Soc. 360 (2008), 5589-5602 Request permission

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.

References
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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