## Dimension of escaping geodesics

HTML articles powered by AMS MathViewer

- 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

- I. Agol,
*Tameness of Hyperbolic $3$-manifolds*, preprint, available at arXiv:math.GT/0405568, 2004. - Lars V. Ahlfors,
*Möbius transformations in several dimensions*, Ordway Professorship Lectures in Mathematics, University of Minnesota, School of Mathematics, Minneapolis, Minn., 1981. MR**725161** - Alan F. Beardon,
*The geometry of discrete groups*, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR**698777**, DOI 10.1007/978-1-4612-1146-4 - Christopher J. Bishop and Peter W. Jones,
*The law of the iterated logarithm for Kleinian groups*, Lipa’s legacy (New York, 1995) Contemp. Math., vol. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 17–50. MR**1476980**, DOI 10.1090/conm/211/02813 - C. J. Bishop and Y. Peres,
*Hausdorff dimension and Fractal Sets*, to appear. - Danny Calegari and David Gabai,
*Shrinkwrapping and the taming of hyperbolic 3-manifolds*, J. Amer. Math. Soc.**19**(2006), no. 2, 385–446. MR**2188131**, DOI 10.1090/S0894-0347-05-00513-8 - Richard D. Canary,
*Ends of hyperbolic $3$-manifolds*, J. Amer. Math. Soc.**6**(1993), no. 1, 1–35. MR**1166330**, DOI 10.1090/S0894-0347-1993-1166330-8 - S.-Y. A. Chang, J. M. Wilson, and T. H. Wolff,
*Some weighted norm inequalities concerning the Schrödinger operators*, Comment. Math. Helv.**60**(1985), no. 2, 217–246. MR**800004**, DOI 10.1007/BF02567411 - José L. Fernández and María V. Melián,
*Escaping geodesics of Riemannian surfaces*, Acta Math.**187**(2001), no. 2, 213–236. MR**1879849**, DOI 10.1007/BF02392617 - Z. Gönye,
*The Dimension of Escaping Points*, Ph.D. thesis, State University of New York at Stony Brook, 2001. - P. Hall and C. C. Heyde,
*Martingale limit theory and its application*, Probability and Mathematical Statistics, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR**624435** - G. J. Hungerford,
*Boundaries of smooth sets and singular sets of Blaschke products in the little Bloch space*, Master’s thesis, California Institute of Technology, 1988. - Bernard Maskit,
*Kleinian groups*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR**959135** - Curtis T. McMullen,
*Renormalization and 3-manifolds which fiber over the circle*, Annals of Mathematics Studies, vol. 142, Princeton University Press, Princeton, NJ, 1996. MR**1401347**, DOI 10.1515/9781400865178 - Peter J. Nicholls,
*The ergodic theory of discrete groups*, London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. MR**1041575**, DOI 10.1017/CBO9780511600678 - S. Rohde,
*The boundary behavior of Bloch functions*, J. London Math. Soc. (2)**48**(1993), no. 3, 488–499. MR**1241783**, DOI 10.1112/jlms/s2-48.3.488 - Dennis Sullivan,
*Growth of positive harmonic functions and Kleinian group limit sets of zero planar measure and Hausdorff dimension two*, Geometry Symposium, Utrecht 1980 (Utrecht, 1980) Lecture Notes in Math., vol. 894, Springer, Berlin-New York, 1981, pp. 127–144. MR**655423**

## 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