Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Bounded geodesics of Riemann surfaces and hyperbolic manifolds

Authors: J. L. Fernández and M. V. Melián
Journal: Trans. Amer. Math. Soc. 347 (1995), 3533-3549
MSC: Primary 30F35; Secondary 30F40, 53C22
MathSciNet review: 1297524
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the set of bounded geodesics of hyperbolic manifolds. For general Riemann surfaces and for hyperbolic manifolds with some finiteness assumption on their geometry we determine its Hausdorff dimension. Some applications to diophantine approximation are included.

References [Enhancements On Off] (What's this?)

  • [A] L. V. Ahlfors, Möbius transformations, Lecture Notes, Univ. of Minnesota, Minneapolis, 1981.
  • [B] A. F. Beardon, The geometry of discrete groups, Springer-Verlag, 1983. MR 698777 (85d:22026)
  • [Be] L. Bers, An inequality for Riemann surfaces, Differential Geometry and Complex Analysis H.E. Rauch memorial volume, Springer-Verlag, 1985. MR 780038 (86h:30076)
  • [BJ] C. Bishop and P. Jones, Hausdorff dimension and Kleinian groups, preprint. MR 1484767 (98k:22043)
  • [C] L. Carleson, Selected problems on exceptional sets, Van Nostrand, Princeton, NJ, 1967. MR 0225986 (37:1576)
  • [D] S. G. Dani, On badly approximable numbers, Schmidt games and bounded orbits of flows, LMS Lecture Notes, 134, Cambridge Univ. Press, 1989. MR 1043706 (91d:58200)
  • [FP] J. L. Fernández and D. Pestana, Radial images by holomorphic mappings, Proc. Amer. Math. Soc. (to appear). MR 1283549 (96d:30007)
  • [J] V. Jarník, Zur metrischen theorie der diophantischen approximationen, Prace Mat-Fiz. 36 (1928-1929), 91-106.
  • [K] I. Kra, Automorphic forms and Kleinian groups, Benjamin, Reading, MA, 1972. MR 0357775 (50:10242)
  • [M] A. Manning, Dynamics of geodesic and horocycle flows on surfaces of constant negative curvature, Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford Univ. Press, 1991. MR 1130173
  • [N] P. Nicholls, The ergodic theory of discrete groups, LMS Lecture Notes, 143, Cambridge Univ. Press, 1989. MR 1041575 (91i:58104)
  • [P1] S. J. Patterson, Diophantine approximation in Fuchsian groups, Philos. Trans. Royal Soc. London 282 (1976). MR 0568140 (58:27872)
  • [P2] -, Some examples of Fuchsian groups, Proc. London Math. Soc. 39 (1979), 276-298. MR 548981 (80j:30070)
  • [P3] -, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241-373. MR 0450547 (56:8841)
  • [PS] R.S. Phillips and P. Sarnak, On the spectrum of the Hecke groups, Duke Math. J. 52 (1985), 211-221. MR 791299 (86j:11042)
  • [R] C.A. Rogers, Hausdorff measures, Cambridge Univ. Press, 1970. MR 0281862 (43:7576)
  • [Ro] D. Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J. 21 (1954), 549-563. MR 0065632 (16:458d)
  • [S1] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. Inst. Hautes Etude. Sci. 50 (1979), 171-202. MR 556586 (81b:58031)
  • [S2] -, Disjoint spheres, approximation by imaginary quadratic numbers and the logarithm law for geodesics, Acta Math. 149 (1982), 215-237. MR 688349 (84j:58097)
  • [S3] -, Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups, Acta Math. 153 (1984), 259-277. MR 766265 (86c:58093)
  • [T] B. Thurston, Geometry and topology of $ 3$-manifolds, Notes from Princeton University, 1978.
  • [Tu1] P. Tukia, On the dimension of limit sets of geometrically finite Möbius groups, Ann. Acad. Sci. Fenn. Ser. A I 19 (1994), 11-24. MR 1246883 (95i:30035)
  • [Tu2] -, On isomorphisms of geometrically finite Kleinian groups, Publ. Math. Inst. Hautes Ètude Sci. 61 (1985), 171-214.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30F35, 30F40, 53C22

Retrieve articles in all journals with MSC: 30F35, 30F40, 53C22

Additional Information

Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society