Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Excursion and return times of a geodesic to a subset of a hyperbolic Riemann surface


Author: Andrew Haas
Journal: Proc. Amer. Math. Soc. 141 (2013), 3957-3967
MSC (2010): Primary 30F35; Secondary 32Q45, 37E35, 53D25
DOI: https://doi.org/10.1090/S0002-9939-2013-11767-3
Published electronically: July 30, 2013
MathSciNet review: 3091786
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We calculate the asymptotic average rate at which a generic geodesic on a finite area hyperbolic 2-orbifold returns to a subsurface with geodesic boundary. As a consequence we get the average time a generic geodesic spends in such a subsurface. Related results are obtained for excursions into a collar neighborhood of a simple closed geodesic and the associated distribution of excursion depths.


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

  • 1. A. Basmajian, Constructing pairs of pants, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), no. 1, 65-74. MR 1050782 (91g:57041)
  • 2. A. F. Beardon, The Geometry of Discrete Groups, Springer-Verlag, Berlin, 1983. MR 698777 (85d:22026)
  • 3. Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, T. Bedford, M. Keane, C. Series, eds., Oxford Univ. Press, Oxford, 1991. MR 1130170 (93e:58002)
  • 4. W. Bosma, Approximation by mediants, Math. of Computation 54 (1990), 421-432. MR 995207 (90m:11119)
  • 5. W. Bosma, H. Jager and F. Wiedijk, Some metrical observations on the approximation by continued fractions, Indag. Math. 45 (1983), 281-299. MR 718069 (85f:11059)
  • 6. M. Bridgeman and D. Dumas, Distribution of intersection lengths of a random geodesic with a geodesic lamination, Ergodic Theory Dynam. Systems 27 (2007), 1055-1072. MR 2342965 (2008h:37035)
  • 7. P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Progress in Math., vol. 106, Birkhäuser, Boston, 1992. MR 1183224 (93g:58149)
  • 8. K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, 2002. MR 1917322 (2003f:37014)
  • 9. I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1982. MR 832433 (87f:28019)
  • 10. A. Haas, The distribution of geodesic excursions out the end of a hyperbolic orbifold and approximation with respect to a Fuchsian group, Geom. Dedicata 116 (2005), no. 1, 129-155. MR 2195445 (2006j:30077)
  • 11. A. Haas, Geodesic cusp excursions and metric diophantine approximation, Math. Res. Lett. 16 (2009), no. 1, 67-85. MR 2480562 (2009m:30080)
  • 12. A. Haas, Geodesic excursions into an embedded disc on a hyperbolic Riemann surface,
    Conform. Geom. Dyn. 13 (2009), 1-5. MR 2476655 (2010h:37065)
  • 13. S. Hersonsky and F. Paulin, On the almost sure spiraling of geodesics in negatively curved manifolds, J. Differential Geom. 85 (2010), no. 2, 271-314. MR 2732978 (2011i:53138)
  • 14. H. Nakada, On metrical theory of Diophantine approximation over imaginary quadratic field. Acta Arith. 51 (1988), no. 4, 399-403. MR 971089 (89m:11070)
  • 15. P. Nicholls, The Ergodic Theory of Discrete Groups, London Math. Soc. Lecture Note Series, 143. Cambridge Univ. Press, 1989. MR 1041575 (91i:58104)
  • 16. B. Stratmann, A note on counting cuspidal excursions. Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995) no. 2, 359-372. MR 1346819 (96k:58174)
  • 17. D. Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers and the logarithm law for geodesics. Acta Math. 149 (1982), 215-273. MR 688349 (84j:58097)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30F35, 32Q45, 37E35, 53D25

Retrieve articles in all journals with MSC (2010): 30F35, 32Q45, 37E35, 53D25


Additional Information

Andrew Haas
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: haas@math.uconn.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11767-3
Keywords: Hyperbolic surface, Fuchsian group, geodesic flow.
Received by editor(s): February 12, 2011
Received by editor(s) in revised form: January 28, 2012
Published electronically: July 30, 2013
Communicated by: Michael Wolf
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society