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 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Angular self-intersections for closed geodesics on surfaces


Authors: Mark Pollicott and Richard Sharp
Journal: Proc. Amer. Math. Soc. 134 (2006), 419-426
MSC (2000): Primary 37C27, 37D20, 37D35, 37D40
Posted: September 20, 2005
MathSciNet review: 2176010
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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we consider asymptotic results for self-intersections of closed geodesics on surfaces for which the angle of the intersection occurs in a given arc. We do this by extending Bonahon's definition of intersection forms for surfaces.

References

  • [An] N. Anantharaman, Distribution of closed geodesics on a surface, under homological constraints, preprint, 1999.
  • [As] D. Anosov, Geodesic flows of closed Riemann manifolds with negative curvature, Proceedings of the Steklov Institute of Mathematics, Vol. 90, Amer. Math. Soc., Providence, RI, 1969. MR 0242194 (39:3527)
  • [BS] J. Birman and C. Series, Geodesics with bounded intersection number on surfaces are sparsely distributed, Topology 24 (1985), 217-225. MR 0793185 (87f:57012)
  • [Bo1] F. Bonahon, Bouts des variétés hyperboliques de dimension $3$, Annals of Math. 124 (1986), 71-158. MR 0847953 (88c:57013)
  • [Bo2] F. Bonahon, The geometry of Teichmüller spaces via geodesic currents, Invent. Math. 92 (1988), 139-162. MR 0931208 (90a:32025)
  • [Do] D. Dolgopyat, On statistical properties of geodesic flows on negatively curved surfaces, Ph.D. thesis, Princeton, 1997.
  • [Hu1] H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen, Math. Ann. 138 (1959), 1-26. MR 0109212 (22:99)
  • [Hu2] H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen, II, Math. Ann. 142 (1961), 385-398. MR 0126549 (23:A3845)
  • [Ke] J. Keating, Periodic orbits, spectral statistics, and the Riemann zeros, Supersymmetry and trace formulae (I. Lerner, J. Keating and D. Khmelnitskii, eds.), Kluwer, New York, 1999, pp. 1-15.
  • [Ki] Y. Kifer, Large deviations, averaging and periodic orbits of dynamical systems, Comm. Math. Phys. 162 (1994), 33-46. MR 1272765 (95b:58091)
  • [La] S. Lalley, Self-intersections of closed geodesics on a negatively curved surface: statistical regularities, Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996, pp. 263-272. MR 1412610 (97h:58103)
  • [Ma] G. Margulis, Certain applications of ergodic theory to the study of manifolds of negative curvature, Functional Anal. Appl. 3 (1969), 89-90. MR 0257933 (41:2582)
  • [Ot] J.-P. Otal, Le théorème d'hyperbolisation pour less variétés fibreés de dimension $3$, Asterisque 235 (1996), 1-159. MR 1402300 (97e:57013)
  • [Po] M. Pollicott, Asymptotic distribution of closed geodesics, Israel J. Math. 52 (1985), 209-224. MR 0815810 (87g:58105)
  • [PS] M. Pollicott and R. Sharp, Exponential error terms for growth functions on negatively curved surfaces, Amer. J. Math. 120 (1998), 1019-1042. MR 1646052 (99h:58148)
  • [SR] M. Sieber and K. Richter, Correlations between periodic orbits and their rôle in spectral statistics, Physica Scripta T90 (2001), 128-133.

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Additional Information

Mark Pollicott
Affiliation: Department of Mathematics, Manchester University, Oxford Road, Manchester M13 9PL, United Kingdom
Address at time of publication: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Richard Sharp
Affiliation: Department of Mathematics, Manchester University, Oxford Road, Manchester M13 9PL, United Kingdom

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08382-6
PII: S 0002-9939(05)08382-6
Received by editor(s): October 15, 2003
Received by editor(s) in revised form: September 4, 2004
Posted: September 20, 2005
Additional Notes: The second author was supported by an EPSRC Advanced Research Fellowship
Communicated by: Michael Handel
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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