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)

 
 

 

A local limit theorem for closed geodesics and homology


Author: Richard Sharp
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 4897-4908
MSC (2000): Primary 37C27, 37C30, 37D20, 37D40, 53C22
DOI: https://doi.org/10.1090/S0002-9947-04-03534-2
Published electronically: January 16, 2004
MathSciNet review: 2084404
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study the distribution of closed geodesics on a compact negatively curved manifold. We concentrate on geodesics lying in a prescribed homology class and, under certain conditions, obtain a local limit theorem to describe the asymptotic behaviour of the associated counting function as the homology class varies.


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

  • 1. N. Anantharaman, Precise counting results for closed orbits of Anosov flows, Ann. Sci. École Norm. Sup. 33 (2000), 33-56. MR 2002c:37048
  • 2. M. Babillot and F. Ledrappier, Lalley's theorem on periodic orbits of hyperbolic flows, Ergodic Theory Dyn. Syst. 18 (1998), 17-39. MR 99a:58128
  • 3. M. Babillot and M. Peigné, Homologie des géodésiques fermées sur des variétés hyperboliques avec bouts cuspidaux, Ann. Sci. École Norm. Sup. 33 (2000), 81-120. MR 2001b:37043
  • 4. D. Dolgopyat, On decay of correlations for Anosov flows, Annals of Math. 147 (1998), 357-390. MR 99g:58073
  • 5. D. Dolgopyat, Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dyn. Syst. 18 (1998), 1097-1114. MR 2000a:37014
  • 6. C. Epstein, Asymptotics for closed geodesics in a homology class, the finite volume case, Duke Math. J. 55 (1987), 717-757. MR 89c:58098
  • 7. A. Ingham, The Distribution of Prime Numbers, Cambridge University Press, Cambridge, 1990; Reprint of the 1932 original. MR 91f:11064
  • 8. A. Katsuda, Density theorems for closed orbits, Geometry and analysis on manifolds (T. Sunada, ed.), Lecture Notes in Mathematics 1339, Springer, Berlin, 1988, pp. 182-202. MR 89j:58066
  • 9. A. Katsuda and T. Sunada, Homology and closed geodesics in a compact Riemann surface, Amer. J. Math. 110 (1988), 145-155. MR 89e:58093
  • 10. A. Katsuda and T. Sunada, Closed orbits in homology classes, Inst. Hautes Études Sci. Publ. Math. 71 (1990), 5-32. MR 92m:58102
  • 11. M. Kotani, A note on asymptotic expansions for closed geodesics in homology classes, Math. Ann. 320 (2001), 507-529. MR 2002h:58044
  • 12. S. Lalley, Closed geodesics in homology classes on surfaces of variable negative curvature, Duke Math. J. 58 (1989), 795-821. MR 91a:58143
  • 13. W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188 (1990), 1-268. MR 92f:58141
  • 14. R. Phillips and P. Sarnak, Geodesics in homology classes, Duke Math. J. 55 (1987), 287-297. MR 88g:58151
  • 15. M. Pollicott, Homology and closed geodesics in a compact negatively curved surface, Amer. J. Math. 113 (1991), 379-385. MR 92e:58158
  • 16. M. Pollicott and R. Sharp, Exponential error terms for growth functions on negatively curved surfaces, Amer. J. Math. 120 (1998), 1019-1042. MR 99h:58148
  • 17. M. Pollicott and R. Sharp, Asymptotic expansions for closed orbits in homology classes, Geom. Dedicata 87 (2001), 123-160. MR 2003b:37051
  • 18. J. Rousseau-Egele, Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. Probab. 11 (1983), 772-788. MR 84m:60032
  • 19. R. Sharp, Closed orbits in homology classes for Anosov flows, Ergodic Theory Dyn. Syst. 13 (1993), 387-408. MR 94g:58169
  • 20. G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, Cambridge, 1995. MR 97e:11005b

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37C27, 37C30, 37D20, 37D40, 53C22

Retrieve articles in all journals with MSC (2000): 37C27, 37C30, 37D20, 37D40, 53C22


Additional Information

Richard Sharp
Affiliation: Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
Email: sharp@maths.man.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-04-03534-2
Received by editor(s): March 28, 2003
Received by editor(s) in revised form: June 6, 2003
Published electronically: January 16, 2004
Additional Notes: The author was supported by an EPSRC Advanced Research Fellowship
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society