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Chebotarev-type theorems in homology classes
Authors:
Mark Pollicott and Richard Sharp
Journal:
Proc. Amer. Math. Soc. 135 (2007), 3887-3894
MSC (2000):
Primary 37C27, 37C30, 37D40
Published electronically:
August 30, 2007
MathSciNet review:
2354151
Full-text PDF Free Access
Abstract |
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Additional Information
Abstract: We describe how closed geodesics lying in a prescribed homology class on a negatively curved manifold split when lifted to a finite cover. This generalizes a result of Zelditch in the case of compact hyperbolic surfaces.
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(86i:58104)
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Steven
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(89k:58236), http://dx.doi.org/10.1090/S0002-9939-1989-0946640-8
- 1.
- N. Anantharaman, Precise counting results for closed orbits of Anosov flows, Ann. Sci. École Norm. Sup. 33 (2000), 33-56. MR 1743718 (2002c:37048)
- 2.
- M. Babillot and F. Ledrappier, Lalley's theorem on periodic orbits of hyperbolic flows, Ergodic Theory Dynam. Systems 18 (1998), 17-39. MR 1609507 (99a:58128)
- 3.
- A. Katsuda, Density theorems for closed orbits, Geometry and analysis on manifolds (Katata-Kyoto, 1987), Lecture Notes in Math., 1339, Springer, Berlin, 1988, pp. 182-202. MR 961481 (89j:58066)
- 4.
- A. Katsuda and T. Sunada, Homology and closed geodesics in a compact Riemann surface, Amer. J. Math. 110 (1988), 145-155. MR 926741 (89e:58093)
- 5.
- A. Katsuda and T. Sunada, Closed orbits in homology classes, Inst. Hautes Études Sci. Publ. Math. 71 (1990), 5-32. MR 1079641 (92m:58102)
- 6.
- M. Kotani, A note on asymptotic expansions for closed geodesics in homology classes, Math. Ann. 320 (2001), 507-529. MR 1846775 (2002h:58044)
- 7.
- S. Lalley, Closed geodesics in homology classes on surfaces of variable negative curvature, Duke Math. J. 58 (1989), 795-821. MR 1016446 (91a:58143)
- 8.
- W. Parry and M. Pollicott, The Chebotarev theorem for Galois coverings of Axiom A flows, Ergodic Theory Dynam. Systems 6 (1986), 133-148. MR 837980 (88k:58124)
- 9.
- R. Phillips and P. Sarnak, Geodesics in homology classes, Duke Math. J. 55 (1987), 287-297. MR 894581 (88g:58151)
- 10.
- M. Pollicott, Homology and closed geodesics in a compact negatively curved surface, Amer. J. Math 113 (1991), 379-385. MR 1109342 (92e:58158)
- 11.
- M. Pollicott and R. Sharp, Asymptotic expansions for closed orbits in homology classes, Geom. Ded. 87 (2001), 123-160. MR 1866845 (2003b:37051)
- 12.
- R. Sharp, Closed orbits in homology classes for Anosov flows, Ergodic Theory Dynam. Systems 13 (1993), 387-408. MR 1235480 (94g:58169)
- 13.
- T. Sunada, Geodesic flows and geodesic random walks, Geometry of geodesics and related topics (Tokyo, 1982), Adv. Stud. Pure Math., 3, North-Holland, Amsterdam, 1984, pp. 47-85. MR 758647 (86i:58104)
- 14.
- S. Zelditch, Splitting of geodesics in homology classes, Proc. Amer. Math. Soc. 105 (1989), 1015-1019. MR 946640 (89k:58236)
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Additional Information
Mark Pollicott
Affiliation:
Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email:
mpollic@maths.warwick.ac.uk
Richard Sharp
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
Email:
sharp@maths.man.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08923-X
PII:
S 0002-9939(07)08923-X
Received by editor(s):
August 16, 2006
Received by editor(s) in revised form:
September 1, 2006
Published electronically:
August 30, 2007
Communicated by:
Jane M. Hawkins
Article copyright:
© Copyright 2007
American Mathematical Society
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