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Pairs of periodic orbits with fixed homology difference

  • Morten S. Risager (a1) and Richard Sharp (a2)

We obtain an asymptotic formula for the number of pairs of closed orbits of a weak-mixing transitive Anosov flow whose homology classes have a fixed difference.

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1. Anantharaman, N., Counting geodesics which are optimal in homology, Ergod. Theory Dynam. Syst. 23 (2003), 353388.
2. Babillot, M. and Ledrappier, F., Lalley's theorem on periodic orbits of hyperbolic flows, Ergod. Theory Dynam. Syst. 18 (1998), 1739.
3. Collier, D. and Sharp, R., Directions and equidistribution in homology for periodic orbits, Ergod. Theory Dynam. Syst. 27 (2007), 405415.
4. Fried, D., The geometry of cross sections to flows, Topology 21 (1982), 353371.
5. Katsuda, A., Density theorems for closed orbits, in Geometry and analysis on manifolds (ed. Sunada, T.), Lecture Notes in Mathematics, Volume 1339, pp. 182202 (Springer, 1988).
6. Katsuda, A. and Sunada, T., Homology and closed geodesics in a compact Riemann surface, Am. J. Math. 110 (1988), 145156.
7. Katsuda, A. and Sunada, T., Closed orbits in homology classes, Publ. Math. IHES 71 (1990), 532.
8. Kifer, Y., Large deviations, averaging and periodic orbits of dynamical systems, Commun. Math. Phys. 162 (1994), 3346.
9. Lalley, S., Distribution of periodic orbits of symbolic and Axiom A flows, Adv. Appl. Math. 8 (1987), 154193.
10. Lalley, S., Closed geodesics in homology classes on surfaces of variable negative curvature, Duke Math. J. 58 (1989), 795821.
11. Margulis, G., On some applications of ergodic theory to the study of manifolds on negative curvature, Funct. Analysis Applic. 3 (1969), 8990.
12. Margulis, G., On some aspects of the theory of Anosov systems (with a survey by Richard Sharp: ‘Periodic orbits of hyperbolic flows’), Springer Monographs in Mathematics (Springer, 2004).
13. Massart, G., Stable norms of surfaces: local structure of the unit ball of rational directions, Geom. Funct. Analysis 7 (1997), 9961010.
14. Parry, W. and Pollicott, M., An analogue of the prime number theorem for closed orbits of Axiom A flows, Annals Math. 118 (1983), 573591.
15. Parry, W. and Pollicott, M., Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187–188 (1990), 1268.
16. Petridis, Y. and Risager, M., Equidistribution of geodesics on homology classes and analogues for free groups, Forum Math. 20 (2008), 783815.
17. Phillips, R. and Sarnak, P., Geodesics in homology classes, Duke Math. J. 55 (1987), 287297.
18. Pollicott, M., Homology and closed geodesics in a compact negatively curved surface, Am. J. Math. 113 (1991), 379385.
19. Risager, M., On pairs of prime geodesics with fixed homology difference, preprint (arXiv:math.NT/0604275; 2006).
20. Sharp, R., Closed orbits in homology classes for Anosov flows, Ergod. Theory Dynam. Syst. 13 (1993), 387408.
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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