Advertisement

The Fried conjecture in small dimensions

  • Nguyen Viet DangEmail author
  • Colin Guillarmou
  • Gabriel Rivière
  • Shu Shen
Article
  • 99 Downloads

Abstract

We study the twisted Ruelle zeta function \(\zeta _X(s)\) for smooth Anosov vector fields X acting on flat vector bundles over smooth compact manifolds. In dimension 3, we prove the Fried conjecture, relating Reidemeister torsion and \(\zeta _X(0)\). In higher dimensions, we show more generally that \(\zeta _X(0)\) is locally constant with respect to the vector field X under a spectral condition. As a consequence, we also show the Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic 3-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flows in variable negative curvature where the Fried conjecture holds true.

Notes

Acknowledgements

We would like to thank V. Baladi for pointing out to us the papers [62, 63] which are used in the last part of Theorem 1. We also would like to thank Y. Bonthonneau, N.T. Dang, P. Dehornoy, F. Faure, S. Gouëzel, B. Hasselblatt, B. Kuester, F. Naud, H. H. Rugh, H. Sanchez-Morgado, T. Weich for discussions, answers to our questions and crucial remarks on this project. We also thank the referees for their detailed and useful comments that helped us to improve the presentation of our proofs. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 725967). CG and GR were partially supported by the ANR project GERASIC (ANR-13-BS01-0007-01) and GR also acknowledges the support of the Labex CEMPI (ANR-11-LABX-0007-01).

References

  1. 1.
    Baladi, V.: Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps. A Functional Approach. Springer, Ergebnisse (2018)zbMATHCrossRefGoogle Scholar
  2. 2.
    Baladi, V., Tsujii, M.: Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier (Grenoble) 57, 127–154 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Baladi, V., Tsujii, M.: Dynamical determinants and spectrum for hyperbolic diffeomorphisms, Contemp. Math. 469 (Amer. Math. Soc.), volume in honour of M. Brin’s 60th birthday, 29–68 (2008)Google Scholar
  4. 4.
    Bismut, J.M., Zhang, W.: An Extension of a Theorem of Cheeger and Müller, Astérisque 205. Société Math. de France, Paris (1992)Google Scholar
  5. 5.
    Blank, M., Keller, G., Liverani, C.: Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15, 1905–1973 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bunke, U., Olbrich, M.: Selberg Zeta and Theta Functions: A Differential Operator Approach. Mathematical Research, vol. 83. Akademie Verlag, Berlin (1995)Google Scholar
  7. 7.
    Butterley, O., Liverani, C.: Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1, 301–322 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cheeger, J.: Analytic torsion and the heat equation. Ann. Math. 109(2), 259–321 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cieliebak, K., Eliashberg, Y.: From Stein to Weinstein and back: symplectic geometry of affine complex manifolds, vol. 59. American Mathematical Society, Providence (2012) zbMATHGoogle Scholar
  10. 10.
    Dang, N.V., Rivière, G.: Topology of Pollicott–Ruelle resonant states. Annali della Scuola normale di Pisa.  https://doi.org/10.2422/2036-2145.201804_010
  11. 11.
    de la Llave, R., Marco, J.M., Moriyon, R.: Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation. Ann. Math. 123(3), 537–611 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    de Rham, G.: Sur les nouveaux invariants de M. Reidemeister. Math. Sb. 1, 737–743 (1936)zbMATHGoogle Scholar
  13. 13.
    Dyatlov, S., Faure, F., Guillarmou, C.: Power spectrum of the geodesic flow on hyperbolic manifolds. Anal. PDE 8, 923–1000 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dyatlov, S., Guillarmou, C.: Pollicott–Ruelle resonances for open systems. Ann. Henri Poincaré 17, 3089–3146 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Dyatlov, S., Guillarmou, C.: Dynamical zeta functions for Axiom A flows. Bull. AMS 55, 337–342 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Dyatlov, S., Zworski, M.: Dynamical zeta functions for Anosov flows via microlocal analysis. Ann. Sci. ENS 49, 543–577 (2016)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Dyatlov, S., Zworski, M.: Ruelle zeta function at zero for surfaces. Invent. Math. 210, 211–229 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Dyatlov, S., Zworski, M.: Mathematical Theory of Scattering Resonances. Graduate Studies in Mathematics, vol. 200. American Mathematical Society, Providence (2019)zbMATHGoogle Scholar
  19. 19.
    Faure, F., Roy, N., Sjöstrand, J.: Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances. Open Math. J. 1, 35–81 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Faure, F., Tsujii, M.: The semiclassical zeta function for geodesic flows on negatively curved manifolds. Invent. Math. 208, 851–998 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Faure, F., Sjöstrand, J.: Upper bound on the density of Ruelle resonances for Anosov flows. Commun. Math. Phys. 308, 325–364 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Fedosova, K., Rowlett, J., Zhang, G.: Second variation of Selberg zeta functions and curvature asymptotics, preprint arXiv: 1709.03841 (2017)
  23. 23.
    Franz, W.: Uber die Torsion einer Uberdeckung. J. Reine Angew. Math. 173, 245–254 (1935)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Fried, D.: Homological identities for closed orbits. Invent. Math. 71, 419–442 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Fried, D.: Analytic torsion and closed geodesics on hyperbolic manifolds. Invent. Math. 84, 523–540 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Fried, D.: The zeta functions of Ruelle and Selberg. I. Ann. l’ENS 19(4), 491–517 (1986)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Fried, D.: Lefschetz formulas for flows. Contemp. Math. 58(Part III), 19–69 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Fried, D.: Meromorphic zeta functions for analytic flows. Commun. Math. Phys. 174, 161–190 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Friedl, S., Nagel, M.: 3-manifolds that can be made acyclic. IMRN 2015, 13360–13378 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Giulietti, P., Liverani, C., Pollicott, M.: Anosov flows and dynamical zeta functions. Ann. Math. 178(2), 687–773 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Goldman, William M., Millson, John J.: The deformation theory of representations of fundamental groups of compact Kähler manifolds. Publ. Math. l’IHÉS 67, 43–96 (1988)zbMATHCrossRefGoogle Scholar
  32. 32.
    Gouëzel, S., Liverani, C.: Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. J. Diff. Geom. 79, 433–477 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Gouëzel, S.: Spectre du flot géodésique en courbure négative [d’après F. Faure et M. Tsujii], Séminaire Bourbaki (2015)Google Scholar
  34. 34.
    Guedes Bonthonneau, Y.: Flow-independent Anisotropic space, and perturbation of resonances, preprint arXiv:1806.08125 (2018)
  35. 35.
    Guedes Bonthonneau, Y., Weich, T.: Ruelle resonances for manifolds with hyperbolic cusps, preprint arXiv:1712.07832 (2017)
  36. 36.
    Guillarmou, C., Knieper, G., Lefeuvre, T.: Geodesic stretch and marked length spectrum rigidity, preprint arXivGoogle Scholar
  37. 37.
    Guillemin, V., Sternberg, S.: Geometric Asymptotics, vol. 14. American Mathematical Society, Providence (1990)zbMATHGoogle Scholar
  38. 38.
    Hadfield, C.: Resonances for symmetric tensors on asymptotically hyperbolic spaces. Anal. PDE 10(8), 1877–1922 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Hadfield, C.: Zeta function at zero for surfaces with boundary, preprint arXiv:1803.10982 (2018)
  40. 40.
    Heil, K., Moroianu, A., Semmelmann, U.: Killing and conformal Killing tensors. J. Geom. Phys. 106, 383–400 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Hirsch, M.W.: Differential Topology, vol. 33. Springer, Berlin (2012)Google Scholar
  42. 42.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis. Springer, Berlin (1983)zbMATHGoogle Scholar
  43. 43.
    Juhl, A.: Cohomological Theory of Dynamical Zeta Functions. Progress in Mathematics. Birkhäuser, Basel (2012)Google Scholar
  44. 44.
    Kitaev, A.Y.: Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness. Nonlinearity 12, 141–179 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Krantz, S.G., Parks, H.R.: A Primer of Real Analytic Functions. Springer, Berlin (2002)zbMATHCrossRefGoogle Scholar
  46. 46.
    Küster, B., Weich, T.: Quantum-classical correspondence on associated vector bundles over locally symmetric spaces. arXiv:1710.04625
  47. 47.
    Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. D. Reidel Publishing Company, Dordrecht (1987)zbMATHCrossRefGoogle Scholar
  48. 48.
    Lin, Y.: Lefschetz contact manifolds and odd dimensional symplectic manifolds. arXiv:1311.1431
  49. 49.
    Liverani, C.: On contact Anosov flows. Ann. Math. 159(3), 1275–1312 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Liverani, C.: Fredholm determinants, Anosov maps and Ruelle resonances. DCDS 13, 1203–1215 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Margulis, G.A.: On Some Aspects of the Theory of Anosov Systems. Springer, Berlin (2004)zbMATHCrossRefGoogle Scholar
  52. 52.
    Mnev, P.: Lecture notes on torsion, preprint arXiv:1406.3705 (2014)
  53. 53.
    Moscovici, H., Stanton, R.: R-torsion and zeta functions for locally symmetric manifolds. Invent. Math. 105(1), 185–216 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Müller, W.: Analytic torsion and R-torsion of Riemannian manifolds. Adv. Math. 28(3), 233–305 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Müller, W.: Analytic torsion and R-torsion for unimodular representations. J. Am. Math. Soc. 6(3), 721–753 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Plante, J.F.: Homology of closed orbits of Anosov flows. Proc. Am. Math. Soc. 37, 297–300 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Ratner, M.: Markov splitting for U-flows in three dimensional manifolds. Math. Notes Acad. Sci. USSR 6, 880–886 (1969)zbMATHGoogle Scholar
  58. 58.
    Ray, D.B., Singer, I.M.: R-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145–210 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Reidemeister, K.: Homotopieringe und Linsenräume. Abh. Math. Sem. Univ. Hambg. 11, 102–109 (1935)MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Ruelle, D.: Zeta functions for expanding maps and Anosov flows. Invent. Math. 34, 231–242 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Rugh, H.H.: Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems. Ergod. Theory Dyn. Syst. 16(4), 805–819 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Sanchez-Morgado, H.: Lefschetz formulae for Anosov flows on 3-manifolds. Ergod. Theory Dyn. Syst. 13(2), 335–347 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Sanchez-Morgado, H.: R-torsion and zeta functions for analytic Anosov flows on 3-manifolds. Trans. AMS 348(3), 963–973 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Shen, S.: Analytic torsion, dynamical zeta functions, and the Fried conjecture. Anal. PDE 11(1), 1–74 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Tsujii, M.: Contact Anosov flows and the Fourier–Bros–Lagolnitzer transform. Ergod. Theory Dyn. Syst. 32(6), 2083–2118 (2012)zbMATHCrossRefGoogle Scholar
  66. 66.
    Vasy, A.: Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov). Invent. Math. 194, 381–513 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Whitney, H.: Differentiable manifolds. Ann. Math. 37, 645–680 (1936)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Yan, D.: Hodge structure on symplectic manifolds. Adv. Math. 120, 143–154 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Zworski, M.: Semiclassical Analysis. Graduate Studies in Mathematics, vol. 138. AMS, Providence (2012)CrossRefGoogle Scholar
  70. 70.
    Zworski, M.: Commentary on “Differentiable dynamical systems” by Stephen Smale. Bull. Am. Math. Soc. 55, 331–336 (2018)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Nguyen Viet Dang
    • 1
    Email author
  • Colin Guillarmou
    • 2
  • Gabriel Rivière
    • 3
  • Shu Shen
    • 4
  1. 1.Institut Camille Jordan (U.M.R. CNRS 5208)Université Claude Bernard Lyon 1Villeurbanne CedexFrance
  2. 2.Département de Mathématiques, (U.M.R. CNRS 8628)Université Paris-Sud, Université Paris-SaclayOrsayFrance
  3. 3.Laboratoire Paul Painlevé (U.M.R. CNRS 8524), Département de mathématiques, Faculté des sciences et technologiesUniversité de LilleVilleneuve d’Ascq CedexFrance
  4. 4.Institut de mathématiques de Jussieu-Paris Rive Gauche (U.M.R. CNRS 7586)Sorbonne UniversitéParis Cedex 05France

Personalised recommendations