Skip to main content
SpringerLink
Log in

On the density of Birkhoff sums for Anosov diffeomorphisms

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

Let f : M → M be an Anosov diffeomorphism on a nilmanifold. We consider Birkhoff sums for a Hölder continuous observation along periodic orbits. We show that if there are two Birkhoff sums distributed at both sides of zero, then the set of Birkhoff sums of all the periodic points is dense in ℝ.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdenur F, Bonatti C, Crovisier S. Nonuniform hyperbolicity for C1-generic diffeomorphisms. Israel J Math, 2011, 183: 1–60

    Article  MathSciNet  Google Scholar 

  2. Anosov D V. Geodesic flows on closed Riemannian manifolds of negative curvature. Tr Mat Inst Steklova, 1967, 90: 3–210

    MathSciNet  Google Scholar 

  3. Dekimpe K. What an infra-nilmanifold endomorphism really should be …. Topol Methods Nonlinear Anal, 2012, 40: 111–136

    MathSciNet  MATH  Google Scholar 

  4. Franks J. Anosov diffeomorphisms on tori. Trans Amer Math Soc, 1969, 145: 117–124

    Article  MathSciNet  Google Scholar 

  5. Franks J. Anosov diffeomorphisms. In: Global Analysis. Proceedings of Symposia in Pure Mathematics, vol. 14. Providence: Amer Math Soc, 1970, 61–93

    Chapter  Google Scholar 

  6. Gan S, Shi Y. Robustly topological mixing of Kan’s map. J Differential Equations, 2019, 266: 7173–7196

    Article  MathSciNet  Google Scholar 

  7. Hammerlindl A. Polynomial global product structure. Proc Amer Math Soc, 2014, 142: 4297–4303

    Article  MathSciNet  Google Scholar 

  8. Katok A. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ Math Inst Hautes Études Sci, 1980, 51: 137–173

    Article  MathSciNet  Google Scholar 

  9. Katok A, Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems. Cambridge: Cambridge University Press, 1995

    Book  Google Scholar 

  10. Katok A, Niţică V. Rigidity in Higher Rank Abelian Group Actions, Volume I. Cambridge: Cambridge University Press, 2011

    Book  Google Scholar 

  11. Livšic A N. Cohomology of dynamical systems. Math USSR Izv, 1972, 6: 1278–1301

    Article  Google Scholar 

  12. Manning A. Anosov diffeomorphisms on nilmanifolds. Proc Amer Math Soc, 1973, 38: 423–426

    Article  MathSciNet  Google Scholar 

  13. Manning A. There are no new Anosov diffeomorphisms on tori. Amer J Math, 1974, 96: 422–429

    Article  MathSciNet  Google Scholar 

  14. Pilyugin S Y. Shadowing in Dynamical Systems. Berlin-Heidelberg: Springer-Verlag, 1999

    MATH  Google Scholar 

  15. Raghunathan M S. Discrete Subgroups of Lie Groups. Berlin-Heidelberg: Springer-Verlag, 1972

    Book  Google Scholar 

  16. Sigmund K. Generic properties of invariant measures for Axiom A-diffeomorphisms. Invent Math, 1970, 11: 99–109

    Article  MathSciNet  Google Scholar 

  17. Smale S. Differentiable dynamical systems. Bull Amer Math Soc NS, 1967, 73: 747–817

    Article  MathSciNet  Google Scholar 

  18. Wang Z, Sun W. Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits. Trans Amer Math Soc, 2010, 362: 4267–4282

    Article  MathSciNet  Google Scholar 

  19. Zhou Y, Sun W. The Lyapunov exponents of C1 hyperbolic systems. Sci China Math, 2010, 53: 1743–1752

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant Nos. 11771025 and 11831001). The second author was supported by National Natural Science Foundation of China (Grant Nos. 12071007 and 11831001). The authors thank Jinpeng An for the valuable discussion and pointing out the reference [15] about the theory of Lie group.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Shaobo Gan, Yi Shi & Mingyang Xia

Authors
  1. Shaobo Gan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mingyang Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi Shi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gan, S., Shi, Y. & Xia, M. On the density of Birkhoff sums for Anosov diffeomorphisms. Sci. China Math. 65, 319–330 (2022). https://doi.org/10.1007/s11425-020-1858-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-020-1858-9

Keywords

MSC(2020)

Search

Navigation