Skip to main content
SpringerLink
Log in

Lyapunov Exponents and Nonadapted Measures for Dispersing Billiards

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

For hyperbolic systems with singularities, such as dispersing billiards, Pesin theory as developed by Katok and Strelcyn applies to measures that are “adapted” in the sense that they do not give too much weight to neighborhoods of the singularity set. The zero-entropy measures supported on grazing periodic orbits are nonadapted, but it has been an open question whether there are nonadapted measures with positive entropy. We construct such measures for any dispersing billiard with a periodic orbit having a single grazing collision; we then use our construction to show that the thermodynamic formalism for such billiards has a phase transition even when one restricts attention to adapted or to positive entropy measures.

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

Fig. 1
Fig. 2

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Notes

  1. See [H, p. 342] for the story of the genesis of this condition, or rather of the slightly stronger condition (a) that appears in [KSLP, P]. The term “adapted” was introduced in [LS], in the context of smooth flows equipped with an invariant measure, as a condition on a Poincaré section (requiring that the boundary \({\mathcal {S}}\) satisfies (1.1) with respect to the induced measure). It has subsequently been used as a condition on measures in systems with singularities [LM, BD1]. For planar dispersing billiards, it is equivalent to assuming the finiteness of the positive Lyapunov exponent almost everywhere (see the proof of Theorem 1.2).

  2. [BD1] constructs a unique measure of maximal entropy under a mild sparse recurrence condition to the singularity set. (If a conjecture of Balint and Toth holds, then this condition is satisfied generically for finite horizon tables [DK].) Yet even without assuming this condition, [BD1] proves that \(P(0) < \infty \). We do not make any assumption of sparse recurrence to singularities in the present paper.

  3. The period 4 is not essential to our construction, yet a second grazing collision along the periodic orbit could complicate the argument and indeed might destroy the horseshoe entirely.

  4. Indeed, since the slopes are \(-{\mathcal {K}}(x_0)\) and \({\mathcal {K}}(x_0)\), respectively, they form an angle equal to 2Arctan\((1/{\mathcal {K}}(x_0))\).

  5. Indeed, a slightly stronger property holds: for every \(x\in R_{k,n,i}\) its local unstable manifold \(W^u(x)\) properly crosses \(R^*\), a proper crossing being a full crossing whose distance from the unstable boundary of \(R^*\) is at least a fixed fraction of the unstable diameter of \(R^*\).

References

  1. Araujo, E., Poletti, M., Lima, Y.: Symbolic dynamics for nonuniformly hyperbolic maps with singularities in high dimension, to appear in Memoirs of the AMS. arXiv:2010.11808

  2. Bunimovich, L., Sinai, Ya. G., Chernov, N.: Markov partitions for two-dimensional hyperbolic billiards. Russ. Math. Surv. 45, 105–152 (1990)

    Article  MathSciNet  Google Scholar 

  3. Bunimovich, L., Sinai, Ya. G., Chernov, N.: Statistical properties of two-dimensional hyperbolic billiards. Russ. Math. Surv. 46, 47–106 (1991)

    Article  MathSciNet  Google Scholar 

  4. Chernov, N., Markarian, R.: Chaotic Billiards, Mathematical Surveys and Monographs, vol. 127. AMS, Providence (2006)

    Book  Google Scholar 

  5. Chernov, N., Troubetzkoy, S.: Measures with infinite Lyapunov exponents for the periodic Lorentz gas. J. Stat. Phys. 83, 193–202 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  6. Chen, J., Wang, F., Zhang, H.-K.: Improved Markov partition and thermodynamic formalism for hyperbolic systems with singularities. arXiv:1709.00527v2 (2019)

  7. Baladi, V., Demers, M.F.: On the measure of maximal entropy for finite horizon Sinai billiard maps. J. Am. Math. Soc. 33(2), 381–449 (2020)

    Article  MathSciNet  Google Scholar 

  8. Baladi, V., Demers, M.F.: Thermodynamic formalism for dispersing billiards. J. Mod. Dyn. 18, 841–923 (2022)

    Article  Google Scholar 

  9. Demers, M.F., Korepanov, A.: Rates of mixing for the measure of maximal entropy of dispersing billiard maps. Proc. London Math. Soc. 128(1), e12578 (2024)

    Article  MathSciNet  Google Scholar 

  10. Demers, M.F., Liverani, C.: Projective cones for sequential dispersing billiards. Commun. Math. Phys. 401(1), 841–923 (2023)

    Article  ADS  MathSciNet  Google Scholar 

  11. Demers, M.F., Zhang, H.-K.: Spectral analysis of the transfer operator for the Lorentz Gas. J. Mod. Dyn. 5(4), 665–709 (2011)

    Article  MathSciNet  Google Scholar 

  12. Hasselblatt, B.: Anatole Katok. Ergod. Theory Dyn. Syst. 42(2), 321–388 (2022)

    Article  MathSciNet  Google Scholar 

  13. Katok, A., Strelcyn, J.-M., Ledrappier, F., Przytycki, F.: Invariant manifolds, entropy and billiards; smooth maps with singularities. In: Lecture Notes in Mathematics, vol. 1222. Springer, Berlin (1986)

  14. Lima, Y., Matheus, C.: Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities. Ann. Sci. Éc. Norm Supér (4) 51(1), 1–38 (2018)

    Article  MathSciNet  Google Scholar 

  15. Lima, Y., Sarig, O.: Symbolic dynamics for three-dimensional flows with positive topological entropy. J. Eur. Math. Soc. 21(1), 199–256 (2019)

    Article  MathSciNet  Google Scholar 

  16. Pesin, Ya.: Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Ergod. Theory Dyn. Syst. 12(1), 123–151 (1992)

    Article  MathSciNet  Google Scholar 

  17. Sarig, O.: Symbolic dynamics for surface diffeomorphisms with positive entropy. J. Am. Math. Soc. 26(2), 341–426 (2013)

    Article  MathSciNet  Google Scholar 

  18. Sinai, Ya. G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surv. 25, 137–189 (1970)

    Article  MathSciNet  Google Scholar 

  19. Young, L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. Math. (2) 147(3), 585–650 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work started as one of the projects proposed in the workshop Equilibrium states for dynamical systems arising from geometry, held at the American Institute of Mathematics in July 2019. The authors are grateful to AIM for its hospitality. VC was partially supported by NSF Grants DMS-1554794 and DMS-2154378, and by a Simons Foundation Fellowship. MD is partially supported by NSF Grant DMS-2055070. YL was supported by CNPq and Instituto Serrapilheira, Grant “Jangada Dinâmica: Impulsionando Sistemas Dinâmicos na Região Nordeste”. HZ is partially supported by NSF Grant DMS-2220211 and Simons Foundation 706383.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77204, USA

    Vaughn Climenhaga

  2. Department of Mathematics, Fairfield University, Fairfield, CT, 06824, USA

    Mark F. Demers

  3. Departamento de Matemática, Centro de Ciências, Campus do Pici, Universidade Federal do Ceará (UFC), Fortaleza, CE, CEP 60455-760, Brazil

    Yuri Lima

  4. Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA, USA

    Hongkun Zhang

Authors
  1. Vaughn Climenhaga
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark F. Demers
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuri Lima
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hongkun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark F. Demers.

Ethics declarations

Conflict of interest

This work was begun during a workshop held at the American Institute of Mathematics in July 2019. V. Climenhaga was partially supported by National Science Foundation (NSF) Grants DMS-1554794 and DMS-2154378, and by a Simons Foundation Fellowship. M. Demers is partially supported by NSF Grant DMS-2055070. Y. Lima was supported by CNPq and Instituto Serrapilheira, Grant “Jangada Dinâmica: Impulsionando Sistemas Dinâmicos na Região Nordeste”. H.-K. Zhang is partially supported by NSF Grant DMS-2220211 and Simons Foundation Grant 706383.

Data sharing

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Additional information

Communicated by C. Liverani.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Climenhaga, V., Demers, M.F., Lima, Y. et al. Lyapunov Exponents and Nonadapted Measures for Dispersing Billiards. Commun. Math. Phys. 405, 24 (2024). https://doi.org/10.1007/s00220-023-04921-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00220-023-04921-4

Search

Navigation