Skip to main content
SpringerLink
Log in

Marked Length Spectrum, Homoclinic Orbits and the Geometry of Open Dispersing Billiards

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

Abstract

We consider billiards obtained by removing three strictly convex obstacles satisfying the non-eclipse condition on the plane. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift on three symbols that provides a natural labeling of all periodic orbits. We study the following inverse problem: does the Marked Length Spectrum (i.e., the set of lengths of periodic orbits together with their labeling), determine the geometry of the billiard table? We show that from the Marked Length Spectrum it is possible to recover the curvature at periodic points of period two, as well as the Lyapunov exponent of each periodic orbit.

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
Fig. 3

Similar content being viewed by others

Notes

  1. For brevity, in most of the exposition we restrict ourselves to the case \(m=3\), yet, all of the results apply to arbitrary \(m\ge 3\).

  2. For more details about this fact, we refer the reader to [M] or to Section III in [GR]. In [GR], the authors consider the case where the obstacles are round discs, but the same construction can be carried out for strictly convex obstacles.

  3. Recall that for \(x=(s,\varphi )\in \mathcal {M}\) and \(x'=(s',\varphi '):=\mathcal {F}(s,\varphi )\), we have \(\det D_{x}\mathcal {F}=\frac{\cos \varphi }{\cos \varphi '}\). Thus, for any periodic orbit \((x_1,x_2,\dots ,x_p)\) of period \(p \ge 2\), we have \(\det D_{x_j}\mathcal {F}^p=1\), for \(j \in \{1,\dots ,p\}\).

  4. As we will see more in detail later, the right hand side of each of the following estimates is negative, because periodic orbits are minimizers of the length functional.

  5. Even for \(p=3\) we have 7 vs 2 free parameters!

  6. Note that the p-length of a dispersing wavefront is uniformly bounded by the length of its trace on the scatterer.

  7. See Footnote 6.

  8. Due to the palindromic symmetry, as in Lemma 3.2, the angle at this point has to vanish.

  9. i.e., such that \(\sigma _{j}\ne \sigma _{j+1}\) for \(j\in \{1,\dots ,p-1\}\) and such that \(\sigma _1 \ne \sigma _p\).

References

  1. Andersson, K.G., Melrose, R.B.: The propagation of singularities along gliding rays. Invent. Math. 41(3), 197–232 (1977)

    Google Scholar 

  2. Bálint, P., De Simoi, J., Kaloshin, V., Leguil, M.: Marked length spectrum and the geometry of open dispersing billiards II, in preparation

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

    Google Scholar 

  4. de Verdière, Y. Colin.: Sur les longueurs des trajectoires périodiques d’un billard, in P. Dazord and N. Desolneux-Moulis (eds.), Géométrie Symplectique et de Contact : Autour du Théorème de Poincaré-Birkhoff, Travaux en Cours, Séminaire Sud-Rhodanien de Géométrie III, Herman, pp. 122–139 (1984)

  5. Croke, C.B.: Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv. 65(1), 150–169 (1990)

    Google Scholar 

  6. Croke, C.B., Sharafutdinov, V.A.: Spectral rigidity of a compact negatively curved manifold. Topology 37(6), 1265–1273 (1998)

    Google Scholar 

  7. De Simoi, J., Kaloshin, V., Leguil, M.: Marked Length Spectral determination of analytic chaotic billiards with axial symmetries, in preparation

  8. De Simoi, J., Kaloshin, V., Wei, Q. (with an appendix co-authored with H. Hezari): Dynamical spectral rigidity among \({\mathbb{Z}}_{2}\)-symmetric strictly convex domains close to a circle. Annals of Mathematics 186, 277–314 (2017)

  9. Duchin, M., Erlandsson, V., Leininger, C. J., Sadanand, C.: You can hear the shape of a billiard table: symbolic dynamics and rigidity for flat surfaces, preprint arXiv

  10. Gaspard, P., Rice, S.A.: Scattering from a classically chaotic repellor. J. Chem. Phys. 90, 2225 (1989)

    Google Scholar 

  11. Guillarmou, C., Lefeuvre, T.: The marked length spectrum of Anosov manifolds, arXiv preprint. arXiv:1806.04218

  12. Hezari, H., Zelditch, S.: \(C^\infty \) spectral rigidity of the ellipse. Anal. PDE 5(5), 1105–1132 (2012)

    Google Scholar 

  13. Hezari, H., Zelditch, S.: Inverse spectral problem for analytic \((\mathbb{Z}/2\mathbb{Z})\) symmetric domains in \(\mathbb{R}^n\). Geom. Funct. Anal. 20(1), 160–191 (2010)

    Google Scholar 

  14. Huang, G., Kaloshin, V., Sorrentino, A.: On the marked length spectrum of generic strictly convex billiard tables. Duke Math. J. 167(1), 175–209 (2018)

    Google Scholar 

  15. Kac, M.: Can one hear the shape of a drum? Am. Math. Mon. 73 (4P2), 1–23 (1966)

    Google Scholar 

  16. Morita, T.: The symbolic representation of billiards without boundary condition. Trans. Am. Math. Soc. 325, 819–828 (1991)

    Google Scholar 

  17. Otal, J.-P.: Le spectre marqué des longueurs des surfaces à courbure négative (French) [The marked spectrum of the lengths of surfaces with negative curvature]. Ann. Math. (2) 131(1), 151162 (1990)

    Google Scholar 

  18. Petkov, V.M., Stoyanov, L.N.: Singularities of the scattering kernel and scattering invariants for several strictly convex obstacles. Trans. Am. Math. Soc. 312(1), 203–235 (1989)

    Google Scholar 

  19. Petkov, V.M., Stoyanov, L.N.: Geometry of the Generalized Geodesic Flow and Inverse Spectral Problems, 2nd edn. Wiley, Chichester (2017)

    Google Scholar 

  20. Stoyanov, L.: A sharp asymptotic for the lengths of certain scattering rays in the exterior of two convex domains. Asymptot. Anal. 35(3,4), 235–255 (2003)

    Google Scholar 

  21. Zelditch, S.: Spectral determination of analytic bi-axisymmetric plane domains. Geom. Funct. Anal. 10(3), 628–677 (2000)

    Google Scholar 

  22. Zelditch, S.: Inverse spectral problem for analytic domains, I. Balian-Bloch trace formula. Commun. Math. Phys. 248(2), 357–407 (2004)

    Google Scholar 

  23. Zelditch, S.: Inverse spectral problem for analytic domains II: domains with one symmetry. Ann. Math. (2) 170(1), 205–269 (2009)

    Google Scholar 

  24. Zelditch, S.: Inverse resonance problem for \(\mathbb{Z}_2\)-symmetric analytic obstacles in the plane. In: Geometric Methods in Inverse Problems and PDE Control, volume 137 of IMA Vol. Math. Appl., pp. 289–321. Springer, New York, NY (2004)

  25. Zworski, M.: A remark on inverse problems for resonances. Inverse Probl. Imaging 1(1), 225–227 (2007)

    Google Scholar 

Download references

Acknowledgements

The authors wish to thank the hospitality of the ETH Institute for Theoretical Studies Zürich and the support of Dr. Max Rssler, the Walter Haefner Foundation and the ETH Zurich Foundation, as well as the Banff International Research Station – where part of this work was carried over. The authors are also indebted to the anonymous referees, to L. Stoyanov and M. Zworski for their most useful comments and suggestions. M.L. is grateful to L. Backes, A. Brown, S. Crovisier, F. Rodriguez-Hertz, D. Obata, A. Wilkinson and D. Xu for useful conversations during visits at the Pennsylvania State University, the University of Chicago, and the Université Paris-Sud.

Author information

Authors and Affiliations

  1. MTA-BME Stochastics Research Group, Budapest University of Technology and Economics, Egry József u. 1, Budapest, 1111, Hungary

    Péter Bálint

  2. Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Egry József u. 1, Budapest, 1111, Hungary

    Péter Bálint

  3. Department of Mathematics, University of Toronto, 40 St George St., Toronto, ON, M5S 2E4, Canada

    Jacopo De Simoi & Martin Leguil

  4. University of Maryland, College Park, MD, USA

    Vadim Kaloshin

Authors
  1. Péter Bálint
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacopo De Simoi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vadim Kaloshin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Martin Leguil
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Leguil.

Additional information

Communicated by C. Liverani

Publisher's Note

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

P.B. is supported in part by Hungarian National Foundation for Scientific Research (NKFIH OTKA) Grants K104745 and K123782. J.D.S. and M.L. are supported by the NSERC Discovery Grant, reference number 502617-2017. V.K. acknowledges partial support of the NSF Grant DMS-1402164.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bálint, P., De Simoi, J., Kaloshin, V. et al. Marked Length Spectrum, Homoclinic Orbits and the Geometry of Open Dispersing Billiards. Commun. Math. Phys. 374, 1531–1575 (2020). https://doi.org/10.1007/s00220-019-03448-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03448-x

Search

Navigation