Duke Mathematical Journal

On the marked length spectrum of generic strictly convex billiard tables

Guan Huang, Vadim Kaloshin, and Alfonso Sorrentino

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper we show that for a generic strictly convex domain, one can recover the eigendata corresponding to Aubry–Mather periodic orbits of the induced billiard map from the (maximal) marked length spectrum of the domain.

Article information

Duke Math. J., Volume 167, Number 1 (2018), 175-209.

Received: 17 June 2016
Revised: 23 June 2017
First available in Project Euclid: 8 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory
Secondary: 37D50: Hyperbolic systems with singularities (billiards, etc.) 37E40: Twist maps 37J50: Action-minimizing orbits and measures

marked length spectrum and length spectrum Laplace spectrum convex billiards Mather beta function action of periodic orbits


Huang, Guan; Kaloshin, Vadim; Sorrentino, Alfonso. On the marked length spectrum of generic strictly convex billiard tables. Duke Math. J. 167 (2018), no. 1, 175--209. doi:10.1215/00127094-2017-0038. https://projecteuclid.org/euclid.dmj/1512702098

Export citation


  • [1] K. G. Andersson and R. Melrose,The propagation of singularities along gliding rays, Invent. Math.4(1977), 23–95.
  • [2] M.-C. Arnaud and P. Berger,The non-hyperbolicity of irrational invariant curves for twist maps and all that follows, Rev. Mat. Iberoam.32(2016), 1295–1310.
  • [3] V. Bangert,Mather Sets for Twist Maps and Geodesics on Tori, Dynam. Report. Ser. Dynam. Systems Appl.1, Wiley, Chichester, 1988, 1–56.
  • [4] G. D. Birkhoff,On the periodic motions of dynamical systems, Acta Math.50(1927), 359–379.
  • [5] J. De Simoi, V. Kaloshin, and Q. Wei,Dynamical spectral rigidity among $\mathbb{Z}_{2}$-symmetric strictly convex domains close to a circle, Ann. of Math. (2)186(2017), 277–314.
  • [6] M. J. Dias Carneiro, S. Oliffson Kamphorst, and S. Pinto-de-Carvalho,Periodic orbits of generic oval billiards, Nonlinearity20(2007), 2453–2462.
  • [7] C. Gordon, D. L. Webb, and S. Wolpert,One cannot hear the shape of a drum, Bull. Amer. Math. Soc.27(1992), 134–138.
  • [8] J. M. Greene,A method for determining a stochastic transition, J. Math. Phys.20(1978), 1183–1201.
  • [9] V. Guillemin and R. Melrose,A Cohomological Invariant of Discrete Dynamical Systems, E. B. Christoffel, Aachen/Monschau, 1979, 672–679; Birkhäuser, Basel, 1981.
  • [10] B. Halpern,Strange billiard tables, Trans. Amer. Math. Soc.232(1977), 297–305.
  • [11] H. Hezari and S. Zelditch,Inverse spectral problem for analytic $(\mathbb{Z}/2\mathbb{Z})^{n}$-symmetric domains in $\mathbb{R}^{n}$, Geom. Funct. Anal.20(2010), 160 –191.
  • [12] M. Kac,Can one hear the shape of a drum?, Amer. Math. Monthly73(4, Part 2) (1966), 1–23.
  • [13] V. F. Lazutkin,Existence of caustics for the billiard problem in a convex domain, (Russian), Izv. Akad. Nauk SSSR Ser. Mat.37(1973), 186–216.
  • [14] R. S. MacKay,Greene’s residue criterion, Nonlinearity5(1992), 161–187.
  • [15] J. N. Mather,Differentiability of the minimal average action as a function of the rotation number, Bol. Soc. Brasil. Mat. (N.S.)21(1990), 59–70.
  • [16] J. N. Mather and G. Forni, “Action minimizing orbits in Hamiltonian systems” inTransition to Chaos in Classical and Quantum Mechanics (Montecatini Terme, 1991), Lecture Notes in Math.1589, 1994, 92–186.
  • [17] J. Milnor,Eigenvalues of the Laplace operator on certain manifolds, Proc. Natl. Acad. Sci. USA15(1964), 275–280.
  • [18] G. Popov,Invariants of the length spectrum and spectral invariants of planar convex domains, Comm. Math. Phys.161(1994), 335–364.
  • [19] G. Popov and P. Topalov,From KAM tori to isospectral invariants and spectral rigidity of billiard tables, preprint,arXiv:1602.03155[math.SP].
  • [20] P. Sarnak, “Determinants of Laplacians; heights and finiteness” inAnalysis, et cetera, Academic Press, Boston, 1990, 601–622.
  • [21] K. F. Siburg,The Principle of Least Action in Geometry and Dynamics, Lecture Notes in Math.1844, Springer, New York, 2004.
  • [22] A. Sorrentino, “Action-minimizing methods in Hamiltonian dynamics” inAn Introduction to Aubry–Mather Theory, Math. Notes Ser.50, Princeton Univ. Press, Princeton, 2015.
  • [23] A. Sorrentino,Computing Mather’s beta-function for Birkhoff billiards, Discrete Contin. Dyn. Syst. Ser. A35(2015), 5055–5082.
  • [24] D. Stowe,Linearization in two dimensions, J. Differential Equations63(1986), 183–226.
  • [25] S. Tabachnikov,Geometry and Billiards, Stud. Math. Libr.30, Amer. Math. Soc., Providence, 2005.
  • [26] Z. Xia and P. Zhang,Homoclinic points for convex billiards, Nonlinearity27(2014), 1181–1192.
  • [27] S. Zelditch,Spectral determination of analytic bi-axisymmetric plane domains, Geom. Funct. Anal.10(2000), 628–677.
  • [28] W. Zhang and W. Zhang,Sharpness for $C^{1}$ linearization of planar hyperbolic diffeomorphisms, J. Differential Equations257(2014), 4470–4502.