Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T08:58:26.330Z Has data issue: false hasContentIssue false
  • English
  • Français

Inverse problems and rigidity questions in billiard dynamics

Part of: Partial differential equations on manifolds; differential operators Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Low-dimensional dynamical systems

Published online by Cambridge University Press:  31 May 2021

VADIM KALOSHIN Open the ORCID record for VADIM KALOSHIN [Opens in a new window]  and
ALFONSO SORRENTINO Open the ORCID record for ALFONSO SORRENTINO [Opens in a new window]
VADIM KALOSHIN*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD, USA
ALFONSO SORRENTINO
Affiliation:
Dipartimento di Matematica, Università degli Studi di Roma ‘Tor Vergata’, Rome, Italy (e-mail: sorrentino@mat.uniroma2.it)
*
e-mail: vadim.kaloshin@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

A Birkhoff billiard is a system describing the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain: while it is evident how the shape determines the dynamics, a more subtle and difficult question is the extent to which the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing inverse problems and unanswered rigidity questions, which have been the focus of very active research in recent decades. In this paper we describe some of these questions, along with their connection to other problems in analysis and geometry, with particular emphasis on recent results obtained by the authors and their collaborators.

Keywords

mathematical billiardsintegrable billiardsBirkhoff conjecturespectral rigiditylength spectrum

MSC classification

Secondary: 37E40: Twist maps 37J35: Completely integrable systems, topological structure of phase space, integration methods 58J50: Spectral problems; spectral geometry; scattering theory
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 3: Anatole Katok Memorial Issue Part 2: Special Issue of Ergodic Theory and Dynamical Systems , March 2022 , pp. 1023 - 1056
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

To the memory of Anatole Katok (1944–2018)

References

Akhiezer, N. I.. Elements of the Theory of Elliptic Functions (Translations of Mathematical Monographs, 79). American Mathematical Society, Providence, RI, 1990.CrossRefGoogle Scholar
Andersson, K. G. and Melrose, R. B.. The propagation of singularities along gliding rays. Invent. Math. 41(3) (1977), 197–232.CrossRefGoogle Scholar
Arnaud, M.-C.. Fibrés de Green et régularité des graphes ${C}^0$ -Lagrangiens invariants par un flot de Tonelli. Ann. Henri Poincaré 9(5) (2008), 881–926.CrossRefGoogle Scholar
Arnold, M. and Bialy, M.. Nonsmooth convex caustics for Birkhoff billiards. Pacific J. Math. 295(2) (2018), 257–269.CrossRefGoogle Scholar
Aubry, S.. The twist map, the extended Frenkel–Kontorova model and the devil’s staircase. Phys. D 7(1–3) (1983), 240–258.CrossRefGoogle Scholar
Aubry, S.. The discrete Frenkel–Kontorova model and its extensions: I. Exact results for the ground-states. Phys. D 8(3) (1983), 381–422.CrossRefGoogle Scholar
Avila, A., De Simoi, J. and Kaloshin, V.. An integrable deformation of an ellipse of small eccentricity is an ellipse. Ann. of Math. (2) 184 (2016), 527–558.CrossRefGoogle Scholar
Baryshnikov, Y. and Zharnitsky, V.. Billiards and nonholonomic distributions. J. Math. Sci. 128 (2005), 2706–2710.CrossRefGoogle Scholar
Berger, M.. Seules les quadriques admettent des caustiques. Bull. Soc. Math. France 123(1) (1995), 107–116.CrossRefGoogle Scholar
Bernard, P.. Symplectic aspects of Mather theory. Duke Math. J. 136(3) (2007), 401–420.CrossRefGoogle Scholar
Besse, A. L.. Manifolds All of Whose Geodesics Are Closed (Ergebnisse der Mathematik und ihrer Grenzgebiete, 93). Springer, Berlin, 1978.CrossRefGoogle Scholar
Bialy, M.. Convex billiards and a theorem by E. Hopf. Math. Z. 124(1) (1993), 147–154.CrossRefGoogle Scholar
Bialy, M. Hopf rigidity for convex billiards on the hemisphere and hyperbolic plane. Discrete Contin. Dyn. Syst. 33(9) (2013), 3903–3913.CrossRefGoogle Scholar
Bialy, M. and Mironov, A.. Cubic and quartic integrals for geodesic flow on 2-torus via a system of the hydrodynamic type. Nonlinearity 24(12) (2011), 3541–3554.CrossRefGoogle Scholar
Bialy, M. and Mironov, A.. Rich quasi-linear system for integrable geodesic flows on $2$ -torus. Discrete Contin. Dyn. Syst. 29(1) (2011), 81–90.CrossRefGoogle Scholar
Bialy, M. and Mironov, A.. Angular billiard and algebraic Birkhoff conjecture. Adv. Math. 313 (2017), 102–126.CrossRefGoogle Scholar
Bialy, M. and Polterovich, L.. Geodesic flows on the two-dimensional torus and phase transitions ‘commensurability–incommensurability’. Funktsional. Anal. i Prilozhen. 20(4) (1986), 9–16.Google Scholar
Birkhoff, G. D.. On the periodic motions of dynamical systems. Acta Math. 50(1) (1927), 359–379.CrossRefGoogle Scholar
Birkhoff, G. D.. Collected Mathematical Papers. Vol. II. American Mathematical Society, Providence, RI, 1950.Google Scholar
Bolotin, S.. Integrable Birkhoff billiards. Mosc. Univ. Mech. Bull. 45(2) (1990), 10–13.Google Scholar
Bolsinov, A. V., Fomenko, A. T. and Matveev, V. S.. Two-dimensional Riemannian metrics with an integrable geodesic flow. Local and global geometries. Mat. Sb. 189(10) (1998), 5–32. Engl. Transl. Sb. Math. 189(9–10) (1998), 1441–1466.Google Scholar
Burago, D. and Ivanov, S.. Riemannian tori without conjugate points are flat. Geom. Funct. Anal. 4(3) (1994), 259–269.CrossRefGoogle Scholar
Chang, S.-J. and Friedberg, R.. Elliptical billiards and Poncelet’s theorem. J. Math. Phys. 29 (1988), 1537–1550.CrossRefGoogle Scholar
Chen, J., Kaloshin, V. and Zhang, H.-K.. Length spectrum rigidity for piecewise analytic Bunimovich billiards. Preprint, 2020, arXiv:1902.07330.Google Scholar
Croke, C. B.. Rigidity for surfaces of nonpositive curvature. Comment. Math. Helv. 65(1) (1990), 150–169.CrossRefGoogle Scholar
Croke, C. B. and Sharafutdinov, V. A.. Spectral rigidity of a compact negatively curved manifold. Topology 37(6) (1998), 1265–1273.CrossRefGoogle Scholar
Damasceno, J., Dias Carneiro, M. J. and Ramírez-Ros, R.. The billiard inside an ellipse deformed by the curvature flow. Proc. Amer. Math. Soc. 145 (2017), 705–719.CrossRefGoogle Scholar
de la Llave, R., Marco, J. M. and Moriyón, R.. Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation. Ann. of Math. (2) 123(3) (1986), 537–611.CrossRefGoogle Scholar
De Simoi, J., Kaloshin, V. and Leguil, M.. Marked length spectral determination of analytic chaotic billiards with axial symmetries. Preprint, 2019, arXiv:1905.00890.Google Scholar
De Simoi, J., Kaloshin, V. and Wei, Q.. Deformational spectral rigidity among ${Z}_2$ -symmetric domains close to the circle (Appendix B coauthored with H. Hezari). Ann. of Math. (2) 186 (2017), 277–314.CrossRefGoogle Scholar
Delshams, A. and Ramírez-Ros, R.. Poincaré–Melnikov–Arnold method for analytic planar maps. Nonlinearity 9(1) (1996), 1–26.CrossRefGoogle Scholar
Duchin, M., Erlandsson, V., Leininger, C. J. and Sadanand, C.. You can hear the shape of a billiard table: symbolic dynamics and rigidity for flat surfaces. Preprint, 2019, arXiv:1804.05690.Google Scholar
Forni, G. and Mather, J. N.. Action minimizing orbits in Hamiltonian systems. Transition to Chaos in Classical and Quantum Mechanics (Montecatini Terme, 1991) (Lecture Notes in Mathematics, 1589). Springer, Berlin, 1994, pp. 92–186.Google Scholar
Forni, G. and Matheus, C.. Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards. J. Mod. Dyn. 8(3/4) (2014), 271–436.CrossRefGoogle Scholar
Glutsyuk, A.. On odd-periodic orbits in complex planar billiards. J. Dyn. Control Syst. 20(3) (2014), 293–306.CrossRefGoogle Scholar
Glutsyuk, A.. On polynomially integrable Birkhoff billiards on surfaces of constant curvature. J. Eur. Math. Soc. 23(3) (2021), 995–1049.CrossRefGoogle Scholar
Glutsyuk, A. A. and Kudryashov, Yu. G.. On quadrilateral orbits in planar billiards. Dokl. Math. 83(3) (2011), 371–373.CrossRefGoogle Scholar
Glutsyuk, A. A. and Kudryashov, Yu. G.. No planar billiard possesses an open set of quadrilateral trajectories. J. Mod. Dyn. 6(3) (2012), 287–326.CrossRefGoogle Scholar
Gordon, C., Webb, D. L. and Wolpert, S.. One cannot hear the shape of a drum. Bull. Amer. Math. Soc. (N.S.) 27(1) (1992), 134–138.CrossRefGoogle Scholar
Gruber, P. M.. Only ellipsoids have caustics. Math. Ann. 303(1) (1995), 185–194.CrossRefGoogle Scholar
Guillarmou, C. and Lefeuvre, T.. The marked length spectrum of Anosov manifolds. Ann. of Math. (2) 190(1) (2019), 321–344.CrossRefGoogle Scholar
Guillemin, V.. The radon transform on Zoll surfaces. Adv. Math. 22(1) (1976), 85–119.CrossRefGoogle Scholar
Guillemin, V. and Kazhdan, D.. Some inverse spectral results for negatively curved 2-manifolds. Topology 19(3) (1980), 301–312.CrossRefGoogle Scholar
Guillemin, V. and Melrose, R.. The Poisson summation formula for manifolds with boundary. Adv. Math. 32(3) (1979), 204–232.CrossRefGoogle Scholar
Gutkin, E.. Billiard dynamics: a survey with the emphasis on open problems. Regul. Chaotic Dyn. 8(1) (2003), 1–13.CrossRefGoogle Scholar
Gutkin, E. and Katok, A.. Caustics for inner and outer billiards. Comm. Math. Phys. 173 (1995), 101–133.CrossRefGoogle Scholar
Halpern, B.. Strange billiard tables. Trans. Amer. Math. Soc. 232 (1977), 297–305.CrossRefGoogle Scholar
Hezari, H. and Zelditch, S.. ${C}^{\infty }$ spectral rigidity of the ellipse. Anal. PDE 5(5) (2012), 1105–1132.CrossRefGoogle Scholar
Hezari, H. and Zelditch, S.. One can hear the shape of ellipses of small eccentricity. Preprint, 2019, arXiv:1907.03882.Google Scholar
Hezari, H. and Zelditch, S.. Eigenfunction asymptotics and spectral rigidity of the ellipse. Preprint, 2020, arXiv:2006.16685.Google Scholar
Hopf, E.. Closed surfaces without conjugate points. Proc. Nat. Acad. Sci. USA 34 (1948), 47–51.CrossRefGoogle ScholarPubMed
Huang, G., Kaloshin, V. and Sorrentino, A. Nearly circular domains which are integrable close to the boundary are ellipses. Geom. Funct. Anal. 28(2) (2018), 334–392.CrossRefGoogle Scholar
Huang, G., Kaloshin, V. and Sorrentino, A. On marked length spectrums of generic strictly convex billiard tables. Duke Math. J. 167(1) (2018), 175–209.CrossRefGoogle Scholar
Innami, N.. Convex curves whose points are vertices of billiard triangles. Kodai Math. J. 11 (1988), 17–24.CrossRefGoogle Scholar
Innami, N.. Geometry of geodesics for convex billiards and circular billiards. Nihonkai Math. J. 13 (2002), 73–120.Google Scholar
Ivrii, V. Ya.. The second term of the spectral asymptotics for a Laplace–Beltrami operator on manifolds with boundary. Funktsional. Anal. i Prilozhen 14(2) (1980), 25–34.CrossRefGoogle Scholar
Kac, M.. Can one hear the shape of a drum? Amer. Math. Monthly 73(4, part II) (1966), 1–23.CrossRefGoogle Scholar
Kaloshin, V. and Sorrentino, A.. On the local Birkhoff conjecture for convex billiards. Ann. of Math. 188(1) (2018), 315–380.CrossRefGoogle Scholar
Katok, A., Strelcyn, J.-M., Ledrappier, F. and Przytycki, F.. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities (Lecture Notes in Mathematics, 1222). Springer, Berlin, 1986.Google Scholar
Knill, O.. On nonconvex caustics of convex billiards. Elem. Math. 53 (1998), 89–106.CrossRefGoogle Scholar
Lazutkin, V. F.. Existence of caustics for the billiard problem in a convex domain. Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 186–216 (in Russian).Google Scholar
LeBrun, C. and Mason, L. J.. Zoll manifolds and complex surfaces. J. Differential Geom. 61(3) (2002), 453–535.CrossRefGoogle Scholar
Levallois, P.. Non-intégrabilité des billiards définis par certaines perturbations algébriques d’une ellipse et du flot géodésique de certaines perturbations algé´briques d’un ellipsoıde. PhD Thesis, Université Paris VII, December 1993.Google Scholar
Levallois, P. and Tabanov, M.. S´paration des séparatrices du billard elliptique pour une perturbation algrique et symétrique de l’ellipse. C. R. Acad. Sci. Paris Sér. I Math. 316(6) (1993), 589–592.Google Scholar
Marvizi, S. and Melrose, R.. Spectral invariants of convex planar regions. J. Differential Geom. 17 (1982), 475–502.CrossRefGoogle Scholar
Massart, D. and Sorrentino, A.. Differentiability of Mather’s average action and integrability on closed surfaces. Nonlinearity 24 (2011), 1777–1793.CrossRefGoogle Scholar
Mather, J. N.. Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology 21(4) (1982), 457–467.CrossRefGoogle Scholar
Mather, J. N.. Glancing billiards. Ergod. Th. & Dynam. Sys. 2(3–4) (1982), 397–403.CrossRefGoogle Scholar
Mather, J. N.. Differentiability of the minimal average action as a function of the rotation number. Bull. Braz. Math. Soc. (N.S.) 21(1) (1990), 59–70.CrossRefGoogle Scholar
Mazzucchelli, M. and Sorrentino, A.. Remarks on the symplectic invariance of Aubry–Mather sets. C. R. Math. Acad. Sci. Paris 354(4) (2016), 419–423.CrossRefGoogle Scholar
Moser, J.. Selected Chapters of the Calculus of Variations (Lectures in Mathematics, ETH Zurich). Birkhäuser, Basel, 2003.CrossRefGoogle Scholar
Osgood, B., Phillips, R. and Sarnak, P.. Compact isospectral sets of surfaces. J. Funct. Anal. 80(1) (1988), 212–234.CrossRefGoogle Scholar
Osgood, B., Phillips, R. and Sarnak, P.. Extremals of determinants of Laplacians. J. Funct. Anal. 80(1) (1988), 148–211.CrossRefGoogle Scholar
Osgood, B., Phillips, R. and Sarnak, P.. Moduli space, heights and isospectral sets of plane domains. Ann. of Math. (2) 129(2) (1989), 293–362.CrossRefGoogle Scholar
Otal, J.-P.. Le spectre marqué des longueurs des surfaces à courbure négative. Ann. of Math. (2) 131(1) (1990), 151–162.CrossRefGoogle Scholar
Petkov, V. M. and Stoyanov, L. N.. Geometry of Reflecting Rays and Inverse Spectral Problems (Pure and Applied Mathematics). Wiley, Chichester, 1992.Google Scholar
Popov, G.. Invariants of the length spectrum and spectral invariants of planar convex domains. Comm. Math. Phys. 161 (1994), 335–364.CrossRefGoogle Scholar
Popov, G. and Topalov, P.. Invariants of isospectral deformations and spectral rigidity. Comm. Partial Differential Equations 37(3) (2012), 369–446.CrossRefGoogle Scholar
Popov, G. and Topalov, P.. From K.A.M. tori to isospectral invariants and spectral rigidity of billiard tables. Preprint, 2019, arXiv:1602.0315.Google Scholar
Poritsky, H.. The billiard ball problem on a table with a convex boundary—an illustrative dynamical problem. Ann. of Math. (2) 51 (1950), 446–470.CrossRefGoogle Scholar
Ramírez-Ros, R.. Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables. Phys. D 214 (2006), 78–87.CrossRefGoogle Scholar
Rychlik, M. R.. Periodic points of the billiard ball map in a convex domain. J. Differential Geom. 30 (1989), 191–205.CrossRefGoogle Scholar
Safarov, Yu. and Vassilev, D.. The Asymptotic Distribution of Eigenvalues of Partial Differential Operators (Translations of Mathematical Monographs, 155). American Mathematical Society, Providence, RI, (1996).CrossRefGoogle Scholar
Sapiro, G. and Tannenbaum, A.. On affine plane curve evolution. J. Funct. Anal. 119 (1994), 79–120.CrossRefGoogle Scholar
Sarnak, P.. Determinants of Laplacians, heights and finiteness. Analysis, et cetera. Ed. Rabinowitz, P. H. and Zehnde, E.. Academic Press, Boston, 1990, pp. 601–622.CrossRefGoogle Scholar
Siburg, K. F.. Aubry–Mather theory and the inverse spectral problem for planar convex domains. Israel J. Math. 113 (1999), 285–304.CrossRefGoogle Scholar
Siburg, K. F.. The Principle of Least Action in Geometry and Dynamics (Lecture Notes in Mathematics, 1844). Springer, Berlin, 2004.CrossRefGoogle Scholar
Sinai, Y. G.. Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russian Math. Surveys 25 (1970), 137–189.CrossRefGoogle Scholar
Sorrentino, A.. Computing Mather’s beta-function for Birkhoff billiards. Discrete Contin. Dyn. Syst. 35(10) (2015), 5055–5082.CrossRefGoogle Scholar
Sorrentino, A.. Action-Minimizing Methods in Hamiltonian Dynamics. An Introduction to Aubry–Mather Theory (Mathematical Notes Series, 50). Princeton University Press, Princeton, NJ, 2015.CrossRefGoogle Scholar
Sorrentino, A. and Viterbo, C.. Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms. Geom. Topol. 14 (2010), 2383–2403.CrossRefGoogle Scholar
Stojanov, L.. Note on the periodic points of the billiard. J. Differential Geom. 34 (1991), 835–837.CrossRefGoogle Scholar
Sunada, T.. Riemannian coverings and isospectral manifolds. Ann. of Math. (2) 121(1) (1985), 169–186.CrossRefGoogle Scholar
Tabachnikov, S.. Billiards (Panoramas et Synthèses, 1). Société Mathématique de France, Paris, 1995.Google Scholar
Tabachnikov, S.. Geometry and Billiards (Student Mathematical Library, 30). American Mathematical Society, Providence, RI, 2005.Google Scholar
Tabanov, M. B.. New ellipsoidal confocal coordinates and geodesics on an ellipsoid. J. Math. Sci. 82(6) (1996), 3851–3858.CrossRefGoogle Scholar
Treschev, D.. Billiard map and rigid rotation. Phys. D 255 (2013), 31–34.CrossRefGoogle Scholar
Vignéras, M.-F.. Variétés riemanniennes isospectrales et non isométriques. Ann. of Math. (2) 112(1) (1980), 21–32.CrossRefGoogle Scholar
Vorobets, Ya. B.. On the measure of the set of periodic points of a billiard. Math. Notes 55 (1994), 455–460.CrossRefGoogle Scholar
Wojtkowski, M. P.. Two applications of Jacobi fields to the billiard ball problem. J. Differential Geom. 40 (1994), 155–164.CrossRefGoogle Scholar
Zelditch, S.. Inverse spectral problem for analytic domains. II. Z2-symmetric domains. Ann. of Math. (2) 170(1) (2009), 205–269.CrossRefGoogle Scholar
Zelditch, S.. Survey on the inverse spectral problem. ICCM Not. 2(2) (2014), 1–20.CrossRefGoogle Scholar
Zelditch, S.. Survey of the inverse spectral problem. Preprint, 2004, arXiv:math/0402356.Google Scholar
Zoll, O.. Über Flächen mit Scharen geschlossener geodätischer Linien. Math. Ann. 57(1903), 108–133 (in German).CrossRefGoogle Scholar