×

zbMATH — the first resource for mathematics

An integrable deformation of an ellipse of small eccentricity is an ellipse. (English) Zbl 1379.37104
A strictly convex domain \(\Omega \subset \mathbb{R}^2\) is \(C^r\) if its boundary is a \(C^r\)-smooth curve. This very interesting paper concerns the billiard problem inside a \(C^r\) domain \(\Omega\), called {billiard table}.
A (possibly not connected) curve \(\Gamma \subset \Omega \) is called {caustic} if any billiard orbit having one segment tangent to \(\Gamma \) has all its segments tangent to \(\Gamma \). Then the billiard \(\Omega \) is called {locally integrable} if the union of all caustics has nonempty interior; \(\Omega \) is called {integrable} if the union of all {smooth convex} caustics has nonempty interior.
It is well known that an ellipse billiard is integrable since its caustics are cofocal ellipses and hyperbolas. A long standing open question is whether or not there exist integrable billiards that are different from ellipses. The Birkhoff conjecture states that integrability implies the fact that \(\partial \Omega \) is ellipse.
The present work proves a version of this conjecture for tables bounded by small perturbations of ellipses of small eccentricity. A main remark is that {infinitesimally} rationally integrable deformations of a circle are tangent to a five-parametric family of ellipses.

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] K. G. Andersson and R. B. Melrose, ”The propagation of singularities along gliding rays,” Invent. Math., vol. 41, iss. 3, pp. 197-232, 1977. · Zbl 0373.35053
[2] M. Bialy, ”Convex billiards and a theorem by E. Hopf,” Math. Z., vol. 214, iss. 1, pp. 147-154, 1993. · Zbl 0790.58023
[3] G. D. Birkhoff, Dynamical Systems, Providence, RI: Amer. Math. Soc., 1966, vol. IX. · Zbl 0171.05402
[4] L. A. Bunimovich, ”On absolutely focusing mirrors,” in Ergodic Theory and Related Topics, III, New York: Springer, 1992, vol. 1514, pp. 62-82. · Zbl 0772.58027
[5] J. Chazarain, ”Formule de Poisson pour les variétés riemanniennes,” Invent. Math., vol. 24, pp. 65-82, 1974. · Zbl 0281.35028
[6] J. J. Duistermaat and V. W. Guillemin, ”The spectrum of positive elliptic operators and periodic bicharacteristics,” Invent. Math., vol. 29, iss. 1, pp. 39-79, 1975. · Zbl 0307.35071
[7] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: Springer-Verlag, 2001. · Zbl 1042.35002
[8] C. Gordon, D. L. Webb, and S. Wolpert, ”One cannot hear the shape of a drum,” Bull. Amer. Math. Soc., vol. 27, iss. 1, pp. 134-138, 1992. · Zbl 0756.58049
[9] V. Guillemin and R. Melrose, ”An inverse spectral result for elliptical regions in \({\mathbf R}^2\),” Adv. in Math., vol. 32, iss. 2, pp. 128-148, 1979. · Zbl 0415.35062
[10] E. Gutkin, ”Billiard dynamics: An updated survey with the emphasis on open problems,” Chaos, vol. 22, iss. 2, p. 026116, 2012. · Zbl 1331.37001
[11] H. Hezari and S. Zelditch, ”\(C^\infty\) spectral rigidity of the ellipse,” Anal. PDE, vol. 5, iss. 5, pp. 1105-1132, 2012. · Zbl 1264.35150
[12] M. Kac, ”Can one hear the shape of a drum?,” Amer. Math. Monthly, vol. 73, iss. 4, part II, pp. 1-23, 1966. · Zbl 0139.05603
[13] V. F. Lazutkin, ”The existence of caustics for a billiard problem in a convex domain,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 37, pp. 186-216, 1973. · Zbl 0256.52001
[14] S. Pinto-de-Carvalho and R. Ram’irez-Ros, ”Non-persistence of resonant caustics in perturbed elliptic billiards,” Ergodic Theory Dynam. Systems, vol. 33, iss. 6, pp. 1876-1890, 2013. · Zbl 1408.37065
[15] G. Popov, ”Invariants of the length spectrum and spectral invariants of planar convex domains,” Comm. Math. Phys., vol. 161, iss. 2, pp. 335-364, 1994. · Zbl 0797.58070
[16] G. Popov and P. Topalov, From K.A.M. tori to isospectral invariants and spectral rigidity of billiard tables, 2016.
[17] H. Poritsky, ”The billiard ball problem on a table with a convex boundary-an illustrative dynamical problem,” Ann. of Math., vol. 51, pp. 446-470, 1950. · Zbl 0037.26802
[18] R. Ram’irez-Ros, ”Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables,” Phys. D, vol. 214, iss. 1, pp. 78-87, 2006. · Zbl 1099.37027
[19] P. Sarnak, ”Determinants of Laplacians; heights and finiteness,” in Analysis, et cetera, Boston: Academic Press, 1990, pp. 601-622. · Zbl 0703.53037
[20] T. Sunada, ”Riemannian coverings and isospectral manifolds,” Ann. of Math., vol. 121, iss. 1, pp. 169-186, 1985. · Zbl 0585.58047
[21] S. Tabachnikov, Geometry and Billiards, Providence, RI: Amer. Math. Soc., 2005, vol. 30. · Zbl 1119.37001
[22] M. B. Tabanov, ”New ellipsoidal confocal coordinates and geodesics on an ellipsoid. Algebra, 3,” J. Math. Sci., vol. 82, iss. 6, pp. 3851-3858, 1996. · Zbl 0889.58062
[23] D. Treschev, ”Billiard map and rigid rotation,” Phys. D, vol. 255, pp. 31-34, 2013. · Zbl 1417.37139
[24] M. Vignéras, ”Variétés riemanniennes isospectrales et non isométriques,” Ann. of Math., vol. 112, iss. 1, pp. 21-32, 1980. · Zbl 0445.53026
[25] M. P. Wojtkowski, ”Two applications of Jacobi fields to the billiard ball problem,” J. Differential Geom., vol. 40, iss. 1, pp. 155-164, 1994. · Zbl 0812.58067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.