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On the local Birkhoff conjecture for convex billiards. (English) Zbl 1394.37093

This paper considers the classical Birkhoff conjecture that the boundary of a strictly convex integrable billiard table must be an ellipse (or, as a special case, a circle). The conjecture is still unresolved. The authors prove a complete local version that a small integrable perturbation of an ellipse must be an ellipse.
The main result is as follows: assume that \({\mathcal E}\) is an ellipse with eccentric \(e_0\), \(0\leq e_0<1\), and semi-focal distance \(c\). Provided that \(k\geq 39\) for every \(K>0\), there is an \(\varepsilon=\varepsilon(e_0,c,K)\) such that if \(\Omega\) is a rationally integrable \(C^k\)-smooth domain with a boundary \(\partial\Omega\) \(C^k\)-\(K\)-closed and \(C^1\)-\(\varepsilon\)-close to \({\mathcal E}\), then \(\Omega\) is an ellipse. Here it is assumed that \(\partial\Omega\) consists of \({\mathcal E}\) plus a \(C^k\)-perturbation \(\mu\) with \(\| \mu\|_{C^k}\leq K\) and \(\|\mu\|_{C^1}<\varepsilon\). (The latter describe \(C^k\)-\(K\)-close and \(C^1\)-\(\varepsilon\)-close.)
The result here parallels similar recent results of A. Avila et al. [Ann. Math. (2) 184, No. 2, 527–558 (2016; Zbl 1379.37104)] and G. Huang et al. [Duke Math. J. 167, No. 1, 175–209 (2018; Zbl 1417.37138)]. A critical idea in this paper enables the authors to move beyond the prior results in the almost-circular case. This was to consider analytic extensions of the action-angle coordinates of elliptic billiards (i.e., the boundary parametrizations that are induced by the integrable caustics) and to carefully evaluate their singularities. The authors express such functions in terms of elliptic integrals and Jacobi elliptic functions.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
37E40 Dynamical aspects of twist maps
33E05 Elliptic functions and integrals
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
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[1] Akhiezer, N. I., Elements of the Theory of Elliptic Functions, Transl. Math. Monogr., 79, viii+237 pp., (1990) · Zbl 0694.33001
[2] Andersson, K. G.; Melrose, R. B., The propagation of singularities along gliding rays, Invent. Math.. Inventiones Mathematicae, 41, 197-232, (1977) · Zbl 0373.35053
[3] Avila, Artur; {De Simoi}, Jacopo; Kaloshin, Vadim, An integrable deformation of an ellipse of small eccentricity is an ellipse, Ann. of Math. (2). Annals of Mathematics. Second Series, 184, 527-558, (2016) · Zbl 1379.37104
[4] Bangert, V., Mather sets for twist maps and geodesics on tori. Dynamics Reported, {V}ol. 1, Dynam. Report. Ser. Dynam. Systems Appl., 1, 1-56, (1988)
[5] Bialy, Misha, Convex billiards and a theorem by {E}. {H}opf, Math. Z.. Mathematische Zeitschrift, 214, 147-154, (1993) · Zbl 0790.58023
[6] Bialy, Misha; Mironov, Andrey E., Angular billiard and algebraic {B}irkhoff conjecture, Adv. Math.. Advances in Mathematics, 313, 102-126, (2017) · Zbl 1364.37124
[7] Birkhoff, George D., On the periodic motions of dynamical systems, Acta Math.. Acta Mathematica, 50, 359-379, (1927) · JFM 53.0733.03
[8] Chang, Shau-Jin; Friedberg, Richard, Elliptical billiards and {P}oncelet’s theorem, J. Math. Phys.. Journal of Mathematical Physics, 29, 1537-1550, (1988) · Zbl 0663.70015
[9] Croke, Christopher B., Rigidity for surfaces of non-positive curvature, Comment. Math. Helv.. Commentarii Mathematici Helvetici, 65, 150-169, (1990) · Zbl 0704.53035
[10] Damasceno, Josu\'e; Dias Carneiro, Mario J.; Ram{\'\i}rez-Ros, Rafael, The billiard inside an ellipse deformed by the curvature flow, Proc. Amer. Math. Soc.. Proceedings of the American Mathematical Society, 145, 705-719, (2017) · Zbl 1368.37048
[11] Delshams, Amadeu; Ram{\'\i}rez-Ros, Rafael, Poincar\'e–{M}elnikov–{A}rnold method for analytic planar maps, Nonlinearity. Nonlinearity, 9, 1-26, (1996) · Zbl 0887.58029
[12] Appendix B. coauthored with H. Hezari, Dynamical spectral rigidity among {\(\Bbb Z_2\)}-symmetric strictly convex domains close to a circle, Ann. of Math. (2). Annals of Mathematics. Second Series, 186, 277-314, (2017) · Zbl 1377.37062
[13] Gilbarg, David; Trudinger, Neil S., Elliptic Partial Differential Equations of Second Order, Classics in Math., xiv+517 pp., (2001) · Zbl 1042.35002
[14] Gordon, Carolyn; Webb, David L.; Wolpert, Scott, One cannot hear the shape of a drum, Bull. Amer. Math. Soc. (N.S.). American Mathematical Society. Bulletin. New Series, 27, 134-138, (1992) · Zbl 0756.58049
[15] Grayson, Matthew A., Shortening embedded curves, Ann. of Math. (2). Annals of Mathematics. Second Series, 129, 71-111, (1989) · Zbl 0686.53036
[16] Guillemin, Victor; Melrose, Richard, A cohomological invariant of discrete dynamical systems. E. {B}. {C}hristoffel, 672-679, (1981) · Zbl 0482.58032
[17] Guillemin, V.; Kazhdan, D., Some inverse spectral results for negatively curved {\(2\)}-manifolds, Topology. Topology. An International Journal of Mathematics, 19, 301-312, (1980) · Zbl 0465.58027
[18] Gutkin, E., Billiard dynamics: a survey with the emphasis on open problems, Regul. Chaotic Dyn.. Regular & Chaotic Dynamics. International Scientific Journal, 8, 1-13, (2003) · Zbl 1023.37022
[19] Halpern, Benjamin, Strange billiard tables, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 232, 297-305, (1977) · Zbl 0374.53001
[20] Hezari, Hamid; Zelditch, Steve, Inverse spectral problem for analytic {\((\Bbb Z/2\Bbb Z)^n\)}-symmetric domains in {\(\Bbb R^n\)}, Geom. Funct. Anal.. Geometric and Functional Analysis, 20, 160-191, (2010) · Zbl 1226.35055
[21] Huang, Guan; Kaloshin, Vadim; Sorrentino, Alfonso, On the marked length spectrum of generic strictly convex billiard tables, Duke Math. J.. Duke Mathematical Journal, 167, 175-209, (2018) · Zbl 1417.37138
[22] Innami, Nobuhiro, Geometry of geodesics for convex billiards and circular billiards, Nihonkai Math. J.. Nihonkai Mathematical Journal, 13, 73-120, (2002) · Zbl 1035.37027
[23] Kac, Mark, Can one hear the shape of a drum?, Amer. Math. Monthly. The American Mathematical Monthly, 73, 1-23, (1966) · Zbl 0139.05603
[24] Lazutkin, V. F., Existence of caustics for the billiard problem in a convex domain, Izv. Akad. Nauk SSSR Ser. Mat.. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 37, 186-216, (1973) · Zbl 0256.52001
[25] Marvizi, Shahla; Melrose, Richard, Spectral invariants of convex planar regions, J. Differential Geom.. Journal of Differential Geometry, 17, 475-502, (1982) · Zbl 0492.53033
[26] Marvizi, Shahla; Melrose, Richard B., Some spectrally isolated convex planar regions, Proc. Nat. Acad. Sci. U.S.A.. Proceedings of the National Academy of Sciences of the United States of America, 79, 7066-7067, (1982) · Zbl 0504.53040
[27] Mather, John N., Glancing billiards, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 2, 397-403, (1982) · Zbl 0525.58021
[28] Mather, John N., Differentiability of the minimal average action as a function of the rotation number, Bol. Soc. Brasil. Mat. (N.S.). Boletim da Sociedade Brasileira de Matem\'atica. Nova S\'erie, 21, 59-70, (1990) · Zbl 0766.58033
[29] Mather, John N.; Forni, Giovanni, Action minimizing orbits in {H}amiltonian systems. Transition to Chaos in Classical and Quantum Mechanics, Lecture Notes in Math., 1589, 92-186, (1994) · Zbl 0822.70011
[30] Milnor, J., Eigenvalues of the {L}aplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A.. Proceedings of the National Academy of Sciences of the United States of America, 51, 542 pp., (1964) · Zbl 0124.31202
[31] Moser, J\`“urgen, Selected Chapters in the Calculus of Variations. Lecture notes by Oliver Knill, Lectures in Math. ETH Z\'”urich, iv+132 pp., (2003) · Zbl 1045.37001
[32] Otal, Jean-Pierre, Le spectre marqu\'e des longueurs des surfaces \`“‘a courbure n\'”’egative, Ann. of Math. (2). Annals of Mathematics. Second Series, 131, 151-162, (1990) · Zbl 0699.58018
[33] Osgood, B.; Phillips, R.; Sarnak, P., Compact isospectral sets of surfaces, J. Funct. Anal.. Journal of Functional Analysis, 80, 212-234, (1988) · Zbl 0653.53021
[34] Osgood, B.; Phillips, R.; Sarnak, P., Extremals of determinants of {L}aplacians, J. Funct. Anal.. Journal of Functional Analysis, 80, 148-211, (1988) · Zbl 0653.53022
[35] Osgood, B.; Phillips, R.; Sarnak, P., Moduli space, heights and isospectral sets of plane domains, Ann. of Math. (2). Annals of Mathematics. Second Series, 129, 293-362, (1989) · Zbl 0677.58045
[36] {Pinto-de-Carvalho}, S\^onia; Ram{\'\i}rez-Ros, Rafael, Non-persistence of resonant caustics in perturbed elliptic billiards, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 33, 1876-1890, (2013) · Zbl 1408.37065
[37] Popov, Georgi, Invariants of the length spectrum and spectral invariants of planar convex domains, Comm. Math. Phys.. Communications in Mathematical Physics, 161, 335-364, (1994) · Zbl 0797.58070
[38] Popov, Georgi; Topalov, Peter, From {KAM} Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables, (2016)
[39] Poritsky, Hillel, The billiard ball problem on a table with a convex boundary—an illustrative dynamical problem, Ann. of Math. (2). Annals of Mathematics. Second Series, 51, 446-470, (1950) · Zbl 0037.26802
[40] Ram{\'\i}rez-Ros, Rafael, Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables, Phys. D. Physica D. Nonlinear Phenomena, 214, 78-87, (2006) · Zbl 1099.37027
[41] Sapiro, Guillermo; Tannenbaum, Allen, On affine plane curve evolution, J. Funct. Anal.. Journal of Functional Analysis, 119, 79-120, (1994) · Zbl 0801.53008
[42] Sarnak, Peter, Determinants of {L}aplacians; heights and finiteness. Analysis, et {C}etera, 601-622, (1990)
[43] Siburg, Karl Friedrich, The Principle of Least Action in Geometry and Dynamics, Lecture Notes in Math., 1844, xii+128 pp., (2004) · Zbl 1056.47002
[44] Sorrentino, Alfonso, Computing {M}ather’s {\(\beta\)}-function for {B}irkhoff billiards, Discrete Contin. Dyn. Syst.. Discrete and Continuous Dynamical Systems. Series A, 35, 5055-5082, (2015) · Zbl 1359.37088
[45] Sorrentino, Alfonso, Action-Minimizing Methods in {H}amiltonian Dynamics: {A}n {I}ntroduction to Aubry-Mather {T}heory, Math. Notes, 50, xii+115 pp., (2015) · Zbl 1373.37002
[46] Tabachnikov, Serge, Billiards, Panor. Synth.. Panoramas et Synth\`eses, vi+142 pp., (1995) · Zbl 0833.58001
[47] Tabachnikov, Serge, Geometry and Billiards, Student Math. Library, 30, xii+176 pp., (2005) · Zbl 1119.37001
[48] Tabanov, M. B., New ellipsoidal confocal coordinates and geodesics on an ellipsoid, J. Math. Sci.. Journal of Mathematical Sciences, 82, 3851-3858, (1996) · Zbl 0889.58062
[49] Treschev, D., Billiard map and rigid rotation, Phys. D. Physica D. Nonlinear Phenomena, 255, 31-34, (2013) · Zbl 1417.37139
[50] Wojtkowski, Maciej P., Two applications of {J}acobi fields to the billiard ball problem, J. Differential Geom.. Journal of Differential Geometry, 40, 155-164, (1994) · Zbl 0812.58067
[51] Zelditch, S., Spectral determination of analytic bi-axisymmetric plane domains, Geom. Funct. Anal.. Geometric and Functional Analysis, 10, 628-677, (2000) · Zbl 0961.58012
[52] Huang, Guan; Kaloshin, Vadim; Sorrentino, Alfonso, Nearly circular domains which are integrable close to the boundary are ellipses, Geom. Funct. Anal.. Geometric and Functional Analysis, 28, 334-392, (2018) · Zbl 1395.37041
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