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Dynamical spectral rigidity among \(\mathbb{Z}_2\)-symmetric strictly convex domains close to a circle. (English) Zbl 1377.37062

The main result of the paper states that any strictly convex domain sufficiently close to a circle, with sufficiently smooth boundary and axial symmetry is dynamically spectrally rigid, i.e., any \(C^1\)-smooth one-parameter dynamically isospectral family is isometric. The paper is densely written. With patience one can get from the proof all needed details and techniques. Moreover, the multilayered story of the problem in question is duly considered.

MSC:

37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
35P05 General topics in linear spectral theory for PDEs
58J53 Isospectrality
35R30 Inverse problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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References:

[1] Andersson, K. G.; Melrose, R. B., The propagation of singularities along gliding rays, Invent. Math.. Inventiones Mathematicae, 41, 197-232, (1977) · Zbl 0373.35053 · doi:10.1007/BF01403048
[2] Avila, Artur; De Simoi, Jacopo; Kaloshin, Vadim, An integrable deformation of an ellipse of small eccentricity is an ellipse, Ann. of Math.. Annals of Mathematics. Second Series, 184, 527-558, (2016) · Zbl 1379.37104 · doi:10.4007/annals.2016.184.2.5
[3] Chazarain, J., Formule de {P}oisson pour les vari\'et\'es riemanniennes, Invent. Math.. Inventiones Mathematicae, 24, 65-82, (1974) · Zbl 0281.35028 · doi:10.1007/BF01418788
[4] Croke, Christopher B., Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv.. Commentarii Mathematici Helvetici, 65, 150-169, (1990) · Zbl 0704.53035 · doi:10.1007/BF02566599
[5] Croke, Christopher B.; Sharafutdinov, Vladimir A., Spectral rigidity of a compact negatively curved manifold, Topology. Topology. An International Journal of Mathematics, 37, 1265-1273, (1998) · Zbl 0936.58013 · doi:10.1016/S0040-9383(97)00086-4
[6] de la Llave, R.; Marco, J. M.; Moriy\'on, R., Canonical perturbation theory of {A}nosov systems and regularity results for the {L}iv\v sic cohomology equation, Ann. of Math.. Annals of Mathematics. Second Series, 123, 537-611, (1986) · Zbl 0603.58016 · doi:10.2307/1971334
[7] Duistermaat, J. J.; Guillemin, V. W., The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math.. Inventiones Mathematicae, 29, 39-79, (1975) · Zbl 0307.35071 · doi:10.1007/BF01405172
[8] Gordon, Carolyn; Webb, David L.; Wolpert, Scott, One cannot hear the shape of a drum, Bull. Amer. Math. Soc.. American Mathematical Society. Bulletin. New Series, 27, 134-138, (1992) · Zbl 0756.58049 · doi:10.1090/S0273-0979-1992-00289-6
[9] 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 · doi:10.1016/0040-9383(80)90015-4
[10] Halpern, Benjamin, Strange billiard tables, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 232, 297-305, (1977) · Zbl 0374.53001 · doi:10.2307/1998942
[11] Hezari, H., Robin spectral rigidity of strictly convex domains with a reflectional symmetry, (2016)
[12] Hezari, Hamid; Zelditch, Steve, {\(C^\infty \)} spectral rigidity of the ellipse, Anal. PDE. Analysis & PDE, 5, 1105-1132, (2012) · Zbl 1264.35150 · doi:10.2140/apde.2012.5.1105
[13] Kac, Mark, Can one hear the shape of a drum?, Amer. Math. Monthly. The American Mathematical Monthly, 73, 1-23, (1966) · Zbl 0139.05603 · doi:10.2307/2313748
[14] 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
[15] 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 · doi:10.1016/0022-1236(88)90071-7
[16] 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 · doi:10.1016/0022-1236(88)90070-5
[17] Osgood, B.; Phillips, R.; Sarnak, P., Moduli space, heights and isospectral sets of plane domains, Ann. of Math.. Annals of Mathematics. Second Series, 129, 293-362, (1989) · Zbl 0677.58045 · doi:10.2307/1971449
[18] Otal, Jean-Pierre, Le spectre marqu\'e des longueurs des surfaces \`“a courbure n\'”egative, Ann. of Math.. Annals of Mathematics. Second Series, 131, 151-162, (1990) · Zbl 0699.58018 · doi:10.2307/1971511
[19] Petkov, Vesselin M.; Stoyanov, Luchezar N., Geometry of Reflecting Rays and Inverse Spectral Problems, Pure Appl. Math., vi+313 pp., (1992) · Zbl 0761.35077
[20] Popov, Georgi; Topalov, Peter, Invariants of isospectral deformations and spectral rigidity, Comm. Partial Differential Equations. Communications in Partial Differential Equations, 37, 369-446, (2012) · Zbl 1245.58015 · doi:10.1080/03605302.2011.641051
[21] Popov, Georgi; Topalov, Peter, From {K.A.M. T}ori to Isospectral Invariants and Spectral Rigidity of Billiard Tables, (2016)
[22] Sarnak, Peter, Analysis, et Cetera. Determinants of {L}aplacians; Heights and Finiteness, 601-622, (1990) · Zbl 0703.53037
[23] Sunada, Toshikazu, Riemannian coverings and isospectral manifolds, Ann. of Math.. Annals of Mathematics. Second Series, 121, 169-186, (1985) · Zbl 0585.58047 · doi:10.2307/1971195
[24] Vign\'eras, Marie-France, Vari\'et\'es riemanniennes isospectrales et non isom\'etriques, Ann. of Math.. Annals of Mathematics. Second Series, 112, 21-32, (1980) · Zbl 0445.53026 · doi:10.2307/1971319
[25] Zelditch, Steve, Surveys in Differential Geometry. {V}ol. {IX}. The inverse spectral problem, Surv. Differ. Geom., 9, 401-467, (2004) · Zbl 1061.58029 · doi:10.4310/SDG.2004.v9.n1.a12
[26] Zelditch, Steve, Inverse spectral problem for analytic domains. {II}. {\( \Bbb Z_2\)}-symmetric domains, Ann. of Math.. Annals of Mathematics. Second Series, 170, 205-269, (2009) · Zbl 1196.58016 · doi:10.4007/annals.2009.170.205
[27] Zelditch, Steve, Survey on the inverse spectral problem, ICCM Not.. ICCM Notices. Notices of the International Congress of Chinese Mathematicians, 2, 1-20, (2014) · doi:10.4310/ICCM.2014.v2.n2.a1
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