Abstract
The classical Birkhoff conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse (or a circle as a special case). In this article we prove a complete local version of this conjecture: a small integrable perturbation of an ellipse must be an ellipse. This extends and completes the result in Avila-De Simoi-Kaloshin, where nearly circular domains were considered. One of the crucial ideas in the proof is to extend action-angle coordinates for elliptic billiards into complex domains (with respect to the angle), and to thoroughly analyze the nature of their complex singularities. As an application, we are able to prove some spectral rigidity results for elliptic domains.
-
[Akhiezer] N. I. Akhiezer, Elements of the Theory of Elliptic Functions, Amer. Math. Soc., Providence, RI, 1990, vol. 79.
@BOOK{Akhiezer,
author = {Akhiezer, N. I.},
title = {Elements of the Theory of Elliptic Functions},
series = {Transl. Math. Monogr.},
volume = {79},
note = {translated from the second Russian edition by H. H. McFaden},
publisher = {Amer. Math. Soc., Providence, RI},
year = {1990},
pages = {viii+237},
isbn = {0-8218-4532-2},
mrclass = {33E05 (11F12 30-02 33C75)},
mrnumber = {1054205},
mrreviewer = {Glenn Stevens},
zblnumber = {0694.33001},
} -
[AM]
K. G. Andersson and R. B. Melrose, "The propagation of singularities along gliding rays," Invent. Math., vol. 41, iss. 3, pp. 197-232, 1977.@ARTICLE{AM,
author = {Andersson, K. G. and Melrose, R. B.},
title = {The propagation of singularities along gliding rays},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {41},
year = {1977},
number = {3},
pages = {197--232},
issn = {0020-9910},
mrclass = {58G15 (35S15)},
mrnumber = {0494322},
mrreviewer = {Bent E. Petersen},
doi = {10.1007/BF01403048},
url = {https://doi.org/10.1007/BF01403048},
zblnumber = {0373.35053},
} -
[ADK] A. Avila, J. De Simoi, and V. Kaloshin, "An integrable deformation of an ellipse of small eccentricity is an ellipse," Ann. of Math. (2), vol. 184, iss. 2, pp. 527-558, 2016.
@ARTICLE{ADK,
author = {Avila, Artur and {De Simoi},
Jacopo and Kaloshin, Vadim},
title = {An integrable deformation of an ellipse of small eccentricity is an ellipse},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {184},
year = {2016},
number = {2},
pages = {527--558},
issn = {0003-486X},
mrclass = {37D50},
mrnumber = {3548532},
mrreviewer = {Nicolas Bedaride},
doi = {10.4007/annals.2016.184.2.5},
url = {https://doi.org/10.4007/annals.2016.184.2.5},
zblnumber = {1379.37104},
} -
[Bangert] V. Bangert, "Mather sets for twist maps and geodesics on tori," in Dynamics Reported, Vol. 1, Wiley, Chichester, 1988, vol. 1, pp. 1-56.
@INCOLLECTION{Bangert,
author = {Bangert, V.},
title = {Mather sets for twist maps and geodesics on tori},
booktitle = {Dynamics Reported, {V}ol. 1},
series = {Dynam. Report. Ser. Dynam. Systems Appl.},
volume = {1},
pages = {1--56},
publisher = {Wiley, Chichester},
year = {1988},
mrclass = {58F17 (58F05 82A68)},
mrnumber = {0945963},
mrreviewer = {Dietrich Flockerzi},
zblnumber = {0664.53021},
} -
[Bialy] M. Bialy, "Convex billiards and a theorem by E. Hopf," Math. Z., vol. 214, iss. 1, pp. 147-154, 1993.
@ARTICLE{Bialy,
author = {Bialy, Misha},
title = {Convex billiards and a theorem by {E}. {H}opf},
journal = {Math. Z.},
fjournal = {Mathematische Zeitschrift},
volume = {214},
year = {1993},
number = {1},
pages = {147--154},
issn = {0025-5874},
mrclass = {58F11 (58F15)},
mrnumber = {1234604},
mrreviewer = {András Krámli},
doi = {10.1007/BF02572397},
url = {https://doi.org/10.1007/BF02572397},
zblnumber = {0790.58023},
} -
[BialyMironov] M. Bialy and A. E. Mironov, "Angular billiard and algebraic Birkhoff conjecture," Adv. Math., vol. 313, pp. 102-126, 2017.
@ARTICLE{BialyMironov,
author = {Bialy, Misha and Mironov, Andrey E.},
title = {Angular billiard and algebraic {B}irkhoff conjecture},
journal = {Adv. Math.},
fjournal = {Advances in Mathematics},
volume = {313},
year = {2017},
pages = {102--126},
issn = {0001-8708},
mrclass = {37D50},
mrnumber = {3649222},
doi = {10.1016/j.aim.2017.04.001},
url = {https://doi.org/10.1016/j.aim.2017.04.001},
zblnumber = {1364.37124},
} -
[Birkhoff] G. D. Birkhoff, "On the periodic motions of dynamical systems," Acta Math., vol. 50, iss. 1, pp. 359-379, 1927.
@ARTICLE{Birkhoff,
author = {Birkhoff, George D.},
title = {On the periodic motions of dynamical systems},
journal = {Acta Math.},
fjournal = {Acta Mathematica},
volume = {50},
year = {1927},
number = {1},
pages = {359--379},
issn = {0001-5962},
mrclass = {DML},
mrnumber = {1555257},
doi = {10.1007/BF02421325},
url = {https://doi.org/10.1007/BF02421325},
zblnumber = {},
jfmnumber = {53.0733.03},
} -
[CF] S. Chang and R. Friedberg, "Elliptical billiards and Poncelet’s theorem," J. Math. Phys., vol. 29, iss. 7, pp. 1537-1550, 1988.
@ARTICLE{CF,
author = {Chang, Shau-Jin and Friedberg, Richard},
title = {Elliptical billiards and {P}oncelet's theorem},
journal = {J. Math. Phys.},
fjournal = {Journal of Mathematical Physics},
volume = {29},
year = {1988},
number = {7},
pages = {1537--1550},
issn = {0022-2488},
mrclass = {58F07 (14K20)},
mrnumber = {0946326},
mrreviewer = {Horst Knörrer},
doi = {10.1063/1.527900},
url = {https://doi.org/10.1063/1.527900},
zblnumber = {0663.70015},
} -
[Cr] C. B. Croke, "Rigidity for surfaces of non-positive curvature," Comment. Math. Helv., vol. 65, iss. 1, pp. 150-169, 1990.
@ARTICLE{Cr,
author = {Croke, Christopher B.},
title = {Rigidity for surfaces of non-positive curvature},
journal = {Comment. Math. Helv.},
fjournal = {Commentarii Mathematici Helvetici},
volume = {65},
year = {1990},
number = {1},
pages = {150--169},
issn = {0010-2571},
mrclass = {53C20 (53C22 58F17)},
mrnumber = {1036134},
mrreviewer = {Chi-Keung Cheung},
doi = {10.1007/BF02566599},
url = {https://doi.org/10.1007/BF02566599},
zblnumber = {0704.53035},
} -
[DCR] J. Damasceno, M. J. Dias Carneiro, and R. Ramírez-Ros, "The billiard inside an ellipse deformed by the curvature flow," Proc. Amer. Math. Soc., vol. 145, iss. 2, pp. 705-719, 2017.
@ARTICLE{DCR,
author = {Damasceno, Josué and Dias Carneiro, Mario J. and Ram{\'\i}rez-Ros, Rafael},
title = {The billiard inside an ellipse deformed by the curvature flow},
journal = {Proc. Amer. Math. Soc.},
fjournal = {Proceedings of the American Mathematical Society},
volume = {145},
year = {2017},
number = {2},
pages = {705--719},
issn = {0002-9939},
mrclass = {37E40 (37B40 37J45 53C44)},
mrnumber = {3577872},
mrreviewer = {Luca Asselle},
doi = {10.1090/proc/13351},
url = {https://doi.org/10.1090/proc/13351},
zblnumber = {1368.37048},
} -
[DRR] A. Delshams and R. Ramírez-Ros, "Poincaré–Melnikov–Arnold method for analytic planar maps," Nonlinearity, vol. 9, iss. 1, pp. 1-26, 1996.
@ARTICLE{DRR,
author = {Delshams, Amadeu and Ram{\'\i}rez-Ros, Rafael},
title = {Poincaré--{M}elnikov--{A}rnold method for analytic planar maps},
journal = {Nonlinearity},
fjournal = {Nonlinearity},
volume = {9},
year = {1996},
number = {1},
pages = {1--26},
issn = {0951-7715},
mrclass = {58F30 (33E05 34C37)},
mrnumber = {1373998},
mrreviewer = {Vassili G. Gelfreich},
doi = {10.1088/0951-7715/9/1/001},
url = {https://doi.org/10.1088/0951-7715/9/1/001},
zblnumber = {0887.58029},
} -
[DKW] J. De Simoi, V. Kaloshin, and Q. Wei, "Dynamical spectral rigidity among $\Bbb Z_2$-symmetric strictly convex domains close to a circle," Ann. of Math. (2), vol. 186, iss. 1, pp. 277-314, 2017.
@ARTICLE{DKW,
author = {{De Simoi},
Jacopo and Kaloshin, Vadim and Wei, Qiaoling},
title = {Dynamical spectral rigidity among {$\Bbb Z_2$}-symmetric strictly convex domains close to a circle},
note = {Appendix B coauthored with H. Hezari},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {186},
year = {2017},
number = {1},
pages = {277--314},
issn = {0003-486X},
mrclass = {37D50 (35J05 35P05 35R30 58J53)},
mrnumber = {3665005},
doi = {10.4007/annals.2017.186.1.7},
url = {https://doi.org/10.4007/annals.2017.186.1.7},
zblnumber = {1377.37062},
} -
[GT] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001.
@BOOK{GT,
author = {Gilbarg, David and Trudinger, Neil S.},
title = {Elliptic Partial Differential Equations of Second Order},
series = {Classics in Math.},
note = {Reprint of the 1998 edition},
publisher = {Springer-Verlag, Berlin},
year = {2001},
pages = {xiv+517},
isbn = {3-540-41160-7},
mrclass = {35-02 (35Jxx)},
mrnumber = {1814364},
zblnumber = {1042.35002},
} -
[GordonWebbWolpert] C. Gordon, D. L. Webb, and S. Wolpert, "One cannot hear the shape of a drum," Bull. Amer. Math. Soc. (N.S.), vol. 27, iss. 1, pp. 134-138, 1992.
@ARTICLE{GordonWebbWolpert,
author = {Gordon, Carolyn and Webb, David L. and Wolpert, Scott},
title = {One cannot hear the shape of a drum},
journal = {Bull. Amer. Math. Soc. (N.S.)},
fjournal = {American Mathematical Society. Bulletin. New Series},
volume = {27},
year = {1992},
number = {1},
pages = {134--138},
issn = {0273-0979},
mrclass = {58G25 (35R30)},
mrnumber = {1136137},
mrreviewer = {Robert Brooks},
doi = {10.1090/S0273-0979-1992-00289-6},
url = {https://doi.org/10.1090/S0273-0979-1992-00289-6},
zblnumber = {0756.58049},
} -
[Grayson] M. A. Grayson, "Shortening embedded curves," Ann. of Math. (2), vol. 129, iss. 1, pp. 71-111, 1989.
@ARTICLE{Grayson,
author = {Grayson, Matthew A.},
title = {Shortening embedded curves},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {129},
year = {1989},
number = {1},
pages = {71--111},
issn = {0003-486X},
mrclass = {53C22 (58E10)},
mrnumber = {0979601},
mrreviewer = {Gudlaugur Thorbergsson},
doi = {10.2307/1971486},
url = {https://doi.org/10.2307/1971486},
zblnumber = {0686.53036},
} -
[GM] V. Guillemin and R. Melrose, "A cohomological invariant of discrete dynamical systems," in E. B. Christoffel, Birkhäuser, Boston, 1981, pp. 672-679.
@INCOLLECTION{GM,
author = {Guillemin, Victor and Melrose, Richard},
title = {A cohomological invariant of discrete dynamical systems},
booktitle = {E. {B}. {C}hristoffel},
venue = {{A}achen/{M}onschau, 1979},
pages = {672--679},
publisher = {Birkhäuser, Boston},
year = {1981},
mrclass = {58G25},
mrnumber = {0661107},
zblnumber = {0482.58032},
doi = {10.1007/978-3-0348-5452-8_53},
} -
[GK] V. Guillemin and D. Kazhdan, "Some inverse spectral results for negatively curved $2$-manifolds," Topology, vol. 19, iss. 3, pp. 301-312, 1980.
@ARTICLE{GK,
author = {Guillemin, V. and Kazhdan, D.},
title = {Some inverse spectral results for negatively curved {$2$}-manifolds},
journal = {Topology},
fjournal = {Topology. An International Journal of Mathematics},
volume = {19},
year = {1980},
number = {3},
pages = {301--312},
issn = {0040-9383},
mrclass = {58G25 (53C20)},
mrnumber = {0579579},
mrreviewer = {H. R. Gluck},
doi = {10.1016/0040-9383(80)90015-4},
url = {https://doi.org/10.1016/0040-9383(80)90015-4},
zblnumber = {0465.58027},
} -
[Gutkin] E. Gutkin, "Billiard dynamics: a survey with the emphasis on open problems," Regul. Chaotic Dyn., vol. 8, iss. 1, pp. 1-13, 2003.
@ARTICLE{Gutkin,
author = {Gutkin, E.},
title = {Billiard dynamics: a survey with the emphasis on open problems},
journal = {Regul. Chaotic Dyn.},
fjournal = {Regular \& Chaotic Dynamics. International Scientific Journal},
volume = {8},
year = {2003},
number = {1},
pages = {1--13},
issn = {1560-3547},
mrclass = {37D50 (37E99 37J10 70H99 82C05)},
mrnumber = {1963964},
mrreviewer = {Michael S. Farber},
doi = {10.1070/RD2003v008n01ABEH000222},
url = {https://doi.org/10.1070/RD2003v008n01ABEH000222},
zblnumber = {1023.37022},
} -
[Halpern] B. Halpern, "Strange billiard tables," Trans. Amer. Math. Soc., vol. 232, pp. 297-305, 1977.
@ARTICLE{Halpern,
author = {Halpern, Benjamin},
title = {Strange billiard tables},
journal = {Trans. Amer. Math. Soc.},
fjournal = {Transactions of the American Mathematical Society},
volume = {232},
year = {1977},
pages = {297--305},
issn = {0002-9947},
mrclass = {58F15 (34C35)},
mrnumber = {0451308},
mrreviewer = {L. A. Bunimovic},
doi = {10.2307/1998942},
url = {https://doi.org/10.2307/1998942},
zblnumber = {0374.53001},
} -
[HZ] H. Hezari and S. Zelditch, "Inverse spectral problem for analytic $(\Bbb Z/2\Bbb Z)^n$-symmetric domains in $\Bbb R^n$," Geom. Funct. Anal., vol. 20, iss. 1, pp. 160-191, 2010.
@ARTICLE{HZ,
author = {Hezari, Hamid and Zelditch, Steve},
title = {Inverse spectral problem for analytic {$(\Bbb Z/2\Bbb Z)^n$}-symmetric domains in {$\Bbb R^n$}},
journal = {Geom. Funct. Anal.},
fjournal = {Geometric and Functional Analysis},
volume = {20},
year = {2010},
number = {1},
pages = {160--191},
issn = {1016-443X},
mrclass = {58J53 (35P05 37D50)},
mrnumber = {2647138},
mrreviewer = {Andrew W. Hassell},
doi = {10.1007/s00039-010-0059-6},
url = {https://doi.org/10.1007/s00039-010-0059-6},
zblnumber = {1226.35055},
} -
[HKS2] G. Huang, V. Kaloshin, and A. Sorrentino, "On the marked length spectrum of generic strictly convex billiard tables," Duke Math. J., vol. 167, iss. 1, pp. 175-209, 2018.
@ARTICLE{HKS2,
author = {Huang, Guan and Kaloshin, Vadim and Sorrentino, Alfonso},
title = {On the marked length spectrum of generic strictly convex billiard tables},
journal = {Duke Math. J.},
fjournal = {Duke Mathematical Journal},
volume = {167},
year = {2018},
number = {1},
pages = {175--209},
issn = {0012-7094},
mrclass = {37D50 (35J05 35P05 37E40 37J50)},
mrnumber = {3743701},
doi = {10.1215/00127094-2017-0038},
url = {https://doi.org/10.1215/00127094-2017-0038},
zblnumber = {06847244},
} -
[Innami] N. Innami, "Geometry of geodesics for convex billiards and circular billiards," Nihonkai Math. J., vol. 13, iss. 1, pp. 73-120, 2002.
@ARTICLE{Innami,
author = {Innami, Nobuhiro},
title = {Geometry of geodesics for convex billiards and circular billiards},
journal = {Nihonkai Math. J.},
fjournal = {Nihonkai Mathematical Journal},
volume = {13},
year = {2002},
number = {1},
pages = {73--120},
issn = {1341-9951},
mrclass = {53D25 (37D50 37J10 53C22)},
mrnumber = {1907072},
mrreviewer = {Roberto Markarian},
zblnumber = {1035.37027},
URL = {https://projecteuclid.org/euclid.nihmj/1273779623},
} -
[Kac] M. Kac, "Can one hear the shape of a drum?," Amer. Math. Monthly, vol. 73, iss. 4, part II, pp. 1-23, 1966.
@ARTICLE{Kac,
author = {Kac, Mark},
title = {Can one hear the shape of a drum?},
journal = {Amer. Math. Monthly},
fjournal = {The American Mathematical Monthly},
volume = {73},
year = {1966},
number = {4, part II},
pages = {1--23},
issn = {0002-9890},
mrclass = {57.50 (00.00)},
mrnumber = {0201237},
mrreviewer = {I. Stakgold},
doi = {10.2307/2313748},
url = {https://doi.org/10.2307/2313748},
zblnumber = {0139.05603},
} -
[Lazutkin] V. F. Lazutkin, "Existence of caustics for the billiard problem in a convex domain," Izv. Akad. Nauk SSSR Ser. Mat., vol. 37, iss. 1, pp. 186-216, 1973.
@ARTICLE{Lazutkin,
author = {Lazutkin, V. F.},
title = {Existence of caustics for the billiard problem in a convex domain},
journal = {Izv. Akad. Nauk SSSR Ser. Mat.},
fjournal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
volume = {37},
year = {1973},
pages = {186--216},
issn = {0373-2436},
mrclass = {34C35 (58F99)},
mrnumber = {0328219},
mrreviewer = {C. Olech},
zblnumber = {0277.52002},
number = {1},
doi = {10.1070/IM1973v007n01ABEH001932},
} -
[MM] S. Marvizi and R. Melrose, "Spectral invariants of convex planar regions," J. Differential Geom., vol. 17, iss. 3, pp. 475-502, 1982.
@ARTICLE{MM,
author = {Marvizi, Shahla and Melrose, Richard},
title = {Spectral invariants of convex planar regions},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {17},
year = {1982},
number = {3},
pages = {475--502},
issn = {0022-040X},
mrclass = {58G25 (35P05)},
mrnumber = {0679068},
mrreviewer = {Hideo Soga},
zblnumber = {0492.53033},
doi = {10.4310/jdg/1214437138},
} -
[MM2] S. Marvizi and R. B. Melrose, "Some spectrally isolated convex planar regions," Proc. Nat. Acad. Sci. U.S.A., vol. 79, iss. 22, pp. 7066-7067, 1982.
@ARTICLE{MM2,
author = {Marvizi, Shahla and Melrose, Richard B.},
title = {Some spectrally isolated convex planar regions},
journal = {Proc. Nat. Acad. Sci. U.S.A.},
fjournal = {Proceedings of the National Academy of Sciences of the United States of America},
volume = {79},
year = {1982},
number = {22},
pages = {7066--7067},
issn = {0027-8424},
mrclass = {58G25 (35P99)},
mrnumber = {0678552},
zblnumber = {0504.53040},
doi = {10.1073/pnas.79.22.7066},
} -
[Mather82] J. N. Mather, "Glancing billiards," Ergodic Theory Dynam. Systems, vol. 2, iss. 3-4, pp. 397-403 (1983), 1982.
@ARTICLE{Mather82,
author = {Mather, John N.},
title = {Glancing billiards},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {2},
year = {1982},
number = {3-4},
pages = {397--403 (1983)},
issn = {0143-3857},
mrclass = {58F11 (34C35 54H20 58F05)},
mrnumber = {0721731},
mrreviewer = {M. I. Brin},
doi = {10.1017/S0143385700001681},
url = {https://doi.org/10.1017/S0143385700001681},
zblnumber = {0525.58021},
} -
[Mather90] J. N. Mather, "Differentiability of the minimal average action as a function of the rotation number," Bol. Soc. Brasil. Mat. (N.S.), vol. 21, iss. 1, pp. 59-70, 1990.
@ARTICLE{Mather90,
author = {Mather, John N.},
title = {Differentiability of the minimal average action as a function of the rotation number},
journal = {Bol. Soc. Brasil. Mat. (N.S.)},
fjournal = {Boletim da Sociedade Brasileira de Matemática. Nova Série},
volume = {21},
year = {1990},
number = {1},
pages = {59--70},
issn = {0100-3569},
mrclass = {58F11 (58F17 58F27)},
mrnumber = {1139556},
mrreviewer = {Helmut Rüssmann},
doi = {10.1007/BF01236280},
url = {https://doi.org/10.1007/BF01236280},
zblnumber = {0766.58033},
} -
[MatherForni] J. N. Mather and G. Forni, "Action minimizing orbits in Hamiltonian systems," in Transition to Chaos in Classical and Quantum Mechanics, Springer-Verlag, Berlin, 1994, vol. 1589, pp. 92-186.
@INCOLLECTION{MatherForni,
author = {Mather, John N. and Forni, Giovanni},
title = {Action minimizing orbits in {H}amiltonian systems},
booktitle = {Transition to Chaos in Classical and Quantum Mechanics},
venue = {{M}ontecatini {T}erme, 1991},
series = {Lecture Notes in Math.},
volume = {1589},
pages = {92--186},
publisher = {Springer-Verlag, Berlin},
year = {1994},
mrclass = {58F05 (58F27)},
mrnumber = {1323222},
mrreviewer = {F. Cardin},
doi = {10.1007/BFb0074076},
url = {https://doi.org/10.1007/BFb0074076},
zblnumber = {0822.70011},
} -
[Milnor] J. Milnor, "Eigenvalues of the Laplace operator on certain manifolds," Proc. Nat. Acad. Sci. U.S.A., vol. 51, p. 542, 1964.
@ARTICLE{Milnor,
author = {Milnor, J.},
title = {Eigenvalues of the {L}aplace operator on certain manifolds},
journal = {Proc. Nat. Acad. Sci. U.S.A.},
fjournal = {Proceedings of the National Academy of Sciences of the United States of America},
volume = {51},
year = {1964},
pages = {542},
issn = {0027-8424},
mrclass = {53.72 (57.50)},
mrnumber = {0162204},
mrreviewer = {J. Eells},
zblnumber = {0124.31202},
} -
[Mo] J. Moser, Selected Chapters in the Calculus of Variations, Birkhäuser Verlag, Basel, 2003.
@BOOK{Mo,
author = {Moser, Jürgen},
title = {Selected Chapters in the Calculus of Variations},
series = {Lectures in Math. ETH Zürich},
titlenote = {Lecture notes by Oliver Knill},
publisher = {Birkhäuser Verlag, Basel},
year = {2003},
pages = {iv+132},
isbn = {3-7643-2185-7},
mrclass = {49-02 (37E40 37J45 37J50 49Q20 58E05)},
mrnumber = {1988457},
mrreviewer = {H. R. Gluck},
doi = {10.1007/978-3-0348-8057-2},
url = {https://doi.org/10.1007/978-3-0348-8057-2},
zblnumber = {1045.37001},
} -
[Ot] J. Otal, "Le spectre marqué des longueurs des surfaces à courbure négative," Ann. of Math. (2), vol. 131, iss. 1, pp. 151-162, 1990.
@ARTICLE{Ot,
author = {Otal, Jean-Pierre},
title = {Le spectre marqué des longueurs des surfaces à courbure négative},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {131},
year = {1990},
number = {1},
pages = {151--162},
issn = {0003-486X},
mrclass = {58E10 (53C22 58F17 58G25)},
mrnumber = {1038361},
mrreviewer = {Carolyn Gordon},
doi = {10.2307/1971511},
url = {https://doi.org/10.2307/1971511},
zblnumber = {0699.58018},
} -
[OPS1] B. Osgood, R. Phillips, and P. Sarnak, "Compact isospectral sets of surfaces," J. Funct. Anal., vol. 80, iss. 1, pp. 212-234, 1988.
@ARTICLE{OPS1,
author = {Osgood, B. and Phillips, R. and Sarnak, P.},
title = {Compact isospectral sets of surfaces},
journal = {J. Funct. Anal.},
fjournal = {Journal of Functional Analysis},
volume = {80},
year = {1988},
number = {1},
pages = {212--234},
issn = {0022-1236},
mrclass = {58G25 (11F72 35P20 47F05 58G11 81E30)},
mrnumber = {0960229},
mrreviewer = {Andreas Juhl},
doi = {10.1016/0022-1236(88)90071-7},
url = {https://doi.org/10.1016/0022-1236(88)90071-7},
zblnumber = {0653.53021},
} -
[OPS2] B. Osgood, R. Phillips, and P. Sarnak, "Extremals of determinants of Laplacians," J. Funct. Anal., vol. 80, iss. 1, pp. 148-211, 1988.
@ARTICLE{OPS2,
author = {Osgood, B. and Phillips, R. and Sarnak, P.},
title = {Extremals of determinants of {L}aplacians},
journal = {J. Funct. Anal.},
fjournal = {Journal of Functional Analysis},
volume = {80},
year = {1988},
number = {1},
pages = {148--211},
issn = {0022-1236},
mrclass = {58G25 (11F72 47F05 81E30)},
mrnumber = {0960228},
mrreviewer = {Andreas Juhl},
doi = {10.1016/0022-1236(88)90070-5},
url = {https://doi.org/10.1016/0022-1236(88)90070-5},
zblnumber = {0653.53022},
} -
[OPS3] B. Osgood, R. Phillips, and P. Sarnak, "Moduli space, heights and isospectral sets of plane domains," Ann. of Math. (2), vol. 129, iss. 2, pp. 293-362, 1989.
@ARTICLE{OPS3,
author = {Osgood, B. and Phillips, R. and Sarnak, P.},
title = {Moduli space, heights and isospectral sets of plane domains},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {129},
year = {1989},
number = {2},
pages = {293--362},
issn = {0003-486X},
mrclass = {58G25 (35P99)},
mrnumber = {0986795},
mrreviewer = {Carolyn Gordon},
doi = {10.2307/1971449},
url = {https://doi.org/10.2307/1971449},
zblnumber = {0677.58045},
} -
[PR] S. Pinto-de-Carvalho and R. Ramírez-Ros, "Non-persistence of resonant caustics in perturbed elliptic billiards," Ergodic Theory Dynam. Systems, vol. 33, iss. 6, pp. 1876-1890, 2013.
@ARTICLE{PR,
author = {{Pinto-de-Carvalho},
Sônia and Ram{\'\i}rez-Ros, Rafael},
title = {Non-persistence of resonant caustics in perturbed elliptic billiards},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {33},
year = {2013},
number = {6},
pages = {1876--1890},
issn = {0143-3857},
mrclass = {37D50 (37E40 37J10)},
mrnumber = {3122155},
mrreviewer = {Ian Melbourne},
doi = {10.1017/S0143385712000417},
url = {https://doi.org/10.1017/S0143385712000417},
zblnumber = {06251103},
} -
[Popov] 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.
@ARTICLE{Popov,
author = {Popov, Georgi},
title = {Invariants of the length spectrum and spectral invariants of planar convex domains},
journal = {Comm. Math. Phys.},
fjournal = {Communications in Mathematical Physics},
volume = {161},
year = {1994},
number = {2},
pages = {335--364},
issn = {0010-3616},
mrclass = {58F19 (58G18 58G25)},
mrnumber = {1266488},
mrreviewer = {Edoh Amiran},
doi = {10.1007/BF02099782},
zblnumber = {0797.58070},
} -
[PT] G. Popov and P. Topalov, From KAM Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables, 2016.
@MISC{PT,
author = {Popov, Georgi and Topalov, Peter},
title = {From {KAM} Tori to Isospectral Invariants and Spectral Rigidity of Billiard Tables},
note = {preprint},
year = {2016},
zblnumber = {},
arxiv = {1602.03155},
} -
[Po] H. Poritsky, "The billiard ball problem on a table with a convex boundary—an illustrative dynamical problem," Ann. of Math. (2), vol. 51, pp. 446-470, 1950.
@ARTICLE{Po,
author = {Poritsky, Hillel},
title = {The billiard ball problem on a table with a convex boundary---an illustrative dynamical problem},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {51},
year = {1950},
pages = {446--470},
issn = {0003-486X},
mrclass = {46.3X},
mrnumber = {0032960},
mrreviewer = {D. C. Lewis},
doi = {10.2307/1969334},
url = {https://doi.org/10.2307/1969334},
zblnumber = {0037.26802},
} -
[RR] R. Ramírez-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.
@ARTICLE{RR,
author = {Ram{\'\i}rez-Ros, Rafael},
title = {Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables},
journal = {Phys. D},
fjournal = {Physica D. Nonlinear Phenomena},
volume = {214},
year = {2006},
number = {1},
pages = {78--87},
issn = {0167-2789},
mrclass = {37J10 (37E40 37J40)},
mrnumber = {2200796},
mrreviewer = {Daniel Massart},
doi = {10.1016/j.physd.2005.12.007},
url = {https://doi.org/10.1016/j.physd.2005.12.007},
zblnumber = {1099.37027},
} -
[SaTa] G. Sapiro and A. Tannenbaum, "On affine plane curve evolution," J. Funct. Anal., vol. 119, iss. 1, pp. 79-120, 1994.
@ARTICLE{SaTa,
author = {Sapiro, Guillermo and Tannenbaum, Allen},
title = {On affine plane curve evolution},
journal = {J. Funct. Anal.},
fjournal = {Journal of Functional Analysis},
volume = {119},
year = {1994},
number = {1},
pages = {79--120},
issn = {0022-1236},
mrclass = {58E10 (53A04)},
mrnumber = {1255274},
mrreviewer = {Anders Linnér},
doi = {10.1006/jfan.1994.1004},
url = {https://doi.org/10.1006/jfan.1994.1004},
zblnumber = {0801.53008},
} -
[Sarnak] P. Sarnak, "Determinants of Laplacians; heights and finiteness," in Analysis, et Cetera, Academic Press, Boston, MA, 1990, pp. 601-622.
@INCOLLECTION{Sarnak,
author = {Sarnak, Peter},
title = {Determinants of {L}aplacians; heights and finiteness},
booktitle = {Analysis, et {C}etera},
pages = {601--622},
publisher = {Academic Press, Boston, MA},
year = {1990},
mrclass = {58G26 (11F72 35P05 53C20)},
mrnumber = {1039364},
mrreviewer = {Carolyn Gordon},
zblnumber = {0703.53037},
doi = {10.1016/B978-0-12-574249-8.50033-X},
} -
[Siburg] K. F. Siburg, The Principle of Least Action in Geometry and Dynamics, Springer-Verlag, Berlin, 2004, vol. 1844.
@BOOK{Siburg,
author = {Siburg, Karl Friedrich},
title = {The Principle of Least Action in Geometry and Dynamics},
series = {Lecture Notes in Math.},
volume = {1844},
publisher = {Springer-Verlag, Berlin},
year = {2004},
pages = {xii+128},
isbn = {3-540-21944-7},
mrclass = {37J50 (37E40 37J10 53D35 58E30)},
mrnumber = {2076302},
mrreviewer = {Renato Iturriaga},
doi = {10.1007/b97327},
url = {https://doi.org/10.1007/b97327},
zblnumber = {1060.37048},
} -
[SorDCDS] A. Sorrentino, "Computing Mather’s $\beta$-function for Birkhoff billiards," Discrete Contin. Dyn. Syst., vol. 35, iss. 10, pp. 5055-5082, 2015.
@ARTICLE{SorDCDS,
author = {Sorrentino, Alfonso},
title = {Computing {M}ather's {$\beta$}-function for {B}irkhoff billiards},
journal = {Discrete Contin. Dyn. Syst.},
fjournal = {Discrete and Continuous Dynamical Systems. Series A},
volume = {35},
year = {2015},
number = {10},
pages = {5055--5082},
issn = {1078-0947},
mrclass = {37D50 (37E40 37J50)},
mrnumber = {3392661},
doi = {10.3934/dcds.2015.35.5055},
url = {https://doi.org/10.3934/dcds.2015.35.5055},
zblnumber = {1359.37088},
} -
[SorLecNotes] A. Sorrentino, Action-Minimizing Methods in Hamiltonian Dynamics: An Introduction to Aubry-Mather Theory, Princeton Univ. Press, Princeton, NJ, 2015, vol. 50.
@BOOK{SorLecNotes,
author = {Sorrentino, Alfonso},
title = {Action-Minimizing Methods in {H}amiltonian Dynamics: {A}n {I}ntroduction to Aubry-Mather {T}heory},
series = {Math. Notes},
volume = {50},
publisher = {Princeton Univ. Press, Princeton, NJ},
year = {2015},
pages = {xii+115},
isbn = {978-0-691-16450-2},
mrclass = {70H08 (37J30 37J35 37J50 70-01)},
mrnumber = {3330134},
mrreviewer = {Héctor Sánchez-Morgado},
doi = {10.1515/9781400866618},
url = {https://doi.org/10.1515/9781400866618},
zblnumber = {1373.37002},
} -
[Tabach1] S. Tabachnikov, "Billiards," Panor. Synth., iss. 1, p. vi, 1995.
@ARTICLE{Tabach1,
author = {Tabachnikov, Serge},
title = {Billiards},
journal = {Panor. Synth.},
fjournal = {Panoramas et Synthèses},
number = {1},
year = {1995},
pages = {vi+142},
mrclass = {58F17 (58F11 58F20)},
mrnumber = {1328336},
mrreviewer = {Roberto Markarian},
zblnumber = {0833.58001},
} -
[Tabach] S. Tabachnikov, Geometry and Billiards, Amer. Math. Soc., Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005, vol. 30.
@BOOK{Tabach,
author = {Tabachnikov, Serge},
title = {Geometry and Billiards},
series = {Student Math. Library},
volume = {30},
publisher = {Amer. Math. Soc., Providence, RI; Mathematics Advanced Study Semesters, University Park, PA},
year = {2005},
pages = {xii+176},
isbn = {0-8218-3919-5},
mrclass = {51-02 (37-01 37D50 37J10 70H05 82C05)},
mrnumber = {2168892},
mrreviewer = {Roberto Markarian},
doi = {10.1090/stml/030},
url = {https://doi.org/10.1090/stml/030},
zblnumber = {1119.37001},
} -
[Taba] M. B. Tabanov, "New ellipsoidal confocal coordinates and geodesics on an ellipsoid," J. Math. Sci., vol. 82, iss. 6, pp. 3851-3858, 1996.
@ARTICLE{Taba,
author = {Tabanov, M. B.},
title = {New ellipsoidal confocal coordinates and geodesics on an ellipsoid},
note = {Algebra, 3},
journal = {J. Math. Sci.},
fjournal = {Journal of Mathematical Sciences},
volume = {82},
year = {1996},
number = {6},
pages = {3851--3858},
issn = {1072-3374},
mrclass = {53A07 (58F17)},
mrnumber = {1431551},
mrreviewer = {Edoh Amiran},
doi = {10.1007/BF02362647},
url = {https://doi.org/10.1007/BF02362647},
zblnumber = {0889.58062},
} -
[Tre] D. Treschev, "Billiard map and rigid rotation," Phys. D, vol. 255, pp. 31-34, 2013.
@ARTICLE{Tre,
author = {Treschev, D.},
title = {Billiard map and rigid rotation},
journal = {Phys. D},
fjournal = {Physica D. Nonlinear Phenomena},
volume = {255},
year = {2013},
pages = {31--34},
issn = {0167-2789},
mrclass = {37D50 (37J35 37M05)},
mrnumber = {3064868},
mrreviewer = {Marco Lenci},
doi = {10.1016/j.physd.2013.04.003},
url = {https://doi.org/10.1016/j.physd.2013.04.003},
zblnumber = {06278202},
} -
[Woi] M. P. Wojtkowski, "Two applications of Jacobi fields to the billiard ball problem," J. Differential Geom., vol. 40, iss. 1, pp. 155-164, 1994.
@ARTICLE{Woi,
author = {Wojtkowski, Maciej P.},
title = {Two applications of {J}acobi fields to the billiard ball problem},
journal = {J. Differential Geom.},
fjournal = {Journal of Differential Geometry},
volume = {40},
year = {1994},
number = {1},
pages = {155--164},
issn = {0022-040X},
mrclass = {58F22 (53C22)},
mrnumber = {1285532},
mrreviewer = {Valery Covachev},
doi = {10.4310/jdg/1214455290},
zblnumber = {0812.58067},
} -
[Zelditch] S. Zelditch, "Spectral determination of analytic bi-axisymmetric plane domains," Geom. Funct. Anal., vol. 10, iss. 3, pp. 628-677, 2000.
@ARTICLE{Zelditch,
author = {Zelditch, S.},
title = {Spectral determination of analytic bi-axisymmetric plane domains},
journal = {Geom. Funct. Anal.},
fjournal = {Geometric and Functional Analysis},
volume = {10},
year = {2000},
number = {3},
pages = {628--677},
issn = {1016-443X},
mrclass = {58J50 (35J05 35P99 35R30 58J37)},
mrnumber = {1779616},
mrreviewer = {Georgi Popov},
doi = {10.1007/PL00001633},
url = {https://doi.org/10.1007/PL00001633},
zblnumber = {0961.58012},
} -
[HKS] G. Huang, V. Kaloshin, and A. Sorrentino, "Nearly circular domains which are integrable close to the boundary are ellipses," Geom. Funct. Anal., vol. 28, iss. 2, pp. 334-392, 2018.
@article{HKS,
author={Huang, Guan and Kaloshin, Vadim and Sorrentino, Alfonso},
title = {Nearly circular domains which are integrable close to the boundary are ellipses},
journal = {Geom. Funct. Anal.},
fjournal = {Geometric and Functional Analysis},
volume = {28},
year = {2018},
number = {2},
pages = {334--392},
issn = {1016-443X},
mrclass = {37D50},
mrnumber = {3788206},
doi = {10.1007/s00039-018-0440-4},
url = {https://doi.org/10.1007/s0039-018-0440-4},
}
Full Article