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 show that a version of this conjecture is true for tables bounded by small perturbations of ellipses of small eccentricity.
-
[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, mrkey = {0494322},
number = {3},
issn = {0020-9910},
author = {Andersson, K. G. and Melrose, R. B.},
mrclass = {58G15 (35S15)},
doi = {10.1007/BF01403048},
journal = {Invent. Math.},
zblnumber = {0373.35053},
volume = {41},
mrnumber = {0494322},
fjournal = {Inventiones Mathematicae},
mrreviewer = {Bent E. Petersen},
title = {The propagation of singularities along gliding rays},
year = {1977},
pages = {197--232},
} -
[Bi] M. Bialy, "Convex billiards and a theorem by E. Hopf," Math. Z., vol. 214, iss. 1, pp. 147-154, 1993.
@ARTICLE{Bi, mrkey = {1234604},
number = {1},
issn = {0025-5874},
author = {Bialy, Misha},
mrclass = {58F11 (58F15)},
doi = {10.1007/BF02572397},
journal = {Math. Z.},
zblnumber = {0790.58023},
volume = {214},
mrnumber = {1234604},
fjournal = {Mathematische Zeitschrift},
mrreviewer = {Andr{á}s Kr{á}mli},
coden = {MAZEAX},
title = {Convex billiards and a theorem by {E}. {H}opf},
year = {1993},
pages = {147--154},
} -
[Bf] G. D. Birkhoff, Dynamical Systems, Providence, RI: Amer. Math. Soc., 1966, vol. IX.
@BOOK{Bf, mrkey = {0209095},
author = {Birkhoff, George D.},
mrclass = {00.00 (70.00)},
series = {Amer. Math. Soc. Colloq. Publ.},
address = {Providence, RI},
publisher = {Amer. Math. Soc.},
volume = {IX},
mrnumber = {0209095},
note = {with an addendum by Jurgen Moser},
title = {Dynamical Systems},
year = {1966},
pages = {xii+305},
zblnumber = {0171.05402},
} -
[Bu] L. A. Bunimovich, "On absolutely focusing mirrors," in Ergodic Theory and Related Topics, III, New York: Springer, 1992, vol. 1514, pp. 62-82.
@INCOLLECTION{Bu, mrkey = {1179172},
author = {Bunimovich, L. A.},
title = {On absolutely focusing mirrors},
BOOKTITLE = {Ergodic {T}heory and {R}elated {T}opics, {III}},
VENUE={{G}üstrow, 1990},
SERIES = {Lecture Notes in Math.},
VOLUME = {1514},
PAGES = {62--82},
PUBLISHER = {Springer},
ADDRESS = {New York},
YEAR = {1992},
MRCLASS = {58F30 (58F11)},
MRNUMBER = {1179172},
MRREVIEWER = {Valery Covachev},
DOI = {10.1007/BFb0097528},
ZBLNUMBER={0772.58027},
} -
[Chaza] J. Chazarain, "Formule de Poisson pour les variétés riemanniennes," Invent. Math., vol. 24, pp. 65-82, 1974.
@ARTICLE{Chaza, mrkey = {0343320},
issn = {0020-9910},
author = {Chazarain, J.},
mrclass = {58G15 (35P05 58E10)},
doi = {10.1007/BF01418788},
journal = {Invent. Math.},
zblnumber = {0281.35028},
volume = {24},
mrnumber = {0343320},
fjournal = {Inventiones Mathematicae},
mrreviewer = {R. S. Palais},
title = {Formule de {P}oisson pour les variétés riemanniennes},
year = {1974},
pages = {65--82},
} -
[Dui] 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.
@ARTICLE{Dui, mrkey = {0405514},
number = {1},
issn = {0020-9910},
author = {Duistermaat, J. J. and Guillemin, V. W.},
mrclass = {58G99 (35P05)},
doi = {10.1007/BF01405172},
journal = {Invent. Math.},
zblnumber = {0307.35071},
volume = {29},
mrnumber = {0405514},
fjournal = {Inventiones Mathematicae},
mrreviewer = {Alan Weinstein},
title = {The spectrum of positive elliptic operators and periodic bicharacteristics},
year = {1975},
pages = {39--79},
} -
[GT] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, New York: Springer-Verlag, 2001.
@BOOK{GT, mrkey = {1814364},
author = {Gilbarg, David and Trudinger, Neil S.},
mrclass = {35-02 (35Jxx)},
series = {Classics in Math.},
address = {New York},
isbn = {3-540-41160-7},
publisher = {Springer-Verlag},
zblnumber = {1042.35002},
mrnumber = {1814364},
note = {reprint of the 1998 edition},
title = {Elliptic Partial Differential Equations of Second Order},
year = {2001},
pages = {xiv+517},
} -
[GWW] 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.
@ARTICLE{GWW, mrkey = {1136137},
number = {1},
issn = {0273-0979},
author = {Gordon, Carolyn and Webb, David L. and Wolpert, Scott},
mrclass = {58G25 (35R30)},
doi = {10.1090/S0273-0979-1992-00289-6},
journal = {Bull. Amer. Math. Soc.},
zblnumber = {0756.58049},
volume = {27},
mrnumber = {1136137},
fjournal = {American Mathematical Society. Bulletin. New Series},
mrreviewer = {Robert Brooks},
coden = {BAMOAD},
title = {One cannot hear the shape of a drum},
year = {1992},
pages = {134--138},
} -
[GM1] V. Guillemin and R. Melrose, "An inverse spectral result for elliptical regions in ${\bf R}^{2}$," Adv. in Math., vol. 32, iss. 2, pp. 128-148, 1979.
@ARTICLE{GM1, mrkey = {0535619},
number = {2},
issn = {0001-8708},
author = {Guillemin, Victor and Melrose, Richard},
mrclass = {35P99 (58G25)},
doi = {10.1016/0001-8708(79)90039-2},
journal = {Adv. in Math.},
zblnumber = {0415.35062},
volume = {32},
mrnumber = {0535619},
fjournal = {Advances in Mathematics},
mrreviewer = {V. M. Babich},
coden = {ADMTA4},
title = {An inverse spectral result for elliptical regions in {${\bf R}\sp{2}$}},
year = {1979},
pages = {128--148},
} -
[Gu] E. Gutkin, "Billiard dynamics: An updated survey with the emphasis on open problems," Chaos, vol. 22, iss. 2, p. 026116, 2012.
@ARTICLE{Gu, mrkey = {3388585},
number = {2},
issn = {1054-1500},
author = {Gutkin, Eugene},
mrclass = {37D50},
doi = {10.1063/1.4729307},
journal = {Chaos},
volume = {22},
mrnumber = {3388585},
fjournal = {Chaos. An Interdisciplinary Journal of Nonlinear Science},
title = {Billiard dynamics: {A}n updated survey with the emphasis on open problems},
year = {2012},
pages = {026116, 13},
zblnumber = {06550458},
} -
[HZ] H. Hezari and S. Zelditch, "$C^\infty$ spectral rigidity of the ellipse," Anal. PDE, vol. 5, iss. 5, pp. 1105-1132, 2012.
@ARTICLE{HZ, mrkey = {3022850},
number = {5},
issn = {2157-5045},
author = {Hezari, Hamid and Zelditch, Steve},
mrclass = {35P05},
doi = {10.2140/apde.2012.5.1105},
journal = {Anal. PDE},
zblnumber = {1264.35150},
volume = {5},
mrnumber = {3022850},
fjournal = {Analysis \& PDE},
mrreviewer = {J. B. Kennedy},
title = {{$C\sp \infty$} spectral rigidity of the ellipse},
year = {2012},
pages = {1105--1132},
} -
[Ka] M. Kac, "Can one hear the shape of a drum?," Amer. Math. Monthly, vol. 73, iss. 4, part II, pp. 1-23, 1966.
@ARTICLE{Ka, mrkey = {0201237},
number = {4, part II},
issn = {0002-9890},
author = {Kac, Mark},
mrclass = {57.50 (00.00)},
doi = {10.2307/2313748},
journal = {Amer. Math. Monthly},
zblnumber = {0139.05603},
volume = {73},
mrnumber = {0201237},
fjournal = {The American Mathematical Monthly},
mrreviewer = {I. Stakgold},
title = {Can one hear the shape of a drum?},
year = {1966},
pages = {1--23},
} -
[L] 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.
@ARTICLE{L, mrkey = {0328219},
issn = {0373-2436},
author = {Lazutkin, V. F.},
mrclass = {34C35 (58F99)},
journal = {Izv. Akad. Nauk SSSR Ser. Mat.},
volume = {37},
mrnumber = {0328219},
fjournal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
mrreviewer = {C. Olech},
title = {The existence of caustics for a billiard problem in a convex domain},
year = {1973},
pages = {186--216},
zblnumber = {0256.52001},
doi = {10.1070/IM1973v007n01ABEH001932},
} -
[PR] 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.
@ARTICLE{PR, mrkey = {3122155},
number = {6},
issn = {0143-3857},
author = {Pinto-de-Carvalho, S{ô}nia and Ram{\'ı}rez-Ros, Rafael},
mrclass = {37D50 (37E40 37J10)},
doi = {10.1017/S0143385712000417},
journal = {Ergodic Theory Dynam. Systems},
zblnumber = {06251103},
volume = {33},
mrnumber = {3122155},
fjournal = {Ergodic Theory and Dynamical Systems},
mrreviewer = {Ian Melbourne},
title = {Non-persistence of resonant caustics in perturbed elliptic billiards},
year = {2013},
pages = {1876--1890},
} -
[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, mrkey = {1266488},
number = {2},
issn = {0010-3616},
author = {Popov, Georgi},
mrclass = {58F19 (58G18 58G25)},
journal = {Comm. Math. Phys.},
zblnumber = {0797.58070},
volume = {161},
mrnumber = {1266488},
fjournal = {Communications in Mathematical Physics},
mrreviewer = {Edoh Amiran},
coden = {CMPHAY},
title = {Invariants of the length spectrum and spectral invariants of planar convex domains},
year = {1994},
pages = {335--364},
doi = {10.1007/BF02099782},
} -
[PT] G. Popov and P. Topalov, From K.A.M. tori to isospectral invariants and spectral rigidity of billiard tables, 2016.
@MISC{PT,
author = {Popov, Georgi and Topalov, P.},
title = {From {K.A.M.} tori to isospectral invariants and spectral rigidity of billiard tables},
year = {2016},
arxiv= {1602.03155},
} -
[Po] 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.
@ARTICLE{Po, mrkey = {0032960},
issn = {0003-486X},
author = {Poritsky, Hillel},
mrclass = {46.3X},
doi = {10.2307/1969334},
journal = {Ann. of Math.},
zblnumber = {0037.26802},
volume = {51},
mrnumber = {0032960},
fjournal = {Annals of Mathematics. Second Series},
mrreviewer = {D. C. Lewis},
title = {The billiard ball problem on a table with a convex boundary---an illustrative dynamical problem},
year = {1950},
pages = {446--470},
} -
[RR] 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.
@ARTICLE{RR, mrkey = {2200796},
number = {1},
issn = {0167-2789},
author = {Ram{\'ı}rez-Ros, Rafael},
mrclass = {37J10 (37E40 37J40)},
doi = {10.1016/j.physd.2005.12.007},
journal = {Phys. D},
zblnumber = {1099.37027},
volume = {214},
mrnumber = {2200796},
fjournal = {Physica D. Nonlinear Phenomena},
mrreviewer = {Daniel Massart},
coden = {PDNPDT},
title = {Break-up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables},
year = {2006},
pages = {78--87},
} -
[Sa] P. Sarnak, "Determinants of Laplacians; heights and finiteness," in Analysis, et cetera, Boston: Academic Press, 1990, pp. 601-622.
@INCOLLECTION{Sa, mrkey = {1039364},
author = {Sarnak, Peter},
mrclass = {58G26 (11F72 35P05 53C20)},
address = {Boston},
publisher = {Academic Press},
zblnumber = {0703.53037},
mrnumber = {1039364},
booktitle = {Analysis, et cetera},
mrreviewer = {Carolyn Gordon},
title = {Determinants of {L}aplacians; heights and finiteness},
pages = {601--622},
year = {1990},
} -
[Sunada] T. Sunada, "Riemannian coverings and isospectral manifolds," Ann. of Math., vol. 121, iss. 1, pp. 169-186, 1985.
@ARTICLE{Sunada, mrkey = {0782558},
number = {1},
issn = {0003-486X},
author = {Sunada, Toshikazu},
mrclass = {58G25 (11F72 53C20 57M99)},
doi = {10.2307/1971195},
journal = {Ann. of Math.},
zblnumber = {0585.58047},
volume = {121},
mrnumber = {0782558},
fjournal = {Annals of Mathematics. Second Series},
mrreviewer = {Scott Wolpert},
coden = {ANMAAH},
title = {Riemannian coverings and isospectral manifolds},
year = {1985},
pages = {169--186},
} -
[Taba] S. Tabachnikov, Geometry and Billiards, Providence, RI: Amer. Math. Soc., 2005, vol. 30.
@BOOK{Taba, mrkey = {2168892},
author = {Tabachnikov, Serge},
mrclass = {51-02 (37-01 37D50 37J10 70H05 82C05)},
series = {Student Math. Lib.},
isbn = {0-8218-3919-5},
address = {Providence, RI},
publisher = {Amer. Math. Soc.},
doi = {10.1090/stml/030},
zblnumber = {1119.37001},
volume = {30},
mrnumber = {2168892},
mrreviewer = {Roberto Markarian},
title = {Geometry and Billiards},
year = {2005},
pages = {xii+176},
} -
[Ta] 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.
@ARTICLE{Ta, mrkey = {1431551},
number = {6},
issn = {1072-3374},
author = {Tabanov, M. B.},
mrclass = {53A07 (58F17)},
doi = {10.1007/BF02362647},
journal = {J. Math. Sci.},
zblnumber = {0889.58062},
volume = {82},
mrnumber = {1431551},
fjournal = {Journal of Mathematical Sciences},
mrreviewer = {Edoh Amiran},
coden = {JMTSEW},
title = {New ellipsoidal confocal coordinates and geodesics on an ellipsoid. {A}lgebra, 3},
year = {1996},
pages = {3851--3858},
} -
[Tre] D. Treschev, "Billiard map and rigid rotation," Phys. D, vol. 255, pp. 31-34, 2013.
@ARTICLE{Tre, mrkey = {3064868},
issn = {0167-2789},
author = {Treschev, D.},
mrclass = {37D50 (37J35 37M05)},
doi = {10.1016/j.physd.2013.04.003},
journal = {Phys. D},
zblnumber = {06278202},
volume = {255},
mrnumber = {3064868},
fjournal = {Physica D. Nonlinear Phenomena},
mrreviewer = {Marco Lenci},
title = {Billiard map and rigid rotation},
year = {2013},
pages = {31--34},
} -
[Vi] M. Vignéras, "Variétés riemanniennes isospectrales et non isométriques," Ann. of Math., vol. 112, iss. 1, pp. 21-32, 1980.
@ARTICLE{Vi, mrkey = {0584073},
number = {1},
issn = {0003-486X},
author = {Vign{é}ras, Marie-France},
mrclass = {58G25 (10J25 12A70)},
doi = {10.2307/1971319},
journal = {Ann. of Math.},
zblnumber = {0445.53026},
volume = {112},
mrnumber = {0584073},
fjournal = {Annals of Mathematics. Second Series},
mrreviewer = {P. Wintgen},
coden = {ANMAAH},
title = {Variétés riemanniennes isospectrales et non isométriques},
year = {1980},
pages = {21--32},
} -
[Woj] 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{Woj, mrkey = {1285532},
number = {1},
issn = {0022-040X},
author = {Wojtkowski, Maciej P.},
mrclass = {58F22 (53C22)},
url = {http://projecteuclid.org/euclid.jdg/1214455290},
journal = {J. Differential Geom.},
zblnumber = {0812.58067},
volume = {40},
mrnumber = {1285532},
fjournal = {Journal of Differential Geometry},
mrreviewer = {Valery Covachev},
coden = {JDGEAS},
title = {Two applications of {J}acobi fields to the billiard ball problem},
year = {1994},
pages = {155--164},
}
Full Article