Dynamical spectral rigidity among $\mathbb{Z}_2$-symmetric strictly convex domains close to a circle (Appendix B coauthored with H. Hezari)

Abstract

We show that any sufficiently (finitely) smooth $\mathbb{Z}_2$-symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid; i.e., all deformations among domains in the same class that preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak.

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  • [AM] Go to document 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},
      }
  • [ADK] Go to document A. Avila, J. De Simoi, and V. Kaloshin, "An integrable deformation of an ellipse of small eccentricity is an ellipse," Ann. of Math., vol. 184, iss. 2, pp. 527-558, 2016.
    @ARTICLE{ADK, mrkey = {3548532},
      number = {2},
      issn = {0003-486X},
      author = {Avila, Artur and De Simoi, Jacopo and Kaloshin, Vadim},
      mrclass = {37D50},
      doi = {10.4007/annals.2016.184.2.5},
      journal = {Ann. of Math.},
      zblnumber = {06662219},
      volume = {184},
      mrnumber = {3548532},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {Nicolas Bedaride},
      title = {An integrable deformation of an ellipse of small eccentricity is an ellipse},
      year = {2016},
      pages = {527--558},
      }
  • [Chaza] Go to document 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},
      }
  • [Cr] Go to document C. B. Croke, "Rigidity for surfaces of nonpositive curvature," Comment. Math. Helv., vol. 65, iss. 1, pp. 150-169, 1990.
    @ARTICLE{Cr, mrkey = {1036134},
      number = {1},
      issn = {0010-2571},
      author = {Croke, Christopher B.},
      mrclass = {53C20 (53C22 58F17)},
      doi = {10.1007/BF02566599},
      journal = {Comment. Math. Helv.},
      zblnumber = {0704.53035},
      volume = {65},
      mrnumber = {1036134},
      fjournal = {Commentarii Mathematici Helvetici},
      mrreviewer = {Chi-Keung Cheung},
      title = {Rigidity for surfaces of nonpositive curvature},
      year = {1990},
      pages = {150--169},
      }
  • [CS] Go to document C. B. Croke and V. A. Sharafutdinov, "Spectral rigidity of a compact negatively curved manifold," Topology, vol. 37, iss. 6, pp. 1265-1273, 1998.
    @ARTICLE{CS, mrkey = {1632920},
      number = {6},
      issn = {0040-9383},
      author = {Croke, Christopher B. and Sharafutdinov, Vladimir A.},
      mrclass = {58G25 (53C20)},
      doi = {10.1016/S0040-9383(97)00086-4},
      journal = {Topology},
      zblnumber = {0936.58013},
      volume = {37},
      mrnumber = {1632920},
      fjournal = {Topology. An International Journal of Mathematics},
      mrreviewer = {Carolyn Gordon},
      title = {Spectral rigidity of a compact negatively curved manifold},
      year = {1998},
      pages = {1265--1273},
      }
  • [dMM] Go to document R. de la Llave, J. M. Marco, and R. Moriyón, "Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation," Ann. of Math., vol. 123, iss. 3, pp. 537-611, 1986.
    @ARTICLE{dMM, mrkey = {0840722},
      number = {3},
      issn = {0003-486X},
      author = {de la Llave, R. and Marco, J. M. and Moriyón, R.},
      mrclass = {58F15 (34C35 58F30)},
      doi = {10.2307/1971334},
      journal = {Ann. of Math.},
      zblnumber = {0603.58016},
      volume = {123},
      mrnumber = {0840722},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {Tong Ren Ding},
      title = {Canonical perturbation theory of {A}nosov systems and regularity results for the {L}ivšic cohomology equation},
      year = {1986},
      pages = {537--611},
      }
  • [Dui] Go to document 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},
      }
  • [GWW] Go to document 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},
      title = {One cannot hear the shape of a drum},
      year = {1992},
      pages = {134--138},
      }
  • [GKaz] Go to document V. Guillemin and D. Kazhdan, "Some inverse spectral results for negatively curved $2$-manifolds," Topology, vol. 19, iss. 3, pp. 301-312, 1980.
    @ARTICLE{GKaz, mrkey = {0579579},
      number = {3},
      issn = {0040-9383},
      author = {Guillemin, V. and Kazhdan, D.},
      mrclass = {58G25 (53C20)},
      doi = {10.1016/0040-9383(80)90015-4},
      journal = {Topology},
      zblnumber = {0465.58027},
      volume = {19},
      mrnumber = {0579579},
      fjournal = {Topology. An International Journal of Mathematics},
      mrreviewer = {H. R. Gluck},
      title = {Some inverse spectral results for negatively curved {$2$}-manifolds},
      year = {1980},
      pages = {301--312},
      }
  • [Halpern] Go to document B. Halpern, "Strange billiard tables," Trans. Amer. Math. Soc., vol. 232, pp. 297-305, 1977.
    @ARTICLE{Halpern, mrkey = {0451308},
      issn = {0002-9947},
      author = {Halpern, Benjamin},
      mrclass = {58F15 (34C35)},
      doi = {10.2307/1998942},
      journal = {Trans. Amer. Math. Soc.},
      zblnumber = {0374.53001},
      volume = {232},
      mrnumber = {0451308},
      fjournal = {Transactions of the American Mathematical Society},
      mrreviewer = {L. A. Bunimovic},
      title = {Strange billiard tables},
      year = {1977},
      pages = {297--305},
      }
  • [He] H. Hezari, Robin spectral rigidity of strictly convex domains with a reflectional symmetry, 2016.
    @MISC{He,
      author = {Hezari, H.},
      arxiv = {1609.01011},
      title = {Robin spectral rigidity of strictly convex domains with a reflectional symmetry},
      year = {2016},
      }
  • [HZ] Go to document 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 = {{$C^\infty$} spectral rigidity of the ellipse},
      year = {2012},
      pages = {1105--1132},
      }
  • [Ka] Go to document 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},
      }
  • [Lazutkin] Go to document V. F. Lazutkin, "Existence of caustics for the billiard problem in a convex domain," Izv. Akad. Nauk SSSR Ser. Mat., vol. 37, pp. 186-216, 1973.
    @ARTICLE{Lazutkin, 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 = {Existence of caustics for the billiard problem in a convex domain},
      year = {1973},
      pages = {186--216},
      zblnumber = {0256.52001},
      doi = {10.1070/IM1973v007n01ABEH001932},
     }
  • [OPS2] Go to document B. Osgood, R. Phillips, and P. Sarnak, "Compact isospectral sets of surfaces," J. Funct. Anal., vol. 80, iss. 1, pp. 212-234, 1988.
    @ARTICLE{OPS2, mrkey = {0960229},
      number = {1},
      issn = {0022-1236},
      author = {Osgood, B. and Phillips, R. and Sarnak, P.},
      mrclass = {58G25 (11F72 35P20 47F05 58G11 81E30)},
      doi = {10.1016/0022-1236(88)90071-7},
      journal = {J. Funct. Anal.},
      zblnumber = {0653.53021},
      volume = {80},
      mrnumber = {0960229},
      fjournal = {Journal of Functional Analysis},
      mrreviewer = {Andreas Juhl},
      title = {Compact isospectral sets of surfaces},
      year = {1988},
      pages = {212--234},
      }
  • [OPS1] Go to document B. Osgood, R. Phillips, and P. Sarnak, "Extremals of determinants of Laplacians," J. Funct. Anal., vol. 80, iss. 1, pp. 148-211, 1988.
    @ARTICLE{OPS1, mrkey = {0960228},
      number = {1},
      issn = {0022-1236},
      author = {Osgood, B. and Phillips, R. and Sarnak, P.},
      mrclass = {58G25 (11F72 47F05 81E30)},
      doi = {10.1016/0022-1236(88)90070-5},
      journal = {J. Funct. Anal.},
      zblnumber = {0653.53022},
      volume = {80},
      mrnumber = {0960228},
      fjournal = {Journal of Functional Analysis},
      mrreviewer = {Andreas Juhl},
      title = {Extremals of determinants of {L}aplacians},
      year = {1988},
      pages = {148--211},
      }
  • [OPS3] Go to document B. Osgood, R. Phillips, and P. Sarnak, "Moduli space, heights and isospectral sets of plane domains," Ann. of Math., vol. 129, iss. 2, pp. 293-362, 1989.
    @ARTICLE{OPS3, mrkey = {0986795},
      number = {2},
      issn = {0003-486X},
      author = {Osgood, B. and Phillips, R. and Sarnak, P.},
      mrclass = {58G25 (35P99)},
      doi = {10.2307/1971449},
      journal = {Ann. of Math.},
      zblnumber = {0677.58045},
      volume = {129},
      mrnumber = {0986795},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {Carolyn Gordon},
      title = {Moduli space, heights and isospectral sets of plane domains},
      year = {1989},
      pages = {293--362},
      }
  • [Ot] Go to document J. Otal, "Le spectre marqué des longueurs des surfaces à courbure négative," Ann. of Math., vol. 131, iss. 1, pp. 151-162, 1990.
    @ARTICLE{Ot, mrkey = {1038361},
      number = {1},
      issn = {0003-486X},
      author = {Otal, Jean-Pierre},
      mrclass = {58E10 (53C22 58F17 58G25)},
      doi = {10.2307/1971511},
      journal = {Ann. of Math.},
      zblnumber = {0699.58018},
      volume = {131},
      mrnumber = {1038361},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {Carolyn Gordon},
      title = {Le spectre marqué des longueurs des surfaces à courbure négative},
      year = {1990},
      pages = {151--162},
      }
  • [PS] V. M. Petkov and L. N. Stoyanov, Geometry of Reflecting Rays and Inverse Spectral Problems, New York: John Wiley & Sons, Ltd., 1992.
    @BOOK{PS, mrkey = {1172998},
      author = {Petkov, Vesselin M. and Stoyanov, Luchezar N.},
      mrclass = {58G25 (35R30 58G17 78A05)},
      series = {Pure Appl. Math.},
      isbn = {0-471-93174-8},
      address = {New York},
      publisher = {John Wiley \& Sons, Ltd.},
      zblnumber = {0761.35077},
      mrnumber = {1172998},
      mrreviewer = {Hideo Soga},
      title = {Geometry of Reflecting Rays and Inverse Spectral Problems},
      year = {1992},
      pages = {vi+313},
      }
  • [PT1] Go to document G. Popov and P. Topalov, "Invariants of isospectral deformations and spectral rigidity," Comm. Partial Differential Equations, vol. 37, iss. 3, pp. 369-446, 2012.
    @ARTICLE{PT1, mrkey = {2889558},
      number = {3},
      issn = {0360-5302},
      author = {Popov, Georgi and Topalov, Peter},
      mrclass = {47A10 (35P05 35R01 37J40 58J53)},
      doi = {10.1080/03605302.2011.641051},
      journal = {Comm. Partial Differential Equations},
      zblnumber = {1245.58015},
      volume = {37},
      mrnumber = {2889558},
      fjournal = {Communications in Partial Differential Equations},
      mrreviewer = {Yuri A. Kordyukov},
      title = {Invariants of isospectral deformations and spectral rigidity},
      year = {2012},
      pages = {369--446},
      }
  • [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, Peter},
      note = {preprint},
      title = {From {K.A.M. T}ori to Isospectral Invariants and Spectral Rigidity of Billiard Tables},
      year = {2016},
      arxiv = {1602.03155},
      }
  • [Sa] Go to document 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},
      doi = {10.1016/B978-0-12-574249-8.50033-X},
     }
  • [Sunada] Go to document 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},
      title = {Riemannian coverings and isospectral manifolds},
      year = {1985},
      pages = {169--186},
      }
  • [Vi] Go to document 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 = {Vignéras, 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},
      title = {Variétés riemanniennes isospectrales et non isométriques},
      year = {1980},
      pages = {21--32},
      }
  • [ZelditchSurvey04] Go to document S. Zelditch, "The inverse spectral problem," in Surveys in Differential Geometry. Vol. IX, Somerville, MA: Int. Press, 2004, vol. 9, pp. 401-467.
    @INCOLLECTION{ZelditchSurvey04, mrkey = {2195415},
      author = {Zelditch, Steve},
      mrclass = {58J53 (35P20 35R30 58J50)},
      series = {Surv. Differ. Geom.},
      address = {Somerville, MA},
      publisher = {Int. Press},
      doi = {10.4310/SDG.2004.v9.n1.a12},
      zblnumber = {1061.58029},
      volume = {9},
      mrnumber = {2195415},
      booktitle = {Surveys in Differential Geometry. {V}ol. {IX}},
      mrreviewer = {David Borthwick},
      title = {The inverse spectral problem},
      pages = {401--467},
      year = {2004},
      }
  • [ZelditchAnalytic] Go to document S. Zelditch, "Inverse spectral problem for analytic domains. II. $\Bbb Z_2$-symmetric domains," Ann. of Math., vol. 170, iss. 1, pp. 205-269, 2009.
    @ARTICLE{ZelditchAnalytic, mrkey = {2521115},
      number = {1},
      issn = {0003-486X},
      author = {Zelditch, Steve},
      mrclass = {58J53 (35P20 35R30 37D50)},
      doi = {10.4007/annals.2009.170.205},
      journal = {Ann. of Math.},
      zblnumber = {1196.58016},
      volume = {170},
      mrnumber = {2521115},
      fjournal = {Annals of Mathematics. Second Series},
      mrreviewer = {David Borthwick},
      title = {Inverse spectral problem for analytic domains. {II}. {$\Bbb Z_2$}-symmetric domains},
      year = {2009},
      pages = {205--269},
      }
  • [ZelditchSurvey15] Go to document S. Zelditch, "Survey on the inverse spectral problem," ICCM Not., vol. 2, iss. 2, pp. 1-20, 2014.
    @ARTICLE{ZelditchSurvey15, mrkey = {3314780},
      number = {2},
      issn = {2326-4810},
      author = {Zelditch, Steve},
      mrclass = {58J50 (58J53)},
      doi = {10.4310/ICCM.2014.v2.n2.a1},
      journal = {ICCM Not.},
      volume = {2},
      mrnumber = {3314780},
      fjournal = {ICCM Notices. Notices of the International Congress of Chinese Mathematicians},
      mrreviewer = {Ruth Gornet},
      title = {Survey on the inverse spectral problem},
      year = {2014},
      pages = {1--20},
      }

Authors

Jacopo De Simoi

University of Toronto, Toronto, ON, Canada

Vadim Kaloshin

University of Maryland, College Park, MD

Qiaoling Wei

Capital Normal University, Beijing, PR China