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No planar billiard possesses an open set of quadrilateral trajectories

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  • The article is devoted to a particular case of Ivriĭ's conjecture on periodic orbits of billiards. The general conjecture states that the set of periodic orbits of the billiard in a domain with smooth boundary in the Euclidean space has measure zero. In this article we prove that for any domain with piecewise $C^4$-smooth boundary in the plane the set of quadrilateral trajectories of the corresponding billiard has measure zero.
    Mathematics Subject Classification: Primary: 58F22; Secondary: 34C25.

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  • [1]

    A. Aleksenko and A. Plakhov, Bodies of zero resistance and bodies invisible in one direction, Nonlinearity, 22 (2009), 1247-1258.doi: 10.1088/0951-7715/22/6/001.

    [2]

    V. I. Avakumovi ć, Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten, Math. Z., 65 (1956), 327-344.doi: 10.1007/BF01473886.

    [3]

    V. Babich and B. Levitan, The focussing problem and the asymptotics of the spectral function of the Laplace-Beltrami operator, Dokl. Akad. Nauk SSSR, 230 (1976), 1017-1020.

    [4]

    Y. Baryshnikov and V. Zharnitsky, Billiards and nonholonomic distributions, J. Math. Sciences, 128 (2005), 2706-2710.doi: 10.1007/s10958-005-0220-1.

    [5]

    É. Cartan, "Les Systèmes Différentiels Extérieurs et Leur Applications Géométriques," Actualités Sci. Ind., No. 994, Hermann et Cie., Paris, 1945.

    [6]

    R. Courant, Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik, Math. Z., 7 (1920), 1-57.doi: 10.1007/BF01199396.

    [7]

    J. J. Duistermaat and V. W. Guilleman, The spectrum of positive elliptic operators and periodic bi-characteristics, Invent. Math., 2 (1975), 39-79.doi: 10.1007/BF01405172.

    [8]

    N. Filonov and Y. Safarov, Asymptotic estimates for the difference between the Dirichlet and Neumann counting functions, (Russian) Funktsional. Anal. i Prilozhen., 44 (2010), 54-64; translation in Funct. Anal. Appl., 44 (2010), 286-294.

    [9]

    L. Hörmander, Fourier integral operators. I, Acta Math., 127 (1971), 79-183.doi: 10.1007/BF02392052.

    [10]

    L. Hörmander, The spectral function of an elliptic operator, Acta Math., 121 (1968), 193-218.

    [11]

    V. Y. Ivriĭ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary, Func. Anal. Appl., 14 (1980), 98-106.doi: 10.1007/BF01086550.

    [12]

    V. Y. IvriĭEverything started from Weyl, presentation slides, http://weyl.math.toronto.edu:8888/victor2/preprints/GradTalk.pdf

    [13]

    M. Kuranishi, On E. Cartan's prolongation theorem of exterior differential systems, American Journal of Mathematics, 79 (1957), 1-47.doi: 10.2307/2372692.

    [14]

    V. Petkov and L. Stojanov, On the number of periodic reflecting rays in generic domains, Erg. Theor. & Dyn. Sys., 8 (1988), 81-91.doi: 10.1017/S0143385700004338.

    [15]

    A. Plakhov and V. Roshchina, Invisibility in billiards, Nonlinearity, 24 (2011), 847-854.doi: 10.1088/0951-7715/24/3/007.

    [16]

    P. K. Raševskiĭ, "Geometrical Theory of Partial Differential Equations," OGIZ, Moscow-Leningrad, 1947.

    [17]

    M. R. Rychlik, Periodic points of the billiard ball map in a convex domain, J. Diff. Geom., 30 (1989), 191-205.

    [18]

    Yu. Safarov, Precise spectral asymptotics and inverse problems, in "Integral Equations and Inverse Problems" (Varna, 1989), Pitman Res. Notes Math. Ser., 235, Longman Sci. Tech., Harlow, (1991), 239-240.

    [19]

    R. Seeley, A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain in $\mathbb R^3$, Adv. Math., 29 (1978), 244-269.doi: 10.1016/0001-8708(78)90013-0.

    [20]

    L. Stojanov, Note on the periodic points of the billiard, J. Differential Geom., 34 (1991), 835-837.

    [21]

    D. Vasil'ev, Two-term asymptotics of the spectrum of a boundary value problem in interior reflection of general form, (Russian) Funktsional. Anal. i Prilozhen., 18 (1984), 1-13, 96; English translation, Functional Anal. Appl., 18 (1984), 267-277.

    [22]

    D. Vasil'ev, Two-term asymptotics of the spectrum of a boundary value problem in the case of a piecewise smooth boundary, (Russian) Dokl. Akad. Nauk SSSR, 286 (1986), 1043-1046; English translation, Soviet Math. Dokl., 33 (1986), 227-230.

    [23]

    Ya. B. Vorobets, On the measure of the set of periodic points of a billiard, Math. Notes, 55 (1994), 455-460.doi: 10.1007/BF02110371.

    [24]

    Hermann Weyl, "Über die asymptotische Verteilung der Eigenwerte," Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, (1911), 110-117.

    [25]

    M. P. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom., 40 (1994), 155-164.

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