No planar billiard possesses an open set of quadrilateral trajectories

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July 2012, 6(3): 287-326. doi: 10.3934/jmd.2012.6.287

No planar billiard possesses an open set of quadrilateral trajectories

Alexey Glutsyuk ^1, and Yury Kudryashov ^2,

1.	CNRS, Unité de Mathématiques Pures et Appliquées, M.R., École Normale Supérieure de Lyon, 46 allée d’Italie, 69364, Lyon 07, France
2.	National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russian Federation

Received January 2011 Revised May 2012 Published October 2012

Abstract
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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.

Keywords: billiards, periodic orbits., Dynamical systems.

Mathematics Subject Classification: Primary: 58F22; Secondary: 34C2.

Citation: Alexey Glutsyuk, Yury Kudryashov. No planar billiard possesses an open set of quadrilateral trajectories. Journal of Modern Dynamics, 2012, 6 (3) : 287-326. doi: 10.3934/jmd.2012.6.287

References:

[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. Ivriĭ, Everything started from Weyl,, presentation slides, ().
[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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References:

[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. Ivriĭ, Everything started from Weyl,, presentation slides, ().
[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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