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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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[12]

V. Y. Ivriĭ, Everything started from Weyl,, presentation slides, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

show all references

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[12]

V. Y. Ivriĭ, Everything started from Weyl,, presentation slides, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

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