# American Institute of Mathematical Sciences

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

## No planar billiard possesses an open set of quadrilateral trajectories

 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.
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:
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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. 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. 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. Google Scholar [4] Y. Baryshnikov and V. Zharnitsky, Billiards and nonholonomic distributions,, J. Math. Sciences, 128 (2005), 2706. doi: 10.1007/s10958-005-0220-1. Google Scholar [5] É. Cartan, "Les Systèmes Différentiels Extérieurs et Leur Applications Géométriques,", Actualités Sci. Ind., (1945). Google Scholar [6] R. Courant, Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik,, Math. Z., 7 (1920), 1. 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. 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. Google Scholar [9] L. Hörmander, Fourier integral operators. I,, Acta Math., 127 (1971), 79. doi: 10.1007/BF02392052. Google Scholar [10] L. Hörmander, The spectral function of an elliptic operator,, Acta Math., 121 (1968), 193. 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. 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. 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. doi: 10.1017/S0143385700004338. Google Scholar [15] A. Plakhov and V. Roshchina, Invisibility in billiards,, Nonlinearity, 24 (2011), 847. doi: 10.1088/0951-7715/24/3/007. Google Scholar [16] P. K. Raševskiĭ, "Geometrical Theory of Partial Differential Equations,", OGIZ, (1947). Google Scholar [17] M. R. Rychlik, Periodic points of the billiard ball map in a convex domain,, J. Diff. Geom., 30 (1989), 191. Google Scholar [18] Yu. Safarov, Precise spectral asymptotics and inverse problems,, in, 235 (1991), 239. 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. 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. 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. 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. Google Scholar [23] Ya. B. Vorobets, On the measure of the set of periodic points of a billiard,, Math. Notes, 55 (1994), 455. doi: 10.1007/BF02110371. Google Scholar [24] Hermann Weyl, "Über die asymptotische Verteilung der Eigenwerte,", Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, (1911), 110. Google Scholar [25] M. P. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem,, J. Differential Geom., 40 (1994), 155. Google Scholar
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