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
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show all references

References:
[1]

Nonlinearity, 22 (2009), 1247-1258. doi: 10.1088/0951-7715/22/6/001.  Google Scholar

[2]

Math. Z., 65 (1956), 327-344. doi: 10.1007/BF01473886.  Google Scholar

[3]

Dokl. Akad. Nauk SSSR, 230 (1976), 1017-1020.  Google Scholar

[4]

J. Math. Sciences, 128 (2005), 2706-2710. doi: 10.1007/s10958-005-0220-1.  Google Scholar

[5]

Actualités Sci. Ind., No. 994, Hermann et Cie., Paris, 1945.  Google Scholar

[6]

Math. Z., 7 (1920), 1-57. doi: 10.1007/BF01199396.  Google Scholar

[7]

Invent. Math., 2 (1975), 39-79. doi: 10.1007/BF01405172.  Google Scholar

[8]

(Russian) Funktsional. Anal. i Prilozhen., 44 (2010), 54-64; translation in Funct. Anal. Appl., 44 (2010), 286-294.  Google Scholar

[9]

Acta Math., 127 (1971), 79-183. doi: 10.1007/BF02392052.  Google Scholar

[10]

Acta Math., 121 (1968), 193-218.  Google Scholar

[11]

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]

American Journal of Mathematics, 79 (1957), 1-47. doi: 10.2307/2372692.  Google Scholar

[14]

Erg. Theor. & Dyn. Sys., 8 (1988), 81-91. doi: 10.1017/S0143385700004338.  Google Scholar

[15]

Nonlinearity, 24 (2011), 847-854. doi: 10.1088/0951-7715/24/3/007.  Google Scholar

[16]

OGIZ, Moscow-Leningrad, 1947.  Google Scholar

[17]

J. Diff. Geom., 30 (1989), 191-205.  Google Scholar

[18]

in "Integral Equations and Inverse Problems" (Varna, 1989), Pitman Res. Notes Math. Ser., 235, Longman Sci. Tech., Harlow, (1991), 239-240.  Google Scholar

[19]

Adv. Math., 29 (1978), 244-269. doi: 10.1016/0001-8708(78)90013-0.  Google Scholar

[20]

J. Differential Geom., 34 (1991), 835-837.  Google Scholar

[21]

(Russian) Funktsional. Anal. i Prilozhen., 18 (1984), 1-13, 96; English translation, Functional Anal. Appl., 18 (1984), 267-277.  Google Scholar

[22]

(Russian) Dokl. Akad. Nauk SSSR, 286 (1986), 1043-1046; English translation, Soviet Math. Dokl., 33 (1986), 227-230.  Google Scholar

[23]

Math. Notes, 55 (1994), 455-460. doi: 10.1007/BF02110371.  Google Scholar

[24]

Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, (1911), 110-117. Google Scholar

[25]

J. Differential Geom., 40 (1994), 155-164.  Google Scholar

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