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:
[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

show all references

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

[1]

Michel L. Lapidus, Robert G. Niemeyer. Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3719-3740. doi: 10.3934/dcds.2013.33.3719

[2]

Dmitri Scheglov. Growth of periodic orbits and generalized diagonals for typical triangular billiards. Journal of Modern Dynamics, 2013, 7 (1) : 31-44. doi: 10.3934/jmd.2013.7.31

[3]

Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331

[4]

Daniel Genin, Serge Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. Journal of Modern Dynamics, 2007, 1 (2) : 155-173. doi: 10.3934/jmd.2007.1.155

[5]

Daniel Franco, J. R. L. Webb. Collisionless orbits of singular and nonsingular dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 747-757. doi: 10.3934/dcds.2006.15.747

[6]

Chao Ma, Baowei Wang, Jun Wu. Diophantine approximation of the orbits in topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2455-2471. doi: 10.3934/dcds.2019104

[7]

Richard Evan Schwartz. Unbounded orbits for outer billiards I. Journal of Modern Dynamics, 2007, 1 (3) : 371-424. doi: 10.3934/jmd.2007.1.371

[8]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109

[9]

P.E. Kloeden, Desheng Li, Chengkui Zhong. Uniform attractors of periodic and asymptotically periodic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 213-232. doi: 10.3934/dcds.2005.12.213

[10]

Flaviano Battelli, Ken Palmer. Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 367-387. doi: 10.3934/dcdsb.2010.14.367

[11]

Chen-Chang Peng, Kuan-Ju Chen. Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1181-1197. doi: 10.3934/dcdsb.2010.14.1181

[12]

Daniel Genin. Research announcement: Boundedness of orbits for trapezoidal outer billiards. Electronic Research Announcements, 2008, 15: 71-78. doi: 10.3934/era.2008.15.71

[13]

Richard Evan Schwartz. Research announcement: unbounded orbits for outer billiards. Electronic Research Announcements, 2007, 14: 1-6. doi: 10.3934/era.2007.14.1

[14]

Misha Bialy. Maximizing orbits for higher-dimensional convex billiards. Journal of Modern Dynamics, 2009, 3 (1) : 51-59. doi: 10.3934/jmd.2009.3.51

[15]

Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59

[16]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807

[17]

B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217

[18]

Gabriele Benedetti, Kai Zehmisch. On the existence of periodic orbits for magnetic systems on the two-sphere. Journal of Modern Dynamics, 2015, 9: 141-146. doi: 10.3934/jmd.2015.9.141

[19]

Armengol Gasull, Héctor Giacomini, Maite Grau. On the stability of periodic orbits for differential systems in $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 495-509. doi: 10.3934/dcdsb.2008.10.495

[20]

Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1177-1199. doi: 10.3934/dcds.2013.33.1177

2018 Impact Factor: 0.295

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]