-
Previous Article
The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$
- JMD Home
- This Issue
- Next Article
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 |
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. |
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. |
[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. |
[1] |
Armengol Gasull, Víctor Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 651-670. doi: 10.3934/dcdsb.2019259 |
[2] |
Michel L. Lapidus, Robert G. Niemeyer. Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3719-3740. doi: 10.3934/dcds.2013.33.3719 |
[3] |
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 |
[4] |
Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331 |
[5] |
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 |
[6] |
Daniel Franco, J. R. L. Webb. Collisionless orbits of singular and nonsingular dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 747-757. doi: 10.3934/dcds.2006.15.747 |
[7] |
Chao Ma, Baowei Wang, Jun Wu. Diophantine approximation of the orbits in topological dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2455-2471. doi: 10.3934/dcds.2019104 |
[8] |
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 |
[9] |
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 |
[10] |
Francesco Fassò, Simone Passarella, Marta Zoppello. Control of locomotion systems and dynamics in relative periodic orbits. Journal of Geometric Mechanics, 2020, 12 (3) : 395-420. doi: 10.3934/jgm.2020022 |
[11] |
P.E. Kloeden, Desheng Li, Chengkui Zhong. Uniform attractors of periodic and asymptotically periodic dynamical systems. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 213-232. doi: 10.3934/dcds.2005.12.213 |
[12] |
Flaviano Battelli, Ken Palmer. Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 367-387. doi: 10.3934/dcdsb.2010.14.367 |
[13] |
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 |
[14] |
Richard Evan Schwartz. Research announcement: unbounded orbits for outer billiards. Electronic Research Announcements, 2007, 14: 1-6. doi: 10.3934/era.2007.14.1 |
[15] |
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 |
[16] |
Chen-Chang Peng, Kuan-Ju Chen. Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1181-1197. doi: 10.3934/dcdsb.2010.14.1181 |
[17] |
Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59 |
[18] |
Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807 |
[19] |
Jan Bouwe van den Berg, Elena Queirolo. A general framework for validated continuation of periodic orbits in systems of polynomial ODEs. Journal of Computational Dynamics, 2021, 8 (1) : 59-97. doi: 10.3934/jcd.2021004 |
[20] |
B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217 |
2020 Impact Factor: 0.848
Tools
Metrics
Other articles
by authors
[Back to Top]