October  2017, 37(10): 5163-5190. doi: 10.3934/dcds.2017224

Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability

1. 

Mathematical Institute SANU, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000 Belgrade, Serbia

2. 

Faculty of Sciences, University of Banja Luka, Mladena Stojanovića 2, 51000 Banja Luka, Bosnia and Herzegovina

* Corresponding author

Received  October 2016 Revised  May 2017 Published  June 2017

The aim of the paper is to unify the efforts in the study of integrable billiards within quadrics in flat and curved spaces and to explore further the interplay of symplectic and contact integrability. As a starting point in this direction, we consider virtual billiard dynamics within quadrics in pseudo-Euclidean spaces. In contrast to the usual billiards, the incoming velocity and the velocity after the billiard reflection can be at opposite sides of the tangent plane at the reflection point. In the symmetric case we prove noncommutative integrability of the system and give a geometrical interpretation of integrals, an analog of the classical Chasles and Poncelet theorems and we show that the virtual billiard dynamics provides a natural framework in the study of billiards within quadrics in projective spaces, in particular of billiards within ellipsoids on the sphere ${\mathbb{S}^{n - 1}}$ and the Lobachevsky space $\mathbb H^{n-1}$.

Citation: Božidar Jovanović, Vladimir Jovanović. Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5163-5190. doi: 10.3934/dcds.2017224
References:
[1]

V. I. Arnol'd, Matematicheskie Metody Klassicheskoy Mehaniki Moskva, Nauka 1974 (Russian). English translation: V. I. Arnol'd, Mathematical Methods of Classical Mechanics Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989.

[2]

M. Audin, Courbes algébriques et systémes intégrables: Géodésiques des quadriques, Expo. Math., 12 (1994), 193-226. 

[3]

A. Avila, J. De Simoi and V. Kaloshin, An integrable deformation of an ellipse of small eccentricity is an ellipse, Annals of Mathematics, 184 (2016), 527-558, arXiv: 1412.2853. doi: 10.4007/annals.2016.184.2.5.

[4]

A. Banyaga and P. Molino, Géométrie des formes de contact complétement intégrables de type torique, Séminare Gaston Darboux, Montpellier (1991-92), 1-25 (French). English translation: Complete integrability in contact geometry, Penn State preprint PM 197,1996.

[5]

M. Bialy and A. E. Mironov, Angular billiard and algebraic Birkhoff conjecture, Adv. Math., 313 (2017), 102-126, arXiv: 1601.03196. doi: 10.1016/j.aim.2017.04.001.

[6]

M. Bialy and A. E. Mironov, Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane, Journal of Geometry and Physics, 115 (2017), 150-156, arXiv: 1602.05698. doi: 10.1016/j.geomphys.2016.04.017.

[7]

S. V. Bolotin, Integriruemye bil'yardy na poverhnostyah postoyannoy krivizny, Mat. Zametki, 51 (1992), 20-28, (Russian); English translation: S. V. Bolotin, Integrable billiards on constant curvature surfaces, Math. Notes, Math. Notes, 51 (1992), 117-123. doi: 10.1007/BF02102114.

[8]

V. Dragović, B. Jovanović and M. Radnović, On elliptic billiards in the Lobachevsky space and associated geodesic hierarchies, J. Geom. Phys., 47 (2003), 221-234, arXiv: math-ph/0210019. doi: 10.1016/S0393-0440(02)00219-X.

[9]

V. Dragović and M. Radnović, Geometry of integrable billiards and pencils of quadrics, J. Math. Pures Appl., 85 (2006), 758-790, arXiv: math-ph/0512049. doi: 10.1016/j.matpur.2005.12.002.

[10]

V. Dragović and M. Radnović, Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: beyond Poncelet porisms, Adv. Math., 219 (2008), 1577-1607, arXiv: 0710.3656. doi: 10.1016/j.aim.2008.06.021.

[11]

V. Dragović and M. Radnović, Integriruemye billiardy, kvadriki i mnogomernye porizmy Ponsele, Moskva-Izhevsk, Regulyarnaya i hoiticheskaya dinamika, 2010. (Russian); English translation: V. Dragović and M. Radnović, Poncelet Porisms and Beyond. Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics, Frontiers in Mathematics. Birkhauser/Springer Basel AG, Basel, 2011.

[12]

V. Dragović and M. Radnović, Ellipsoidal billiards in pseudo-euclidean spaces and relativistic quadrics, Adv. Math., 231 (2012), 1173-1201, arXiv: 1108.4552. doi: 10.1016/j.aim.2012.06.004.

[13]

V. Dragović and M. Radnović, Billiard algebra, integrable line congruences, and double reflection nets, Journal of Nonlinear Mathematical Physics 19 (2012), 1250019, 18pp, arXiv: 1112.5860. doi: 10.1142/S1402925112500192.

[14]

V. Dragović and M. Radnović, Bicentennial of the great Poncelet theorem (1813-2013): Current advances, Bulletin AMS, 51 (2014), 373-445, arXiv: 1212.6867. doi: 10.1090/S0273-0979-2014-01437-5.

[15]

Yu. N. Fedorov, Integrable systems, Lax representation and confocal quadrics, Dynamical systems in classical mechanics, Amer. Math. Soc. Transl.(2), 168 (1995), 173-199.  doi: 10.1090/trans2/168/07.

[16]

Yu. N. Fedorov, Ellipsoidal'ny billiard s kvadratichnym potentsialom, Funkc. analiz i ego prilozh., 35 (2001), 48-59, 95-96 (Russian); English translation: Yu. N. Fedorov, An ellipsoidal billiard with quadratic potential, Funct. Anal. Appl., 35 (2001), 199-208. doi: 10.1023/A:1012326828456.

[17]

Yu. N. Fedorov and B. Jovanović, Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi-Mumford systems, preprint, arXiv: 1503.07053.

[18]

A. Glutsyuk, On quadrilateral orbits in complex algebraic planar billiards, Moscow Math. J., 14 (2014), 239-289, arXiv: 1309.1843.

[19]

B. Jovanović, Noncommutative integrability and action angle variables in contact geometry, Journal of Symplectic Geometry, 10 (2012), 535-561, arXiv: 1103.3611. doi: 10.4310/JSG.2012.v10.n4.a3.

[20]

B. Jovanović, The Jacobi-Rosochatius problem on an ellipsoid: The Lax representations and billiards, Arch. Rational Mech. Anal. 210 (2013), 101-131, arXiv: 1303.6204. doi: 10.1007/s00205-013-0638-4.

[21]

B. Jovanović, Heisenberg model in pseudo-Euclidean spaces, Regular and Chaotic Dynamics, 19 (2014), 245-250, arXiv: 1405.0905. doi: 10.1134/S1560354714020075.

[22]

B. Jovanović and V. Jovanović, Contact flows and integrable systems, Journal of Geometry and Physics, 87 (2015), 217-232, arXiv: 1212.2918. doi: 10.1016/j.geomphys.2014.07.030.

[23]

B. Jovanović and V. Jovanović, Geodesic and billiard flows on quadrics in pseudo-Euclidean spaces: L-A pairs and Chasles theorem, International Mathematics Research Notices, 15 (2015), 6618-6638, arXiv: 1407.0555. doi: 10.1093/imrn/rnu141.

[24]

B. Khesin and S. Tabachnikov, Pseudo-Riemannian geodesics and billiards, Adv. Math., 221 (2009), 1364-1396, arXiv: math/0608620. doi: 10.1016/j.aim.2009.02.010.

[25]

B. Khesin and S. Tabachnikov, Contact complete integrability, Regular and Chaotic Dynamics, 15 (2010), 504-520, arXiv: 0910.0375. doi: 10.1134/S1560354710040076.

[26]

V. V. Kozlov i D. V. Treshchev, Billiardy, Geneticheskoe vvedenie v dinamiku sistem s udarami, Izd-vo Mosk. un-ta, Moskva, 1991. English translation: V. V. Kozlov and D. V. Treshchev, Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts Transl. Math. Monogr., 89, Amer. Math. Soc., Providence, RI, 1991.

[27]

P. Libermann and C. Marle, Symplectic Geometry and Analytical Mechanics Riedel, Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.

[28]

A. S. Mishchenko and A. T. Fomenko, Obobshchenny metod Liuvillya integrirovaniya gamiltonovyh sistem, Funkc. Analiz i ego Prilozh., 12 (1978), 46-56 (Russian); English translation: A. S. Mishchenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121.

[29]

J. Moser, Geometry of quadric and spectral theory, Chern Symposium 1979, Berlin-Heidelberg-New York, (1980), 147-188.

[30]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243.  doi: 10.1007/BF02352494.

[31]

N. N. Nekhoroshev, Peremennye deystvie-ugol i ih obobshcheniya Tr. Mosk. Mat. O.-va., 26 (1972), 181-198 (Russian). English translation: N. N. Nekhoroshev, Action-angle variables and their generalization, Trans. Mosc. Math. Soc., 26 (1972), 180-198.

[32]

M. Radnović, Topology of the elliptical billiard with the Hooke's potential, Theoretical and Applied Mehanics, 42 (2015), 1-9, arXiv: 1508.01025.

[33]

Yu. B. Suris, The Problem of Integrable Discretization: Hamiltonian Approach Progress in Mathematics, 219. Birkhäuser Verlag, Basel, 2003. doi: 10.1007/978-3-0348-8016-9.

[34]

S. L. Tabachnikov, Ellipsoids, complete integrability and hyperbolic geometry, Mosc. Math. J., 2 (2002), 183-196. 

[35]

S. Tabachnikov, Geometry and Billiards volume 30 of Student Mathematical Library. American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005. doi: 10.1090/stml/030.

[36]

A. P. Veselov, Integriruemye sistemy s diskretnym vremenem i raznostnye operatory, Funkc. Analiz i ego Prilozh. 22 (1988), 1-13 (Russian); English translation: A. P. Veselov, Integrable systems with discrete time, and difference operators, Funct. Anal. Appl. 22 (1988), 83-93. doi: 10.1007/BF01077598.

[37]

A. P. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107.  doi: 10.1016/0393-0440(90)90021-T.

[38]

A. P. Veselov, Integriruemye otobrazheniya, Uspekhi Mat. Nauk, 46 (1991), 3-45 (Russian); English translation: A. P. Veselov, Integrable mappings, Russ. Math. Surv., 46 (1991), 1-51.

[39]

A. P. Veselov, Growth and integrability in the dynamics of mappings, Comm. Math. Phys., 145 (1992), 181-193.  doi: 10.1007/BF02099285.

show all references

References:
[1]

V. I. Arnol'd, Matematicheskie Metody Klassicheskoy Mehaniki Moskva, Nauka 1974 (Russian). English translation: V. I. Arnol'd, Mathematical Methods of Classical Mechanics Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989.

[2]

M. Audin, Courbes algébriques et systémes intégrables: Géodésiques des quadriques, Expo. Math., 12 (1994), 193-226. 

[3]

A. Avila, J. De Simoi and V. Kaloshin, An integrable deformation of an ellipse of small eccentricity is an ellipse, Annals of Mathematics, 184 (2016), 527-558, arXiv: 1412.2853. doi: 10.4007/annals.2016.184.2.5.

[4]

A. Banyaga and P. Molino, Géométrie des formes de contact complétement intégrables de type torique, Séminare Gaston Darboux, Montpellier (1991-92), 1-25 (French). English translation: Complete integrability in contact geometry, Penn State preprint PM 197,1996.

[5]

M. Bialy and A. E. Mironov, Angular billiard and algebraic Birkhoff conjecture, Adv. Math., 313 (2017), 102-126, arXiv: 1601.03196. doi: 10.1016/j.aim.2017.04.001.

[6]

M. Bialy and A. E. Mironov, Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane, Journal of Geometry and Physics, 115 (2017), 150-156, arXiv: 1602.05698. doi: 10.1016/j.geomphys.2016.04.017.

[7]

S. V. Bolotin, Integriruemye bil'yardy na poverhnostyah postoyannoy krivizny, Mat. Zametki, 51 (1992), 20-28, (Russian); English translation: S. V. Bolotin, Integrable billiards on constant curvature surfaces, Math. Notes, Math. Notes, 51 (1992), 117-123. doi: 10.1007/BF02102114.

[8]

V. Dragović, B. Jovanović and M. Radnović, On elliptic billiards in the Lobachevsky space and associated geodesic hierarchies, J. Geom. Phys., 47 (2003), 221-234, arXiv: math-ph/0210019. doi: 10.1016/S0393-0440(02)00219-X.

[9]

V. Dragović and M. Radnović, Geometry of integrable billiards and pencils of quadrics, J. Math. Pures Appl., 85 (2006), 758-790, arXiv: math-ph/0512049. doi: 10.1016/j.matpur.2005.12.002.

[10]

V. Dragović and M. Radnović, Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: beyond Poncelet porisms, Adv. Math., 219 (2008), 1577-1607, arXiv: 0710.3656. doi: 10.1016/j.aim.2008.06.021.

[11]

V. Dragović and M. Radnović, Integriruemye billiardy, kvadriki i mnogomernye porizmy Ponsele, Moskva-Izhevsk, Regulyarnaya i hoiticheskaya dinamika, 2010. (Russian); English translation: V. Dragović and M. Radnović, Poncelet Porisms and Beyond. Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics, Frontiers in Mathematics. Birkhauser/Springer Basel AG, Basel, 2011.

[12]

V. Dragović and M. Radnović, Ellipsoidal billiards in pseudo-euclidean spaces and relativistic quadrics, Adv. Math., 231 (2012), 1173-1201, arXiv: 1108.4552. doi: 10.1016/j.aim.2012.06.004.

[13]

V. Dragović and M. Radnović, Billiard algebra, integrable line congruences, and double reflection nets, Journal of Nonlinear Mathematical Physics 19 (2012), 1250019, 18pp, arXiv: 1112.5860. doi: 10.1142/S1402925112500192.

[14]

V. Dragović and M. Radnović, Bicentennial of the great Poncelet theorem (1813-2013): Current advances, Bulletin AMS, 51 (2014), 373-445, arXiv: 1212.6867. doi: 10.1090/S0273-0979-2014-01437-5.

[15]

Yu. N. Fedorov, Integrable systems, Lax representation and confocal quadrics, Dynamical systems in classical mechanics, Amer. Math. Soc. Transl.(2), 168 (1995), 173-199.  doi: 10.1090/trans2/168/07.

[16]

Yu. N. Fedorov, Ellipsoidal'ny billiard s kvadratichnym potentsialom, Funkc. analiz i ego prilozh., 35 (2001), 48-59, 95-96 (Russian); English translation: Yu. N. Fedorov, An ellipsoidal billiard with quadratic potential, Funct. Anal. Appl., 35 (2001), 199-208. doi: 10.1023/A:1012326828456.

[17]

Yu. N. Fedorov and B. Jovanović, Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi-Mumford systems, preprint, arXiv: 1503.07053.

[18]

A. Glutsyuk, On quadrilateral orbits in complex algebraic planar billiards, Moscow Math. J., 14 (2014), 239-289, arXiv: 1309.1843.

[19]

B. Jovanović, Noncommutative integrability and action angle variables in contact geometry, Journal of Symplectic Geometry, 10 (2012), 535-561, arXiv: 1103.3611. doi: 10.4310/JSG.2012.v10.n4.a3.

[20]

B. Jovanović, The Jacobi-Rosochatius problem on an ellipsoid: The Lax representations and billiards, Arch. Rational Mech. Anal. 210 (2013), 101-131, arXiv: 1303.6204. doi: 10.1007/s00205-013-0638-4.

[21]

B. Jovanović, Heisenberg model in pseudo-Euclidean spaces, Regular and Chaotic Dynamics, 19 (2014), 245-250, arXiv: 1405.0905. doi: 10.1134/S1560354714020075.

[22]

B. Jovanović and V. Jovanović, Contact flows and integrable systems, Journal of Geometry and Physics, 87 (2015), 217-232, arXiv: 1212.2918. doi: 10.1016/j.geomphys.2014.07.030.

[23]

B. Jovanović and V. Jovanović, Geodesic and billiard flows on quadrics in pseudo-Euclidean spaces: L-A pairs and Chasles theorem, International Mathematics Research Notices, 15 (2015), 6618-6638, arXiv: 1407.0555. doi: 10.1093/imrn/rnu141.

[24]

B. Khesin and S. Tabachnikov, Pseudo-Riemannian geodesics and billiards, Adv. Math., 221 (2009), 1364-1396, arXiv: math/0608620. doi: 10.1016/j.aim.2009.02.010.

[25]

B. Khesin and S. Tabachnikov, Contact complete integrability, Regular and Chaotic Dynamics, 15 (2010), 504-520, arXiv: 0910.0375. doi: 10.1134/S1560354710040076.

[26]

V. V. Kozlov i D. V. Treshchev, Billiardy, Geneticheskoe vvedenie v dinamiku sistem s udarami, Izd-vo Mosk. un-ta, Moskva, 1991. English translation: V. V. Kozlov and D. V. Treshchev, Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts Transl. Math. Monogr., 89, Amer. Math. Soc., Providence, RI, 1991.

[27]

P. Libermann and C. Marle, Symplectic Geometry and Analytical Mechanics Riedel, Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.

[28]

A. S. Mishchenko and A. T. Fomenko, Obobshchenny metod Liuvillya integrirovaniya gamiltonovyh sistem, Funkc. Analiz i ego Prilozh., 12 (1978), 46-56 (Russian); English translation: A. S. Mishchenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121.

[29]

J. Moser, Geometry of quadric and spectral theory, Chern Symposium 1979, Berlin-Heidelberg-New York, (1980), 147-188.

[30]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243.  doi: 10.1007/BF02352494.

[31]

N. N. Nekhoroshev, Peremennye deystvie-ugol i ih obobshcheniya Tr. Mosk. Mat. O.-va., 26 (1972), 181-198 (Russian). English translation: N. N. Nekhoroshev, Action-angle variables and their generalization, Trans. Mosc. Math. Soc., 26 (1972), 180-198.

[32]

M. Radnović, Topology of the elliptical billiard with the Hooke's potential, Theoretical and Applied Mehanics, 42 (2015), 1-9, arXiv: 1508.01025.

[33]

Yu. B. Suris, The Problem of Integrable Discretization: Hamiltonian Approach Progress in Mathematics, 219. Birkhäuser Verlag, Basel, 2003. doi: 10.1007/978-3-0348-8016-9.

[34]

S. L. Tabachnikov, Ellipsoids, complete integrability and hyperbolic geometry, Mosc. Math. J., 2 (2002), 183-196. 

[35]

S. Tabachnikov, Geometry and Billiards volume 30 of Student Mathematical Library. American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005. doi: 10.1090/stml/030.

[36]

A. P. Veselov, Integriruemye sistemy s diskretnym vremenem i raznostnye operatory, Funkc. Analiz i ego Prilozh. 22 (1988), 1-13 (Russian); English translation: A. P. Veselov, Integrable systems with discrete time, and difference operators, Funct. Anal. Appl. 22 (1988), 83-93. doi: 10.1007/BF01077598.

[37]

A. P. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107.  doi: 10.1016/0393-0440(90)90021-T.

[38]

A. P. Veselov, Integriruemye otobrazheniya, Uspekhi Mat. Nauk, 46 (1991), 3-45 (Russian); English translation: A. P. Veselov, Integrable mappings, Russ. Math. Surv., 46 (1991), 1-51.

[39]

A. P. Veselov, Growth and integrability in the dynamics of mappings, Comm. Math. Phys., 145 (1992), 181-193.  doi: 10.1007/BF02099285.

Figure 1.  A segment of a virtual billiard trajectory within hyperbola ($a_1>0, a_2<0$) in the Euclidean space $\mathbb E^{2, 0}$. The caustic is an ellipse
Figure 2.  Families of pseudo-confocal quadrics for $a_1>0, a_2<0$ in $\mathbb E^{1, 1}$ (with $\alpha_1=-a_2>\alpha_2=a_1$) and $\mathbb E^{2, 0}$, respectively
Figure 3.  Family of pseudo-confocal quadrics for $a_1>0, a_2<0$ in $\mathbb E^{1, 1}$, where $\alpha_1=a_1>\alpha_2=-a_2$
Figure 4.  The segments of time-like and space-like billiard trajectories for $a_1>0, a_2<0$, $\alpha_1=-a_2>\alpha_2=a_1$ in $\mathbb E^{1, 1}$. The caustics are hyperbolas
Figure 5.  The segment of a space-like billiard trajectory for $a_1>0, a_2<0$, $\alpha_1=a_1>\alpha_2=-a_2$ in $\mathbb E^{1, 1}$. The caustic is an ellipse
[1]

Pedro Duarte, José Pedro GaivÃo, Mohammad Soufi. Hyperbolic billiards on polytopes with contracting reflection laws. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3079-3109. doi: 10.3934/dcds.2017132

[2]

Vladimir P. Burskii, Alexei S. Zhedanov. On Dirichlet, Poncelet and Abel problems. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1587-1633. doi: 10.3934/cpaa.2013.12.1587

[3]

David M. McClendon. An Ambrose-Kakutani representation theorem for countable-to-1 semiflows. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 251-268. doi: 10.3934/dcdss.2009.2.251

[4]

Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems and Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475

[5]

Hélène Hibon, Ying Hu, Yiqing Lin, Peng Luo, Falei Wang. Quadratic BSDEs with mean reflection. Mathematical Control and Related Fields, 2018, 8 (3&4) : 721-738. doi: 10.3934/mcrf.2018031

[6]

C. D. Ahlbrandt, A. C. Peterson. A general reduction of order theorem for discrete linear symplectic systems. Conference Publications, 1998, 1998 (Special) : 7-18. doi: 10.3934/proc.1998.1998.7

[7]

Hilde De Ridder, Hennie De Schepper, Frank Sommen. The Cauchy-Kovalevskaya extension theorem in discrete Clifford analysis. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1097-1109. doi: 10.3934/cpaa.2011.10.1097

[8]

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

[9]

Renato Iturriaga, Héctor Sánchez Morgado. The Lax-Oleinik semigroup on graphs. Networks and Heterogeneous Media, 2017, 12 (4) : 643-662. doi: 10.3934/nhm.2017026

[10]

Giovanni F. Gronchi, Giacomo Tommei. On the uncertainty of the minimal distance between two confocal Keplerian orbits. Discrete and Continuous Dynamical Systems - B, 2007, 7 (4) : 755-778. doi: 10.3934/dcdsb.2007.7.755

[11]

Federica Dragoni. Metric Hopf-Lax formula with semicontinuous data. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 713-729. doi: 10.3934/dcds.2007.17.713

[12]

W. Patrick Hooper, Richard Evan Schwartz. Billiards in nearly isosceles triangles. Journal of Modern Dynamics, 2009, 3 (2) : 159-231. doi: 10.3934/jmd.2009.3.159

[13]

Serge Tabachnikov. Birkhoff billiards are insecure. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1035-1040. doi: 10.3934/dcds.2009.23.1035

[14]

Simon Castle, Norbert Peyerimhoff, Karl Friedrich Siburg. Billiards in ideal hyperbolic polygons. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 893-908. doi: 10.3934/dcds.2011.29.893

[15]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319

[16]

Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255

[17]

Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048

[18]

Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385

[19]

Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481

[20]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (119)
  • HTML views (117)
  • Cited by (4)

Other articles
by authors

[Back to Top]