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Re-factorising a QRT map
1. | School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia |
2. | Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus |
A QRT map is the composition of two involutions on a biquadratic curve: one switching the $ x $-coordinates of two intersection points with a given horizontal line, and the other switching the $ y $-coordinates of two intersections with a vertical line. Given a QRT map, a natural question is to ask whether it allows a decomposition into further involutions. Here we provide new answers to this question and show how they lead to a new class of maps, as well as known HKY maps and quadrirational Yang-Baxter maps.
References:
[1] |
V. Adler,
Recutting of polygons, Funct. Anal. Appl., 27 (1993), 141-143.
doi: 10.1007/BF01085984. |
[2] |
V. Adler, A. Bobenko and Y. Suris,
Geometry of Yang-Baxter maps: Pencils of conics and quadrirational mappings, Comm. Anal. Geom., 12 (2004), 967-1007.
doi: 10.4310/CAG.2004.v12.n5.a1. |
[3] |
J. Atkinson, Idempotent biquadratics, Yang-Baxter maps and birational representations of Coxeter groups, preprint, arXiv: math/1301.4613. Google Scholar |
[4] |
J. Atkinson, P. Howes, N. Joshi and N. Nakazono, Geometry of an elliptic difference equation
related to Q4, J. Lond. Math. Soc. (2), 93 (2016), 763–784.
doi: 10.1112/jlms/jdw020. |
[5] |
J. Atkinson and M. Nieszporski,
Multi-quadratic quad equations: Integrable cases from a factorised-discriminant hypothesis, Int. Math. Res. Not. IMRN, 2014 (2013), 4215-4240.
doi: 10.1093/imrn/rnt066. |
[6] |
J. Atkinson and F. Nijhoff,
Solutions of Adler's lattice equation associated with 2-cycles of the Bäcklund transformation, J. Nonlinear Math. Phys., 15 (2008), 34-42.
doi: 10.2991/jnmp.2008.15.s3.4. |
[7] |
J. Atkinson and Y. Yamada, Quadrirational Yang-Baxter maps and the elliptic Cremona system, preprint, arXiv: math/1804.01794. Google Scholar |
[8] |
H. Baker, Principles of Geometry, Cambridge Library Collection, 1, Cambridge University
Press, Cambridge, 2010. |
[9] |
R. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc., London,
1982. |
[10] |
V. Bazhanov and S. Sergeev,
Yang-Baxter maps, discrete integrable equations and quantum groups, Nuclear Phys. B, 926 (2018), 509-543.
doi: 10.1016/j.nuclphysb.2017.11.017. |
[11] |
M. Bellon, J.-M. Maillard and C.-M. Viallet,
Integrable Coxeter groups, Phys. Lett. A, 159 (1991), 221-232.
doi: 10.1016/0375-9601(91)90516-B. |
[12] |
M. Bellon, J.-M. Maillard and C.-M. Viallet,
Rational mappings, arborescent iterations, and the symmetries of integrability, Phys. Rev. Lett., 67 (1991), 1373-1376.
doi: 10.1103/PhysRevLett.67.1373. |
[13] |
M. Bruschi, O. Ragnisco, P. Santini and T. Gui-Zhang,
Integrable symplectic maps, Phys. D, 49 (1991), 273-294.
doi: 10.1016/0167-2789(91)90149-4. |
[14] |
W. Burnside, Theory of Groups of Finite Order, Cambridge University Press, Cambridge, 1911. Google Scholar |
[15] |
V. Caudrelier, N. Crampé and Q. Zhang, Set-theoretical reflection equation: Classification of reflection maps, J. Phys. A, 46 (2013), 12pp.
doi: 10.1088/1751-8113/46/9/095203. |
[16] |
H. Coxeter,
Frieze patterns, Acta Arith., 18 (1971), 297-310.
doi: 10.4064/aa-18-1-297-310. |
[17] |
A. Dimakis and F. Müller-Hoissen,
Matrix Kadomtsev-Petviashvili equation: Tropical limit, Yang-Baxter and pentagon maps, Theoret. and Math. Phys., 196 (2018), 1164-1173.
doi: 10.1134/S0040577918080056. |
[18] |
A. Dimakis and F. Müller-Hoissen,
Matrix KP: Tropical limit and Yang-Baxter maps, Lett. Math. Phys., 109 (2019), 799-827.
doi: 10.1007/s11005-018-1127-3. |
[19] |
V. G. Drinfeld, On some unsolved problems in quantum group theory, in Quantum Groups, Lecture Notes in Math., 1510, Springer, Berlin, 1992, 1–8.
doi: 10.1007/BFb0101175. |
[20] |
J. J. Duistermaat, Discrete Integrable Systems. QRT Maps and Elliptic Surfaces, Springer Monographs in Mathematics, Springer, New York, 2010.
doi: 10.1007/978-0-387-72923-7. |
[21] |
P. Etingof,
Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation, Commun. Algebra, 31 (2003), 1961-1973.
doi: 10.1081/AGB-120018516. |
[22] |
P. Etingof, T. Schedler and A. Soloviev,
Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100 (1999), 169-209.
doi: 10.1215/S0012-7094-99-10007-X. |
[23] |
A. Fordy and P. Kassotakis,
Multidimensional maps of QRT type, J. Phys. A, 39 (2006), 10773-10786.
doi: 10.1088/0305-4470/39/34/012. |
[24] |
A. Fordy and P. Kassotakis, Integrable maps which preserve functions with symmetries, J. Phys. A, 46 (2013), 12pp.
doi: 10.1088/1751-8113/46/20/205201. |
[25] |
G. G. Grahovski, S. Konstantinou-Rizos and A. V. Mikhailov, Grassmann extensions of Yang-Baxter maps, J. Phys. A, 49 (2016), 17pp.
doi: 10.1088/1751-8113/49/14/145202. |
[26] |
G. Gubbiotti, N. Joshi, D. T. Tran and C.-M. Viallet, Complexity and integrability in 4D bi-rational maps with two invariants, preprint, arXiv: math/1808.04942. Google Scholar |
[27] |
F. Haggar, G. Byrnes, G. Quispel and H. Capel,
K-integrals and k-Lie symmetries in discrete dynamical systems, Phys. A, 233 (1996), 379-394.
doi: 10.1016/S0378-4371(96)00142-2. |
[28] |
J. Hietarinta,
Permutation-type solutions to the Yang-Baxter and other $n$-simplex equations, J. Phys. A, 30 (1997), 4757-4771.
doi: 10.1088/0305-4470/30/13/024. |
[29] |
J. Hietarinta and C. Viallet, On the parametrization of solutions of the Yang-Baxter equations, preprint, arXiv: math/9504028. Google Scholar |
[30] |
R. Hirota, K. Kimura and H. Yahagi,
How to find conserved quantities of nonlinear discrete equations, J. Phys. A, 34 (2001), 10377-10386.
doi: 10.1088/0305-4470/34/48/304. |
[31] |
A. Iatrou and J. Roberts,
Integrable mappings of the plane preserving biquadratic invariant curves, J. Phys. A, 34 (2001), 6617-6636.
doi: 10.1088/0305-4470/34/34/308. |
[32] |
A. Iatrou and J. Roberts,
Integrable mappings of the plane preserving biquadratic invariant curves Ⅱ, Nonlinearity, 15 (2002), 459-489.
doi: 10.1088/0951-7715/15/2/313. |
[33] |
N. Iyudu and S. Shkarin et al., The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations, Duke Math. J., 164 (2015), 2539–2575.
doi: 10.1215/00127094-3146603. |
[34] |
M. Jimbo (ed.), Yang-Baxter Equation in Integrable Systems, Advanced Series in Mathematical Physics, 10, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
doi: 10.1142/1021. |
[35] |
J. Jogia, D. Roberts and F. Vivaldi,
An algebraic geometric approach to integrable maps of the plane, J. Phys. A, 39 (2006), 1133-1149.
doi: 10.1088/0305-4470/39/5/008. |
[36] |
N. Joshi, B. Grammaticos, T. Tamizhmani and A. Ramani,
From integrable lattices to non-QRT mappings, Lett. Math. Phys., 78 (2006), 27-37.
doi: 10.1007/s11005-006-0103-5. |
[37] |
N. Joshi and C.-M. Viallet, Rational maps with invariant surfaces, J. Integrable Syst., 3 (2018), 14pp.
doi: 10.1093/integr/xyy017. |
[38] |
K. Kajiwara, M. Noumi and Y. Yamada,
Discrete dynamical systems with ${W}({A}_{m-1}^{(1)} \times {A}_{n-1}^{(1)})$ symmetry, Lett. Math. Phys., 60 (2002), 211-219.
doi: 10.1023/A:1016298925276. |
[39] |
P. Kassotakis, The Construction of Discrete Dynamical System, Ph.D thesis, University of Leeds, 2006. Google Scholar |
[40] |
P. Kassotakis, Invariants in separated variables: Yang-Baxter, entwining and transfer maps, SIGMA Symmetry Integrability Geom. Methods Appl., 15 (2019), 36pp.
doi: 10.3842/SIGMA.2019.048. |
[41] |
P. Kassotakis and N. Joshi,
Integrable non-QRT mappings of the plane, Lett. Math. Phys., 91 (2010), 71-81.
doi: 10.1007/s11005-009-0360-1. |
[42] |
P. Kassotakis and M. Nieszporski, Families of integrable equations, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 14pp.
doi: 10.3842/SIGMA.2011.100. |
[43] |
P. Kassotakis and M. Nieszporski,
On non-multiaffine consistent-around-the-cube lattice equations, Phys. Lett. A, 376 (2012), 3135-3140.
doi: 10.1016/j.physleta.2012.10.009. |
[44] |
P. Kassotakis and M. Nieszporski, $2^n-$rational maps, J. Phys. A, 50 (2017), 9pp.
doi: 10.1088/1751-8121/aa6dbd. |
[45] |
P. Kassotakis and M. Nieszporski, Difference systems in bond and face variables and non-potential versions of discrete integrable systems, J. Phys. A, 51 (2018), 21pp.
doi: 10.1088/1751-8121/aad4c4. |
[46] |
K. Kimura, H. Yahagi, R. Hirota, A. Ramani, B. Grammaticos and Y. Ohta,
A new class of integrable discrete systems, J. Phys. A, 35 (2002), 9205-9212.
doi: 10.1088/0305-4470/35/43/315. |
[47] |
M. Kontsevich, Noncommutative identities, preprint, arXiv: math/1109.2469. Google Scholar |
[48] |
T. E. Kouloukas,
Relativistic collisions as Yang-Baxter maps, Phys. Lett. A, 381 (2017), 3445-3449.
doi: 10.1016/j.physleta.2017.09.007. |
[49] |
R. Lyness, Cycles, Math. Gaz., 26 (1942), 62, note 1581. Google Scholar |
[50] |
R. Lyness, Cycles, Math. Gaz., 29 (1945), 231–233, note 1847. Google Scholar |
[51] |
S. Maeda,
Completely integrable symplectic mapping, Proc. Japan Acad. Ser. A Math. Sci., 63 (1987), 198-200.
doi: 10.3792/pjaa.63.198. |
[52] |
J. M. Maillet and F. Nijhoff,
Integrability for multidimensional lattice models, Phys. Lett. B, 224 (1989), 389-396.
doi: 10.1016/0370-2693(89)91466-4. |
[53] |
E. McMillan, A problem in the stability of periodic systems, in Topics in Modern Physics, A Tribute to E. V. Condon, Colorado Assoc. Univ. Press, Boulder, 1971, 219–244. Google Scholar |
[54] |
A. V. Mikhailov, G. Papamikos and J. P. Wang,
Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere, Lett. Math. Phys., 106 (2016), 973-996.
doi: 10.1007/s11005-016-0855-5. |
[55] |
H. Mulholland and C. Smith,
An inequality arising in genetic theory, Amer. Math. Monthly, 66 (1959), 673-683.
doi: 10.1080/00029890.1959.11989387. |
[56] |
M. Nieszporski and P. Kassotakis, Systems of difference equations on a vector valued function that admit a 3D vector space of scalar potentials, preprint, arXiv: math/1908.01706. Google Scholar |
[57] |
M. Noumi and Y. Yamada,
Affine Weyl groups, discrete dynamical systems and Painlevé equations, Commun. Math. Phys., 199 (1998), 281-295.
doi: 10.1007/s002200050502. |
[58] |
V. Papageorgiou, Y. Suris, A. Tongas and A. Veselov, On quadrirational Yang-Baxter maps, SIGMA Symmetry Integrability Geom. Methods Appl., 6 (2010), 9pp.
doi: 10.3842/SIGMA.2010.033. |
[59] |
V. Papageorgiou, A. Tongas and A. Veselov, Yang-Baxter maps and symmetries of integrable equations on quad-graphs, J. Math. Phys., 47 (2006), 16pp.
doi: 10.1063/1.2227641. |
[60] |
R. Penrose and C. Smith,
A quadratic mapping with invariant cubic curve, Math. Proc. Cambridge Philos. Soc., 89 (1981), 89-105.
doi: 10.1017/S0305004100057972. |
[61] |
J. Pettigrew and J. Roberts, Characterizing singular curves in parametrized families of biquadratics, J. Phys. A, 41 (2008), 28pp.
doi: 10.1088/1751-8113/41/11/115203. |
[62] |
G. Quispel,
An alternating integrable map whose square is the QRT map, Phy. Lett. A, 307 (2003), 50-54.
doi: 10.1016/S0375-9601(02)01681-X. |
[63] |
G. Quispel, J. Roberts and C. Thompson,
Integrable mappings and soliton equations, Phys. Lett. A, 126 (1988), 419-421.
doi: 10.1016/0375-9601(88)90803-1. |
[64] |
G. Quispel, J. Roberts and and C. Thompson,
Integrable mappings and soliton equations Ⅱ, Phys. D, 34 (1989), 183-192.
doi: 10.1016/0167-2789(89)90233-9. |
[65] |
J. Roberts and D. Jogia, Birational maps that send biquadratic curves to biquadratic curves, J. Phys. A, 48 (2015), 13pp.
doi: 10.1088/1751-8113/48/8/08FT02. |
[66] |
J. Roberts and G. Quispel,
Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems, Phys. Rep., 216 (1992), 63-177.
doi: 10.1016/0370-1573(92)90163-T. |
[67] |
J. Roberts and G. Quispel, Creating and relating three-dimensional integrable maps, J. Phys. A, 39 (2006), L605–L615.
doi: 10.1088/0305-4470/39/42/L03. |
[68] |
P. Scheuer and S. Mandel,
An inequality in population genetics, Heredity, 13 (1959), 519-524.
doi: 10.1038/hdy.1959.52. |
[69] |
E. K. Sklyanin,
Classical limits of SU(2)-invariant solutions of the Yang-Baxter equation, J. Soviet Math., 40 (1988), 93-107.
doi: 10.1007/BF01084941. |
[70] |
T. Tsuda,
Integrable mappings via rational elliptic surfaces, J. Phys. A, 37 (2004), 2721-2730.
doi: 10.1088/0305-4470/37/7/014. |
[71] |
A. Veselov,
Integrable mappings, Russ. Math. Surv., 46 (1991), 1-51.
doi: 10.1070/RM1991v046n05ABEH002856. |
[72] |
A. Veselov,
Yang-Baxter maps and integrable dynamics, Phys. Lett. A, 314 (2003), 214-221.
doi: 10.1016/S0375-9601(03)00915-0. |
[73] |
C. Viallet,
Integrable lattice maps: ${Q}_v$, a rational version of ${Q}_4$, Glasg. Math. J., 51 (2009), 157-163.
doi: 10.1017/S0017089508004874. |
[74] |
C. Yang,
Some exact results for the many body problem in one dimension repulsive delta-function interaction, Phys. Rev. Lett., 19 (1967), 1312-1315.
doi: 10.1103/PhysRevLett.19.1312. |
show all references
References:
[1] |
V. Adler,
Recutting of polygons, Funct. Anal. Appl., 27 (1993), 141-143.
doi: 10.1007/BF01085984. |
[2] |
V. Adler, A. Bobenko and Y. Suris,
Geometry of Yang-Baxter maps: Pencils of conics and quadrirational mappings, Comm. Anal. Geom., 12 (2004), 967-1007.
doi: 10.4310/CAG.2004.v12.n5.a1. |
[3] |
J. Atkinson, Idempotent biquadratics, Yang-Baxter maps and birational representations of Coxeter groups, preprint, arXiv: math/1301.4613. Google Scholar |
[4] |
J. Atkinson, P. Howes, N. Joshi and N. Nakazono, Geometry of an elliptic difference equation
related to Q4, J. Lond. Math. Soc. (2), 93 (2016), 763–784.
doi: 10.1112/jlms/jdw020. |
[5] |
J. Atkinson and M. Nieszporski,
Multi-quadratic quad equations: Integrable cases from a factorised-discriminant hypothesis, Int. Math. Res. Not. IMRN, 2014 (2013), 4215-4240.
doi: 10.1093/imrn/rnt066. |
[6] |
J. Atkinson and F. Nijhoff,
Solutions of Adler's lattice equation associated with 2-cycles of the Bäcklund transformation, J. Nonlinear Math. Phys., 15 (2008), 34-42.
doi: 10.2991/jnmp.2008.15.s3.4. |
[7] |
J. Atkinson and Y. Yamada, Quadrirational Yang-Baxter maps and the elliptic Cremona system, preprint, arXiv: math/1804.01794. Google Scholar |
[8] |
H. Baker, Principles of Geometry, Cambridge Library Collection, 1, Cambridge University
Press, Cambridge, 2010. |
[9] |
R. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc., London,
1982. |
[10] |
V. Bazhanov and S. Sergeev,
Yang-Baxter maps, discrete integrable equations and quantum groups, Nuclear Phys. B, 926 (2018), 509-543.
doi: 10.1016/j.nuclphysb.2017.11.017. |
[11] |
M. Bellon, J.-M. Maillard and C.-M. Viallet,
Integrable Coxeter groups, Phys. Lett. A, 159 (1991), 221-232.
doi: 10.1016/0375-9601(91)90516-B. |
[12] |
M. Bellon, J.-M. Maillard and C.-M. Viallet,
Rational mappings, arborescent iterations, and the symmetries of integrability, Phys. Rev. Lett., 67 (1991), 1373-1376.
doi: 10.1103/PhysRevLett.67.1373. |
[13] |
M. Bruschi, O. Ragnisco, P. Santini and T. Gui-Zhang,
Integrable symplectic maps, Phys. D, 49 (1991), 273-294.
doi: 10.1016/0167-2789(91)90149-4. |
[14] |
W. Burnside, Theory of Groups of Finite Order, Cambridge University Press, Cambridge, 1911. Google Scholar |
[15] |
V. Caudrelier, N. Crampé and Q. Zhang, Set-theoretical reflection equation: Classification of reflection maps, J. Phys. A, 46 (2013), 12pp.
doi: 10.1088/1751-8113/46/9/095203. |
[16] |
H. Coxeter,
Frieze patterns, Acta Arith., 18 (1971), 297-310.
doi: 10.4064/aa-18-1-297-310. |
[17] |
A. Dimakis and F. Müller-Hoissen,
Matrix Kadomtsev-Petviashvili equation: Tropical limit, Yang-Baxter and pentagon maps, Theoret. and Math. Phys., 196 (2018), 1164-1173.
doi: 10.1134/S0040577918080056. |
[18] |
A. Dimakis and F. Müller-Hoissen,
Matrix KP: Tropical limit and Yang-Baxter maps, Lett. Math. Phys., 109 (2019), 799-827.
doi: 10.1007/s11005-018-1127-3. |
[19] |
V. G. Drinfeld, On some unsolved problems in quantum group theory, in Quantum Groups, Lecture Notes in Math., 1510, Springer, Berlin, 1992, 1–8.
doi: 10.1007/BFb0101175. |
[20] |
J. J. Duistermaat, Discrete Integrable Systems. QRT Maps and Elliptic Surfaces, Springer Monographs in Mathematics, Springer, New York, 2010.
doi: 10.1007/978-0-387-72923-7. |
[21] |
P. Etingof,
Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation, Commun. Algebra, 31 (2003), 1961-1973.
doi: 10.1081/AGB-120018516. |
[22] |
P. Etingof, T. Schedler and A. Soloviev,
Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100 (1999), 169-209.
doi: 10.1215/S0012-7094-99-10007-X. |
[23] |
A. Fordy and P. Kassotakis,
Multidimensional maps of QRT type, J. Phys. A, 39 (2006), 10773-10786.
doi: 10.1088/0305-4470/39/34/012. |
[24] |
A. Fordy and P. Kassotakis, Integrable maps which preserve functions with symmetries, J. Phys. A, 46 (2013), 12pp.
doi: 10.1088/1751-8113/46/20/205201. |
[25] |
G. G. Grahovski, S. Konstantinou-Rizos and A. V. Mikhailov, Grassmann extensions of Yang-Baxter maps, J. Phys. A, 49 (2016), 17pp.
doi: 10.1088/1751-8113/49/14/145202. |
[26] |
G. Gubbiotti, N. Joshi, D. T. Tran and C.-M. Viallet, Complexity and integrability in 4D bi-rational maps with two invariants, preprint, arXiv: math/1808.04942. Google Scholar |
[27] |
F. Haggar, G. Byrnes, G. Quispel and H. Capel,
K-integrals and k-Lie symmetries in discrete dynamical systems, Phys. A, 233 (1996), 379-394.
doi: 10.1016/S0378-4371(96)00142-2. |
[28] |
J. Hietarinta,
Permutation-type solutions to the Yang-Baxter and other $n$-simplex equations, J. Phys. A, 30 (1997), 4757-4771.
doi: 10.1088/0305-4470/30/13/024. |
[29] |
J. Hietarinta and C. Viallet, On the parametrization of solutions of the Yang-Baxter equations, preprint, arXiv: math/9504028. Google Scholar |
[30] |
R. Hirota, K. Kimura and H. Yahagi,
How to find conserved quantities of nonlinear discrete equations, J. Phys. A, 34 (2001), 10377-10386.
doi: 10.1088/0305-4470/34/48/304. |
[31] |
A. Iatrou and J. Roberts,
Integrable mappings of the plane preserving biquadratic invariant curves, J. Phys. A, 34 (2001), 6617-6636.
doi: 10.1088/0305-4470/34/34/308. |
[32] |
A. Iatrou and J. Roberts,
Integrable mappings of the plane preserving biquadratic invariant curves Ⅱ, Nonlinearity, 15 (2002), 459-489.
doi: 10.1088/0951-7715/15/2/313. |
[33] |
N. Iyudu and S. Shkarin et al., The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations, Duke Math. J., 164 (2015), 2539–2575.
doi: 10.1215/00127094-3146603. |
[34] |
M. Jimbo (ed.), Yang-Baxter Equation in Integrable Systems, Advanced Series in Mathematical Physics, 10, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
doi: 10.1142/1021. |
[35] |
J. Jogia, D. Roberts and F. Vivaldi,
An algebraic geometric approach to integrable maps of the plane, J. Phys. A, 39 (2006), 1133-1149.
doi: 10.1088/0305-4470/39/5/008. |
[36] |
N. Joshi, B. Grammaticos, T. Tamizhmani and A. Ramani,
From integrable lattices to non-QRT mappings, Lett. Math. Phys., 78 (2006), 27-37.
doi: 10.1007/s11005-006-0103-5. |
[37] |
N. Joshi and C.-M. Viallet, Rational maps with invariant surfaces, J. Integrable Syst., 3 (2018), 14pp.
doi: 10.1093/integr/xyy017. |
[38] |
K. Kajiwara, M. Noumi and Y. Yamada,
Discrete dynamical systems with ${W}({A}_{m-1}^{(1)} \times {A}_{n-1}^{(1)})$ symmetry, Lett. Math. Phys., 60 (2002), 211-219.
doi: 10.1023/A:1016298925276. |
[39] |
P. Kassotakis, The Construction of Discrete Dynamical System, Ph.D thesis, University of Leeds, 2006. Google Scholar |
[40] |
P. Kassotakis, Invariants in separated variables: Yang-Baxter, entwining and transfer maps, SIGMA Symmetry Integrability Geom. Methods Appl., 15 (2019), 36pp.
doi: 10.3842/SIGMA.2019.048. |
[41] |
P. Kassotakis and N. Joshi,
Integrable non-QRT mappings of the plane, Lett. Math. Phys., 91 (2010), 71-81.
doi: 10.1007/s11005-009-0360-1. |
[42] |
P. Kassotakis and M. Nieszporski, Families of integrable equations, SIGMA Symmetry Integrability Geom. Methods Appl., 7 (2011), 14pp.
doi: 10.3842/SIGMA.2011.100. |
[43] |
P. Kassotakis and M. Nieszporski,
On non-multiaffine consistent-around-the-cube lattice equations, Phys. Lett. A, 376 (2012), 3135-3140.
doi: 10.1016/j.physleta.2012.10.009. |
[44] |
P. Kassotakis and M. Nieszporski, $2^n-$rational maps, J. Phys. A, 50 (2017), 9pp.
doi: 10.1088/1751-8121/aa6dbd. |
[45] |
P. Kassotakis and M. Nieszporski, Difference systems in bond and face variables and non-potential versions of discrete integrable systems, J. Phys. A, 51 (2018), 21pp.
doi: 10.1088/1751-8121/aad4c4. |
[46] |
K. Kimura, H. Yahagi, R. Hirota, A. Ramani, B. Grammaticos and Y. Ohta,
A new class of integrable discrete systems, J. Phys. A, 35 (2002), 9205-9212.
doi: 10.1088/0305-4470/35/43/315. |
[47] |
M. Kontsevich, Noncommutative identities, preprint, arXiv: math/1109.2469. Google Scholar |
[48] |
T. E. Kouloukas,
Relativistic collisions as Yang-Baxter maps, Phys. Lett. A, 381 (2017), 3445-3449.
doi: 10.1016/j.physleta.2017.09.007. |
[49] |
R. Lyness, Cycles, Math. Gaz., 26 (1942), 62, note 1581. Google Scholar |
[50] |
R. Lyness, Cycles, Math. Gaz., 29 (1945), 231–233, note 1847. Google Scholar |
[51] |
S. Maeda,
Completely integrable symplectic mapping, Proc. Japan Acad. Ser. A Math. Sci., 63 (1987), 198-200.
doi: 10.3792/pjaa.63.198. |
[52] |
J. M. Maillet and F. Nijhoff,
Integrability for multidimensional lattice models, Phys. Lett. B, 224 (1989), 389-396.
doi: 10.1016/0370-2693(89)91466-4. |
[53] |
E. McMillan, A problem in the stability of periodic systems, in Topics in Modern Physics, A Tribute to E. V. Condon, Colorado Assoc. Univ. Press, Boulder, 1971, 219–244. Google Scholar |
[54] |
A. V. Mikhailov, G. Papamikos and J. P. Wang,
Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere, Lett. Math. Phys., 106 (2016), 973-996.
doi: 10.1007/s11005-016-0855-5. |
[55] |
H. Mulholland and C. Smith,
An inequality arising in genetic theory, Amer. Math. Monthly, 66 (1959), 673-683.
doi: 10.1080/00029890.1959.11989387. |
[56] |
M. Nieszporski and P. Kassotakis, Systems of difference equations on a vector valued function that admit a 3D vector space of scalar potentials, preprint, arXiv: math/1908.01706. Google Scholar |
[57] |
M. Noumi and Y. Yamada,
Affine Weyl groups, discrete dynamical systems and Painlevé equations, Commun. Math. Phys., 199 (1998), 281-295.
doi: 10.1007/s002200050502. |
[58] |
V. Papageorgiou, Y. Suris, A. Tongas and A. Veselov, On quadrirational Yang-Baxter maps, SIGMA Symmetry Integrability Geom. Methods Appl., 6 (2010), 9pp.
doi: 10.3842/SIGMA.2010.033. |
[59] |
V. Papageorgiou, A. Tongas and A. Veselov, Yang-Baxter maps and symmetries of integrable equations on quad-graphs, J. Math. Phys., 47 (2006), 16pp.
doi: 10.1063/1.2227641. |
[60] |
R. Penrose and C. Smith,
A quadratic mapping with invariant cubic curve, Math. Proc. Cambridge Philos. Soc., 89 (1981), 89-105.
doi: 10.1017/S0305004100057972. |
[61] |
J. Pettigrew and J. Roberts, Characterizing singular curves in parametrized families of biquadratics, J. Phys. A, 41 (2008), 28pp.
doi: 10.1088/1751-8113/41/11/115203. |
[62] |
G. Quispel,
An alternating integrable map whose square is the QRT map, Phy. Lett. A, 307 (2003), 50-54.
doi: 10.1016/S0375-9601(02)01681-X. |
[63] |
G. Quispel, J. Roberts and C. Thompson,
Integrable mappings and soliton equations, Phys. Lett. A, 126 (1988), 419-421.
doi: 10.1016/0375-9601(88)90803-1. |
[64] |
G. Quispel, J. Roberts and and C. Thompson,
Integrable mappings and soliton equations Ⅱ, Phys. D, 34 (1989), 183-192.
doi: 10.1016/0167-2789(89)90233-9. |
[65] |
J. Roberts and D. Jogia, Birational maps that send biquadratic curves to biquadratic curves, J. Phys. A, 48 (2015), 13pp.
doi: 10.1088/1751-8113/48/8/08FT02. |
[66] |
J. Roberts and G. Quispel,
Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems, Phys. Rep., 216 (1992), 63-177.
doi: 10.1016/0370-1573(92)90163-T. |
[67] |
J. Roberts and G. Quispel, Creating and relating three-dimensional integrable maps, J. Phys. A, 39 (2006), L605–L615.
doi: 10.1088/0305-4470/39/42/L03. |
[68] |
P. Scheuer and S. Mandel,
An inequality in population genetics, Heredity, 13 (1959), 519-524.
doi: 10.1038/hdy.1959.52. |
[69] |
E. K. Sklyanin,
Classical limits of SU(2)-invariant solutions of the Yang-Baxter equation, J. Soviet Math., 40 (1988), 93-107.
doi: 10.1007/BF01084941. |
[70] |
T. Tsuda,
Integrable mappings via rational elliptic surfaces, J. Phys. A, 37 (2004), 2721-2730.
doi: 10.1088/0305-4470/37/7/014. |
[71] |
A. Veselov,
Integrable mappings, Russ. Math. Surv., 46 (1991), 1-51.
doi: 10.1070/RM1991v046n05ABEH002856. |
[72] |
A. Veselov,
Yang-Baxter maps and integrable dynamics, Phys. Lett. A, 314 (2003), 214-221.
doi: 10.1016/S0375-9601(03)00915-0. |
[73] |
C. Viallet,
Integrable lattice maps: ${Q}_v$, a rational version of ${Q}_4$, Glasg. Math. J., 51 (2009), 157-163.
doi: 10.1017/S0017089508004874. |
[74] |
C. Yang,
Some exact results for the many body problem in one dimension repulsive delta-function interaction, Phys. Rev. Lett., 19 (1967), 1312-1315.
doi: 10.1103/PhysRevLett.19.1312. |


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