American Institute of Mathematical Sciences

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December  2019, 6(2): 325-343. doi: 10.3934/jcd.2019016

Re-factorising a QRT map

Nalini Joshi 1,, and  Pavlos Kassotakis 2,

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

* Corresponding author: Nalini Joshi

Dedicated to Reinout Quispel on the occasion of his 66th birthday

Received  June 2019 Revised  September 2019 Published  November 2019

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.

Keywords: Integrable maps, QRT/non-QRT maps, Yang-Baxter maps.
Mathematics Subject Classification: 37J35, 37J15.
Citation: Nalini Joshi, Pavlos Kassotakis. Re-factorising a QRT map. Journal of Computational Dynamics, 2019, 6 (2) : 325-343. doi: 10.3934/jcd.2019016
References:
[1]

V. Adler, Recutting of polygons, Funct. Anal. Appl., 27 (1993), 141-143.  doi: 10.1007/BF01085984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

H. Coxeter, Frieze patterns, Acta Arith., 18 (1971), 297-310.  doi: 10.4064/aa-18-1-297-310.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

N. Joshi and C.-M. Viallet, Rational maps with invariant surfaces, J. Integrable Syst., 3 (2018), 14pp. doi: 10.1093/integr/xyy017.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

P. Kassotakis and M. Nieszporski, $2^n-$rational maps, J. Phys. A, 50 (2017), 9pp. doi: 10.1088/1751-8121/aa6dbd.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[68]

P. Scheuer and S. Mandel, An inequality in population genetics, Heredity, 13 (1959), 519-524.  doi: 10.1038/hdy.1959.52.  Google Scholar

[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.  Google Scholar

[70]

T. Tsuda, Integrable mappings via rational elliptic surfaces, J. Phys. A, 37 (2004), 2721-2730.  doi: 10.1088/0305-4470/37/7/014.  Google Scholar

[71]

A. Veselov, Integrable mappings, Russ. Math. Surv., 46 (1991), 1-51.  doi: 10.1070/RM1991v046n05ABEH002856.  Google Scholar

[72]

A. Veselov, Yang-Baxter maps and integrable dynamics, Phys. Lett. A, 314 (2003), 214-221.  doi: 10.1016/S0375-9601(03)00915-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

V. Adler, Recutting of polygons, Funct. Anal. Appl., 27 (1993), 141-143.  doi: 10.1007/BF01085984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

H. Coxeter, Frieze patterns, Acta Arith., 18 (1971), 297-310.  doi: 10.4064/aa-18-1-297-310.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

N. Joshi and C.-M. Viallet, Rational maps with invariant surfaces, J. Integrable Syst., 3 (2018), 14pp. doi: 10.1093/integr/xyy017.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

P. Kassotakis and M. Nieszporski, $2^n-$rational maps, J. Phys. A, 50 (2017), 9pp. doi: 10.1088/1751-8121/aa6dbd.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[68]

P. Scheuer and S. Mandel, An inequality in population genetics, Heredity, 13 (1959), 519-524.  doi: 10.1038/hdy.1959.52.  Google Scholar

[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.  Google Scholar

[70]

T. Tsuda, Integrable mappings via rational elliptic surfaces, J. Phys. A, 37 (2004), 2721-2730.  doi: 10.1088/0305-4470/37/7/014.  Google Scholar

[71]

A. Veselov, Integrable mappings, Russ. Math. Surv., 46 (1991), 1-51.  doi: 10.1070/RM1991v046n05ABEH002856.  Google Scholar

[72]

A. Veselov, Yang-Baxter maps and integrable dynamics, Phys. Lett. A, 314 (2003), 214-221.  doi: 10.1016/S0375-9601(03)00915-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.1.  The QRT map
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Figure 1.2.  A QRT re-factorisation
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Table 3.1.  The $F$-list of Yang-Baxter maps
$\mathcal{A}_0$ $\mathcal{R}:(x, y)\mapsto (X, Y)$ $r:(x, y, p, q)\\ \mapsto (X, Y, P, Q)$
$F_I$ $\left( \begin{array}{lll} 0&p-q&q(1-p)\\ 0&2p(q-1)&0\\ p(1-p)&p(p-q)&0 \end{array}\right)$ $\begin{array}{l} X=p y W(x, y, p, q), \\ Y=q x W(x, y, p, q), \\ W(x, y, p, q)=\\ \frac{(1-q) x+q-p+(p -1)y}{q(1-p) x+(p-q)xy+p (q-1)y} \end{array}$ $\begin{array}{ll} X=q x W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
$F_{II}$ $\left( \begin{array}{lll} 0&0&p\\ 0&-2p&0\\ q&(p-q)&0 \end{array}\right)$ $\begin{array}{l} X=y/p W(x, y, p, q), \\ Y=x/q W(x, y, p, q), \\ W(x, y, p, q)= \frac{p x-q y+q-p}{x-y}, \end{array}$ $\begin{array}{ll} X=x/q W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
$F_{III}$ $\left( \begin{array}{lll} 0&0&p\\ 0&-2p&0\\ q&0&0 \end{array}\right)$ $\begin{array}{l} X=y/p W(x, y, p, q), \\ Y=x/q W(x, y, p, q), \\ W(x, y, p, q)= \frac{p x-q y}{x-y} \end{array}$ $\begin{array}{ll} X=x/q W(y, x, q, p), \\ Y=x, \\ P=q, \\ Q=p \end{array}$
$F_{IV}$ $\left( \begin{array}{lll} 0&0&1\\ 0&-2&0\\ -1&(p-q)&0 \end{array}\right)$ $\begin{array}{l} X=y W(x, y, p, q), \\ Y=x W(x, y, p, q), \\ W(x, y, p, q)= 1-\frac{p-q}{x-y} \end{array}$ $\begin{array}{ll} X=x W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
$F_{V}$ $\left( \begin{array}{lll} 0&0&1\\ 0&-2&0\\ 1&0&q-p \end{array}\right)$ $\begin{array}{l} X=y+ W(x, y, p, q), \\ Y=x+ W(x, y, p, q), \\ W(x, y, p, q)= \frac{p-q}{x-y} \end{array}$ $\begin{array}{ll} X=x+ W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
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$\mathcal{A}_0$ $\mathcal{R}:(x, y)\mapsto (X, Y)$ $r:(x, y, p, q)\\ \mapsto (X, Y, P, Q)$
$F_I$ $\left( \begin{array}{lll} 0&p-q&q(1-p)\\ 0&2p(q-1)&0\\ p(1-p)&p(p-q)&0 \end{array}\right)$ $\begin{array}{l} X=p y W(x, y, p, q), \\ Y=q x W(x, y, p, q), \\ W(x, y, p, q)=\\ \frac{(1-q) x+q-p+(p -1)y}{q(1-p) x+(p-q)xy+p (q-1)y} \end{array}$ $\begin{array}{ll} X=q x W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
$F_{II}$ $\left( \begin{array}{lll} 0&0&p\\ 0&-2p&0\\ q&(p-q)&0 \end{array}\right)$ $\begin{array}{l} X=y/p W(x, y, p, q), \\ Y=x/q W(x, y, p, q), \\ W(x, y, p, q)= \frac{p x-q y+q-p}{x-y}, \end{array}$ $\begin{array}{ll} X=x/q W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
$F_{III}$ $\left( \begin{array}{lll} 0&0&p\\ 0&-2p&0\\ q&0&0 \end{array}\right)$ $\begin{array}{l} X=y/p W(x, y, p, q), \\ Y=x/q W(x, y, p, q), \\ W(x, y, p, q)= \frac{p x-q y}{x-y} \end{array}$ $\begin{array}{ll} X=x/q W(y, x, q, p), \\ Y=x, \\ P=q, \\ Q=p \end{array}$
$F_{IV}$ $\left( \begin{array}{lll} 0&0&1\\ 0&-2&0\\ -1&(p-q)&0 \end{array}\right)$ $\begin{array}{l} X=y W(x, y, p, q), \\ Y=x W(x, y, p, q), \\ W(x, y, p, q)= 1-\frac{p-q}{x-y} \end{array}$ $\begin{array}{ll} X=x W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
$F_{V}$ $\left( \begin{array}{lll} 0&0&1\\ 0&-2&0\\ 1&0&q-p \end{array}\right)$ $\begin{array}{l} X=y+ W(x, y, p, q), \\ Y=x+ W(x, y, p, q), \\ W(x, y, p, q)= \frac{p-q}{x-y} \end{array}$ $\begin{array}{ll} X=x+ W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
Table 3.2.  The dual ${\hat F}$-list of Yang-Baxter maps
$\mathcal{A}_0$ $\mathcal{L}:(x, y)\mapsto (X, Y)$ $l:(x, y, p, q)\\ \mapsto (X, Y, P, Q)$
${\hat F}_I$ $\left( \begin{array}{lll} 0&p-q&q(1-p)\\ 0&2p(q-1)&0\\ p(1-p)&p(p-q)&0 \end{array}\right)$ $\begin{array}{l} X=y+ W(x, y, p, q), \\ Y=x+ W(y, x, q, p), \\ W(x, y, p, q)=\\ \frac{(q-p)\left(q(x+y-2 x y)+y^2(x+y-2)\right)}{q(p-q)2qx(q-1)+y\left(2q-2pq+(p-q)y\right)} \end{array}$ $\begin{array}{ll} X=x+ W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
${\hat F}_{II}$ $\left( \begin{array}{lll} 0&0&p\\ 0&-2p&0\\ q&(p-q)&0 \end{array}\right)$ $\begin{array}{l} X=y+ W(x, y, p, q), \\ Y=x+ W(x, y, p, q) \\ W(x, y, p, q)= \frac{(q-p)(x+y-2xy)}{p-q-2px+2qy} \end{array}$ $\begin{array}{ll} X=x+W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
${\hat F}_{III}$ $\left( \begin{array}{lll} 0&0&p\\ 0&-2p&0\\ q&0&0 \end{array}\right)$ $\begin{array}{l} X=qy W(x, y, p, q), \\ Y=px W(x, y, p, q), \\ W(x, y, p, q)= \frac{x-y}{p x-q y} \end{array}$ $\begin{array}{ll} X=px W(y, x, q, p), \\ Y=x, \\ P=q, \\ Q=p \end{array}$
${\hat F}_{IV}$ $\left( \begin{array}{lll} 0&0&1\\ 0&-2&0\\ -1&(p-q)&0 \end{array}\right)$ $\begin{array}{l} X=y+W(x, y, p, q), \\ Y=x+W(x, y, p, q), \\ W(x, y, p, q)= \frac{(q-p)(x+y)}{p-q-2(x+y)} \end{array}$ $\begin{array}{ll} X=x+W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
${\hat F}_{V}$ $\left( \begin{array}{lll} 0&0&1\\ 0&-2&0\\ 1&0&q-p \end{array}\right)$ $\begin{array}{l} X=y+ W(x, y, p, q), \\ Y=x+ W(x, y, p, q), \\ W(x, y, p, q)= -\frac{p-q}{x-y} \end{array}$ $\begin{array}{ll} X=x+ W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
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$\mathcal{A}_0$ $\mathcal{L}:(x, y)\mapsto (X, Y)$ $l:(x, y, p, q)\\ \mapsto (X, Y, P, Q)$
${\hat F}_I$ $\left( \begin{array}{lll} 0&p-q&q(1-p)\\ 0&2p(q-1)&0\\ p(1-p)&p(p-q)&0 \end{array}\right)$ $\begin{array}{l} X=y+ W(x, y, p, q), \\ Y=x+ W(y, x, q, p), \\ W(x, y, p, q)=\\ \frac{(q-p)\left(q(x+y-2 x y)+y^2(x+y-2)\right)}{q(p-q)2qx(q-1)+y\left(2q-2pq+(p-q)y\right)} \end{array}$ $\begin{array}{ll} X=x+ W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
${\hat F}_{II}$ $\left( \begin{array}{lll} 0&0&p\\ 0&-2p&0\\ q&(p-q)&0 \end{array}\right)$ $\begin{array}{l} X=y+ W(x, y, p, q), \\ Y=x+ W(x, y, p, q) \\ W(x, y, p, q)= \frac{(q-p)(x+y-2xy)}{p-q-2px+2qy} \end{array}$ $\begin{array}{ll} X=x+W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
${\hat F}_{III}$ $\left( \begin{array}{lll} 0&0&p\\ 0&-2p&0\\ q&0&0 \end{array}\right)$ $\begin{array}{l} X=qy W(x, y, p, q), \\ Y=px W(x, y, p, q), \\ W(x, y, p, q)= \frac{x-y}{p x-q y} \end{array}$ $\begin{array}{ll} X=px W(y, x, q, p), \\ Y=x, \\ P=q, \\ Q=p \end{array}$
${\hat F}_{IV}$ $\left( \begin{array}{lll} 0&0&1\\ 0&-2&0\\ -1&(p-q)&0 \end{array}\right)$ $\begin{array}{l} X=y+W(x, y, p, q), \\ Y=x+W(x, y, p, q), \\ W(x, y, p, q)= \frac{(q-p)(x+y)}{p-q-2(x+y)} \end{array}$ $\begin{array}{ll} X=x+W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
${\hat F}_{V}$ $\left( \begin{array}{lll} 0&0&1\\ 0&-2&0\\ 1&0&q-p \end{array}\right)$ $\begin{array}{l} X=y+ W(x, y, p, q), \\ Y=x+ W(x, y, p, q), \\ W(x, y, p, q)= -\frac{p-q}{x-y} \end{array}$ $\begin{array}{ll} X=x+ W(y, x, q, p) , \\ Y=x, \\ P=q, \\ Q=p \end{array}$
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