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:
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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.

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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.

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show all references

References:
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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.

[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.

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J. Atkinson and Y. Yamada, Quadrirational Yang-Baxter maps and the elliptic Cremona system, preprint, arXiv: math/1804.01794.

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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.

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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.

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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.

[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.

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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.

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[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.

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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.

[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.

[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.

[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.

[48]

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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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