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

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

* 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.

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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show all references

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[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]

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[27]

F. HaggarG. ByrnesG. 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. HirotaK. 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]

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[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

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J. JogiaD. 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. JoshiB. GrammaticosT. 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

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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. KajiwaraM. 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

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P. Kassotakis, The Construction of Discrete Dynamical System, Ph.D thesis, University of Leeds, 2006. Google Scholar

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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]

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