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
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[25]

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[27]

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J. Hietarinta and C. Viallet, On the parametrization of solutions of the Yang-Baxter equations, preprint, arXiv: math/9504028. Google Scholar

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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

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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

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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

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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

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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

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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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