# American Institute of Mathematical Sciences

April  2021, 8(2): 183-211. doi: 10.3934/jcd.2021009

## Generalised Manin transformations and QRT maps

 Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia

* Corresponding author: P.VanDerKamp@latrobe.edu.au

Received  September 2020 Revised  January 2021 Published  April 2021 Early access  March 2021

Manin transformations are maps of the plane that preserve a pencil of cubic curves. They are the composition of two involutions. Each involution is constructed in terms of an involution point that is required to be one of the base points of the pencil. We generalise this construction to explicit birational maps of the plane that preserve quadratic resp. certain quartic pencils, and show that they are measure-preserving and hence integrable. In the quartic construction the two involution points are required to be base points of the pencil of multiplicity 2. On the other hand, for the quadratic pencils the involution points can be any two distinct points in the plane (except for base points). We employ Pascal's theorem to show that the maps that preserve a quadratic pencil admit infinitely many symmetries. The full 18-parameter QRT map is obtained as a special instance of the quartic case in a limit where the two involution points go to infinity. We show by construction that each generalised Manin transformation can be brought to QRT form by a fractional affine transformation. We also specify classes of generalised Manin transformations which admit a root.

Citation: Peter H. van der Kamp, David I. McLaren, G. R. W. Quispel. Generalised Manin transformations and QRT maps. Journal of Computational Dynamics, 2021, 8 (2) : 183-211. doi: 10.3934/jcd.2021009
##### References:
 [1] E. Artin, Geometric Algebra, Interscience Publishers, New York, 1957. doi: 10.1002/9781118164518. [2] G. Bastien and M. Rogalski, Global behavior of the solutions of Lyness' difference equation $u_{n + 2}u_{n} = u_{n + 1} + a$, J. Differ. Equ. Appl., 10 (2004), 977-1003.  doi: 10.1080/10236190410001728104. [3] L. Bayle and A. Beauville, Birational involutions of $\mathbb{P}^2$, Asian J. Math., 4 (2000), 11-18.  doi: 10.4310/AJM.2000.v4.n1.a2. [4] E. Bertini, Ricerche sulle trasformazioni univoche involutorie nel piano, Annali di Mat., 8 (1877), 244-286.  doi: 10.1007/BF02420790. [5] A. S. Carstea and T. Takenawa, A classification of two-dimensional integrable mappings and rational elliptic surfaces, J. Phys. A, 45 (2012), 155206. doi: 10.1088/1751-8113/45/15/155206. [6] V. Caudrelier, P. H. van der Kamp and C. Zhang, Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings, preprint, arXiv: 2009.00412. [7] E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Two classes of quadratic vector fields for which the Kahan discretization is integrable, MI Lecture Notes, 74, 60–62. [8] E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Integrability properties of Kahan's method, J. Phys. A, 47 (2014), 365202. doi: 10.1088/1751-8113/47/36/365202. [9] F. Cossec and I. V. Dolgachev, Enriques Surfaces I, Progress in Mathematics, 76, Birkhäuser, Boston, 1989. doi: 10.1007/978-1-4612-3696-2. [10] I. V. Dolgačhev, Rational surfaces with a pencil of elliptic curves, Izv. Akad. Nauk SSSR Ser. Mat., 30 (1966), 1073-1100. [11] J. J. Duistermaat, Discrete integrable systems, QRT Maps and Elliptic Surfaces, Springer, New York, 2010. doi: 10.1007/978-0-387-72923-7. [12] F. A. Haggar, G. B. Byrnes, G. R. W. Quispel and H. W. 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. [13] 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. [14] N. J. Hitchin, N. S. Manton and M. K. Murray, Symmetric monopoles, Nonlinearity, 8 (1995), 661-692.  doi: 10.1088/0951-7715/8/5/002. [15] A. Iatrou and J. A. G. 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. [16] D. Jogia, J. A. G. 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. [17] N. Joshi, B. Grammaticos, T. Tamizhmami and A. Ramani, From integrable lattices to non-QRT mappings, Lett. Math. Phys., 78 (2006), 27-37.  doi: 10.1007/s11005-006-0103-5. [18] 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. [19] 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. [20] R. C. Lyness, Note 1581, Math. Gaz., 26 (1942), 62. [21] Yu. I. Manin, The Tate height of points on an Abelian variety., AMS Translations Ser., 2 (1966), 82-110. [22] E. I. Moody, Notes on the Bertini involution, Bull. Amer. Math. Soc., 49 (1943), 433-436.  doi: 10.1090/S0002-9904-1943-07940-2. [23] [24] M. Petrera, A. Pfadler and Y. B. Suris, On integrability of Hirota-Kimura type discretizations, Regul. Chaotic Dyn., 16 (2011), 245-289.  doi: 10.1134/S1560354711030051. [25] M. Petrera, J. Smirin and Y. B. Suris, Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems, Proc. A., 475 (2019), 20180761. doi: 10.1098/rspa.2018.0761. [26] M. Petrera and Y. B. Suris, Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor, J. Comput. Dyn., 6 (2019), 401-408.  doi: 10.3934/jcd.2019020. [27] M. Petrera and Y. B. Suris, Manin involutions for elliptic pencils and discrete integrable systems, arXiv: 2008.08308. [28] G. R. W. Quispel, J. A. G. Roberts and C. J. Thompson, Integrable mappings and soliton equations, Phys. Lett. A, 126 (1988), 419-421.  doi: 10.1016/0375-9601(88)90803-1. [29] G. R. W. Quispel, J. A. G. Roberts and C. J. Thompson, Integrable mappings and soliton equations Ⅱ, Physica D, 34 (1989), 183-192.  doi: 10.1016/0167-2789(89)90233-9. [30] J. A. G. Roberts and G. R. W. 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. [31] T. Sakkalis and R. Farouki, Singular points of algebraic curves, J. Symbolic Comput., 9 (1990), 405-421.  doi: 10.1016/S0747-7171(08)80019-3. [32] J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4757-4252-7. [33] G. K. Suslov, Theoretical Mechanics, Gostekhizdat, Moskva, 1946, (in Russian). [34] T. Tsuda, Integrable mapping via rational elliptic surfaces, J. Phys. A, 37 (2004), 2721-2730.  doi: 10.1088/0305-4470/37/7/014. [35] P. H. van der Kamp, A new class of integrable maps of the plane: Manin transformations with involution curves, preprint, arXiv: 2009.09854. [36] P. H. van der Kamp, E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Three classes of quadratic vector fields for which the Kahan discretisation is the root of a generalised Manin transformation, J. Phys. A: Math. Theor., 52 (2019), 045204, 10 pp. [37] J. van Yzeren, A simple proof of Pascal's hexagon theorem, Amer. Math. Monthly, 100 (1993), 930-931.  doi: 10.1080/00029890.1993.11990514. [38] A. P. Veselov, Integrable mappings, Russ. Math. Surv., 46 (1991), 1-51.  doi: 10.1070/RM1991v046n05ABEH002856. [39] C. M. Viallet, B. Grammaticos and A. Ramani, On the integrability of correspondences associated to integral curves, Phys. Lett. A, 322 (2004), 186-193.  doi: 10.1016/j.physleta.2004.01.013.

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##### References:
 [1] E. Artin, Geometric Algebra, Interscience Publishers, New York, 1957. doi: 10.1002/9781118164518. [2] G. Bastien and M. Rogalski, Global behavior of the solutions of Lyness' difference equation $u_{n + 2}u_{n} = u_{n + 1} + a$, J. Differ. Equ. Appl., 10 (2004), 977-1003.  doi: 10.1080/10236190410001728104. [3] L. Bayle and A. Beauville, Birational involutions of $\mathbb{P}^2$, Asian J. Math., 4 (2000), 11-18.  doi: 10.4310/AJM.2000.v4.n1.a2. [4] E. Bertini, Ricerche sulle trasformazioni univoche involutorie nel piano, Annali di Mat., 8 (1877), 244-286.  doi: 10.1007/BF02420790. [5] A. S. Carstea and T. Takenawa, A classification of two-dimensional integrable mappings and rational elliptic surfaces, J. Phys. A, 45 (2012), 155206. doi: 10.1088/1751-8113/45/15/155206. [6] V. Caudrelier, P. H. van der Kamp and C. Zhang, Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings, preprint, arXiv: 2009.00412. [7] E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Two classes of quadratic vector fields for which the Kahan discretization is integrable, MI Lecture Notes, 74, 60–62. [8] E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Integrability properties of Kahan's method, J. Phys. A, 47 (2014), 365202. doi: 10.1088/1751-8113/47/36/365202. [9] F. Cossec and I. V. Dolgachev, Enriques Surfaces I, Progress in Mathematics, 76, Birkhäuser, Boston, 1989. doi: 10.1007/978-1-4612-3696-2. [10] I. V. Dolgačhev, Rational surfaces with a pencil of elliptic curves, Izv. Akad. Nauk SSSR Ser. Mat., 30 (1966), 1073-1100. [11] J. J. Duistermaat, Discrete integrable systems, QRT Maps and Elliptic Surfaces, Springer, New York, 2010. doi: 10.1007/978-0-387-72923-7. [12] F. A. Haggar, G. B. Byrnes, G. R. W. Quispel and H. W. 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. [13] 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. [14] N. J. Hitchin, N. S. Manton and M. K. Murray, Symmetric monopoles, Nonlinearity, 8 (1995), 661-692.  doi: 10.1088/0951-7715/8/5/002. [15] A. Iatrou and J. A. G. 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. [16] D. Jogia, J. A. G. 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. [17] N. Joshi, B. Grammaticos, T. Tamizhmami and A. Ramani, From integrable lattices to non-QRT mappings, Lett. Math. Phys., 78 (2006), 27-37.  doi: 10.1007/s11005-006-0103-5. [18] 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. [19] 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. [20] R. C. Lyness, Note 1581, Math. Gaz., 26 (1942), 62. [21] Yu. I. Manin, The Tate height of points on an Abelian variety., AMS Translations Ser., 2 (1966), 82-110. [22] E. I. Moody, Notes on the Bertini involution, Bull. Amer. Math. Soc., 49 (1943), 433-436.  doi: 10.1090/S0002-9904-1943-07940-2. [23] [24] M. Petrera, A. Pfadler and Y. B. Suris, On integrability of Hirota-Kimura type discretizations, Regul. Chaotic Dyn., 16 (2011), 245-289.  doi: 10.1134/S1560354711030051. [25] M. Petrera, J. Smirin and Y. B. Suris, Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems, Proc. A., 475 (2019), 20180761. doi: 10.1098/rspa.2018.0761. [26] M. Petrera and Y. B. Suris, Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor, J. Comput. Dyn., 6 (2019), 401-408.  doi: 10.3934/jcd.2019020. [27] M. Petrera and Y. B. Suris, Manin involutions for elliptic pencils and discrete integrable systems, arXiv: 2008.08308. [28] G. R. W. Quispel, J. A. G. Roberts and C. J. Thompson, Integrable mappings and soliton equations, Phys. Lett. A, 126 (1988), 419-421.  doi: 10.1016/0375-9601(88)90803-1. [29] G. R. W. Quispel, J. A. G. Roberts and C. J. Thompson, Integrable mappings and soliton equations Ⅱ, Physica D, 34 (1989), 183-192.  doi: 10.1016/0167-2789(89)90233-9. [30] J. A. G. Roberts and G. R. W. 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. [31] T. Sakkalis and R. Farouki, Singular points of algebraic curves, J. Symbolic Comput., 9 (1990), 405-421.  doi: 10.1016/S0747-7171(08)80019-3. [32] J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4757-4252-7. [33] G. K. Suslov, Theoretical Mechanics, Gostekhizdat, Moskva, 1946, (in Russian). [34] T. Tsuda, Integrable mapping via rational elliptic surfaces, J. Phys. A, 37 (2004), 2721-2730.  doi: 10.1088/0305-4470/37/7/014. [35] P. H. van der Kamp, A new class of integrable maps of the plane: Manin transformations with involution curves, preprint, arXiv: 2009.09854. [36] P. H. van der Kamp, E. Celledoni, R. I. McLachlan, D. I. McLaren, B. Owren and G. R. W. Quispel, Three classes of quadratic vector fields for which the Kahan discretisation is the root of a generalised Manin transformation, J. Phys. A: Math. Theor., 52 (2019), 045204, 10 pp. [37] J. van Yzeren, A simple proof of Pascal's hexagon theorem, Amer. Math. Monthly, 100 (1993), 930-931.  doi: 10.1080/00029890.1993.11990514. [38] A. P. Veselov, Integrable mappings, Russ. Math. Surv., 46 (1991), 1-51.  doi: 10.1070/RM1991v046n05ABEH002856. [39] C. M. Viallet, B. Grammaticos and A. Ramani, On the integrability of correspondences associated to integral curves, Phys. Lett. A, 322 (2004), 186-193.  doi: 10.1016/j.physleta.2004.01.013.
Ten curves from the quadratic pencil defined by (2) and (13), labeled by the value of $-\beta/\alpha$. The base points are $(1,0)$, $(0,-1)$, $(-2,0)$, $(2,0)$
Lines through opposite sides of a hexagon on a conic meet in three points which lie on a straight line, called the Pascal line
Ten curves from the cubic pencil defined by (2) and (22), labeled by the value of $-\beta/\alpha$
Ten curves from the quartic pencil defined by (2), (25) and (26), labeled by the value of $-\beta/\alpha$
Six iterations of the point $(-\frac32,\frac 3{10})$ under the Manin transformation (27), $\iota_{0,1}\circ\iota_{0,0}$
The base points lie on curves defined by the numerators and denominators of $A$ (pink) and $B$ (grey)
A degree 2 curve, given by (32), which admits the symmetry switch (31). The symmetry switch is a reflection in the line through $(0,0)$ perpendicular to $W = (10,-3)$ (purple), in the direction $(2,-1)$ (dotted)
A degree 5 curve does not intersect a line in 6 points
Any degree 5 curve with two triple points contains the line through the triple points
A product of lines admitting fractional linear symmetries, cf. Example 8
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