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

Generalised Manin transformations and QRT maps

Peter H. van der Kamp , , David I. McLaren and  G. R. W. Quispel

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.

Keywords: QRT map, Manin transformation, pencil of curves, involution, measure preservation, Pascal's theorem, symmetries.
Mathematics Subject Classification: Primary: 14E05, 14H70, 39A20; Secondary: 14H81, 37J35.
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:
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[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.  Google Scholar

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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.  Google Scholar

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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. Google Scholar

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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.  Google Scholar

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[16]

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[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.  Google Scholar

[18]

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[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.  Google Scholar

[20]

R. C. Lyness, Note 1581, Math. Gaz., 26 (1942), 62. Google Scholar

[21]

Yu. I. Manin, The Tate height of points on an Abelian variety., AMS Translations Ser., 2 (1966), 82-110.   Google Scholar

[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.  Google Scholar

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Maple 2016, http://www.maplesoft.com/products/Maple/index.aspx Google Scholar

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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.  Google Scholar

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M. Petrera and Y. B. Suris, Manin involutions for elliptic pencils and discrete integrable systems, arXiv: 2008.08308. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

G. K. Suslov, Theoretical Mechanics, Gostekhizdat, Moskva, 1946, (in Russian). Google Scholar

[34]

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

[35]

P. H. van der Kamp, A new class of integrable maps of the plane: Manin transformations with involution curves, preprint, arXiv: 2009.09854. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[38]

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

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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.  Google Scholar

show all references

References:
[1]

E. Artin, Geometric Algebra, Interscience Publishers, New York, 1957. doi: 10.1002/9781118164518.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

E. Bertini, Ricerche sulle trasformazioni univoche involutorie nel piano, Annali di Mat., 8 (1877), 244-286.  doi: 10.1007/BF02420790.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

I. V. Dolgačhev, Rational surfaces with a pencil of elliptic curves, Izv. Akad. Nauk SSSR Ser. Mat., 30 (1966), 1073-1100.   Google Scholar

[11]

J. J. Duistermaat, Discrete integrable systems, QRT Maps and Elliptic Surfaces, Springer, New York, 2010. doi: 10.1007/978-0-387-72923-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

R. C. Lyness, Note 1581, Math. Gaz., 26 (1942), 62. Google Scholar

[21]

Yu. I. Manin, The Tate height of points on an Abelian variety., AMS Translations Ser., 2 (1966), 82-110.   Google Scholar

[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.  Google Scholar

[23]

Maple 2016, http://www.maplesoft.com/products/Maple/index.aspx Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

M. Petrera and Y. B. Suris, Manin involutions for elliptic pencils and discrete integrable systems, arXiv: 2008.08308. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

G. K. Suslov, Theoretical Mechanics, Gostekhizdat, Moskva, 1946, (in Russian). Google Scholar

[34]

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

[35]

P. H. van der Kamp, A new class of integrable maps of the plane: Manin transformations with involution curves, preprint, arXiv: 2009.09854. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

Figure 1.  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) $
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Figure 2.  Lines through opposite sides of a hexagon on a conic meet in three points which lie on a straight line, called the Pascal line
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Figure 3.  Ten curves from the cubic pencil defined by (2) and (22), labeled by the value of $ -\beta/\alpha $
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Figure 4.  Ten curves from the quartic pencil defined by (2), (25) and (26), labeled by the value of $ -\beta/\alpha $
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Figure 5.  Six iterations of the point $ (-\frac32,\frac 3{10}) $ under the Manin transformation (27), $ \iota_{0,1}\circ\iota_{0,0} $
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Figure 6.  The base points lie on curves defined by the numerators and denominators of $ A $ (pink) and $ B $ (grey)
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Figure 7.  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)
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Figure 8.  A degree 5 curve does not intersect a line in 6 points
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Figure 9.  Any degree 5 curve with two triple points contains the line through the triple points
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Figure 10.  A product of lines admitting fractional linear symmetries, cf. Example 8
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