Generalised Manin transformations and QRT maps

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

doi:10.3934/jcd.2021009

Article Contents

2021, Volume 8, Issue 2: 183-211. Doi: 10.3934/jcd.2021009

This issue Previous Article Chebyshev spectral methods for computing center manifolds Next Article On the influence of cross-diffusion in pattern formation

Generalised Manin transformations and QRT maps

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

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

Received: September 2020

Revised: January 2021

Early access: March 2021

Published: April 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 14E05, 14H70, 39A20; Secondary: 14H81, 37J35.

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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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