A New Class of Integrable Maps of the Plane: Manin Transformations with Involution Curves
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"A New Class of Integrable Maps of t..." refers background or methods in this paper
...to a QRT map [14, 15] through a projective collineation....
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...A rather large (18-parameter) family of integrable maps of the plane was obtained by Quispel, Roberts and Thompson (QRT) [14, 15]....
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...Recall that the QRT map preserves a special N = 4 pencil, namely a biquadratic pencil....
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...The corresponding involutions are the horizontal switch ι1, and the vertical switch ι2, and the QRT map is the composition τ = ι2 ◦ ι1 [5, p. viii]....
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...In [18] it was shown that this construction does not generalise to maps which preserve a pencil of degree > 4 and that all generalised Manin transformations obtained in this way are equivalent to a QRT map [14, 15] through a projective collineation....
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279 citations
"A New Class of Integrable Maps of t..." refers background in this paper
...For planar maps, integrability is equivalent to the preservation of a pencil of curves and measure-preservation [8]....
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200 citations
"A New Class of Integrable Maps of t..." refers methods in this paper
...One way to obtain such maps is by reduction from integrable lattice equations [12, 19]....
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182 citations
"A New Class of Integrable Maps of t..." refers background in this paper
...One clear definition of an integrable map is the notion of Liouville integrability, which requires the existence of sufficiently many invariant functions in involution with each other [6, 20]....
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