Solving the Quispel–Roberts–Thompson maps using Kajiwara–Noumi–Yamada’s representation of elliptic curves

Source abstract

Abstract It is well known that the dynamical system determined by a Quispel–Roberts–Thompson map (a QRT map) preserves a pencil of biquadratic polynomial curves on CP 1 × CP 1 . In most cases this pencil is elliptic, i.e. its generic member is a smooth algebraic curve of genus one, and the system can be solved as a translation on the elliptic fiber to which the initial point belongs. However, this procedure is rather complicated to handle, especially in the normalization process. In this paper, for a given initial point on an invariant elliptic curve, we present a method to construct the solution directly in terms of the Weierstrass sigma function, using Kajiwara–Noumi–Yamada’s parametric representation of elliptic curves.