On the Construction of Elliptic Solutions of Integrable Birational Maps

摘要

We present a systematic technique to find explicit solutions of birational maps, provided that these solutions are given in terms of elliptic functions. The two main ingredients are the following: (i) application of classical addition theorems for elliptic functions and (ii) experimental technique to detect an algebraic curve containing a given sequence of points in a plane. These methods are applied to Kahan-Hirota-Kimura discretizations of the periodic Volterra chains with three and four particles.