An Ultradiscrete QRT mapping from Tropical Elliptic Curves

Abstract

Recently the area of tropical geometry has introduced the concept of the tropical elliptic group law associated with a tropical elliptic curve. This gives rise to a notion of the tropical QRT mapping. We compute the explicit tropically birational expressions that define the tropical QRT mapping for some arbitrary set of parameters. We consider this a new integrable ultradiscrete system. This also induces a new notion of tropical elliptic functions, different to the ultradiscretized elliptic functions.