QRT Map on a Bielliptic Surface

TL;DR

This paper extends QRT maps, which are plane mappings with biquadratic invariants, by replacing shifts with elliptic curve group operations on a bielliptic surface, enriching the map's structure.

Contribution

It introduces a novel extension of QRT maps using elliptic curve group operations on bielliptic surfaces, expanding the class of integrable mappings.

Findings

Extended QRT maps on bielliptic surfaces.

Demonstrated integrability through elliptic curve group operations.

Provides new insights into mappings with biquadratic invariants.

Abstract

The family of mappings of the plane possessing a biquadratic invariant, which is known collectively as QRT maps, is composed of two involutions, one preserving a vertical shift and the other preserving a horizontal shift in the plane. In this paper, we extend the map by replacing each shift by the group operation on each of two families of elliptic curves, whose product forms a bielliptic surface.