On non-QRT Mappings of the Plane

TL;DR

This paper introduces new families of dynamical systems on the plane that generalize QRT maps, mapping bi-quadratic curves to other such curves and returning to the original after two steps, including reductions from integrable lattices.

Contribution

It constructs 9-parameter and 13-parameter dynamical systems that extend QRT maps, broadening the class of integrable mappings of the plane.

Findings

New 9-parameter and 13-parameter maps constructed.

These maps generalize QRT maps and include reductions from integrable lattices.

Maps return to the original bi-quadratic curve after two iterations.

Abstract

We construct 9-parameter and 13-parameter dynamical systems of the plane which map bi-quadratic curves to other bi-quadratic curves and return to the original curve after two iterations. These generalize the QRT maps which map each such curve to itself. The new families of maps include those that were found as reductions of integrable lattices.