A construction of a large family of commuting pairs of integrable symplectic birational 4-dimensional maps

TL;DR

This paper constructs a large family of 4-dimensional birational maps that are symplectic and integrable, derived from Hamiltonian systems with cubic functions, using a discretization scheme that preserves integrability.

Contribution

It introduces a novel method to generate integrable symplectic maps from cubic Hamiltonian systems via the Kahan-Hirota-Kimura discretization, ensuring commutativity and preservation of integrals.

Findings

Maps are symplectic with a perturbed structure.

Maps possess two independent integrals of motion.

Pairs of maps commute and are integrable in the Liouville-Arnold sense.

Abstract

We give a construction of completely integrable 4-dimensional Hamiltonian systems with cubic Hamilton functions. Applying to the corresponding pairs of commuting quadratic Hamiltonian vector fields the so called Kahan-Hirota-Kimura discretization scheme, we arrive at pairs of birational 4-dimensional maps. We show that these maps are symplectic with respect to a symplectic structure that is a perturbation of the standard symplectic structure on $\mathbb R^4$, and possess two independent integrals of motion, which are perturbations of the original Hamilton functions. Thus, these maps are completely integrable in the Liouville-Arnold sense. Moreover, under a suitable normalization of the original pairs of vector fields, the pairs of maps commute and share the invariant symplectic structure and the two integrals of motion.