Integrable Maps which Preserve Functions with Symmetries

TL;DR

This paper introduces a method for constructing integrable maps that preserve functions built from invariants of vector fields, with applications to Yang-Baxter maps, highlighting new integrability techniques.

Contribution

It presents a reduction procedure to derive commuting, symplectic-preserving maps with first integrals, connecting to recent Yang-Baxter map developments.

Findings

Derived commuting maps preserving symplectic forms and integrals

Applied the method to recent Yang-Baxter maps

Provided a systematic reduction procedure for integrable maps

Abstract

We consider maps which preserve functions which are built out of the invariants of some simple vector fields. We give a reduction procedure, which can be used to derive commuting maps of the plane, which preserve the same symplectic form and first integral. We show how our method can be applied to some maps which have recently appeared in the context of Yang-Baxter maps.