Equivalence among orbital equations of polynomial maps

TL;DR

This paper demonstrates that orbital equations from polynomial map iterations can be represented in infinitely many equivalent forms connected by nonlinear transformations, with explicit examples and recursive methods for generating these equivalents.

Contribution

It introduces explicit nonlinear transformations linking different orbital equations and recursive sequences to generate infinitely many equivalent representations.

Findings

Explicit transformations between orbital equations are established.

Infinite recursive sequences of transformations are introduced.

Transformations are applicable to general algebraic dynamical systems.

Abstract

This paper shows that orbital equations generated by iteration of polynomial maps do not have necessarily a unique representation. Remarkably, they may be represented in an infinity of ways, all interconnected by certain nonlinear transformations. Five direct and five inverse transformations are established explicitly between a pair of orbits defined by cyclic quintic polynomials with real roots and minimum discriminant. In addition, infinite sequences of transformations generated recursively are introduced and shown to produce unlimited supplies of equivalent orbital equations. Such transformations are generic, valid for arbitrary dynamics governed by algebraic equations of motion.