Semiconjugate Factorization and Reduction of Order in Difference Equations

TL;DR

This paper introduces a method to reduce the order of difference equations on groups by using semiconjugate relations, enabling simpler analysis and solutions for complex equations.

Contribution

It presents a general technique for transforming higher order difference equations into lower order systems via semiconjugate factorization, applicable to broad classes of equations.

Findings

Reduction of order for many difference equations

Existence of semiconjugate relations in classes of equations

Complete factorization into first order systems possible

Abstract

We discuss a general method by which a higher order difference equation on a group is transformed into an equivalent triangular system of two difference equations of lower orders. This breakdown into lower order equations is based on the existence of a semiconjugate relation between the unfolding map of the difference equation and a lower dimensional mapping that unfolds a lower order difference equation. Substantial classes of difference equations are shown to possess this property and for these types of equations reductions of order are obtained. In some cases a complete semiconjugate factorization into a triangular system of first order equations is possible.