Components of Invariant Variety of Periodic Points and Fundamental Domains of Recurrence Equation

TL;DR

This paper explores the relationship between invariant varieties of periodic points and fundamental domains of recurrence equations, introducing an algorithm to derive all IVPP components for rational maps, demonstrated on 2D and 3D examples.

Contribution

It presents a novel algorithm for deriving all IVPP components of rational maps, utilizing cyclotomic polynomials for the 2D case.

Findings

All IVPP components of the 2D map are completely determined by cyclotomic polynomials.

The duality between IVPP components and fundamental domains is established.

The algorithm applies to maps of any dimension, demonstrated on 2D and 3D examples.

Abstract

In this paper, we discuss duality about components of invariant variety of periodic points(IVPP) and fundamental domain of recurrence equation, and present an algorithm for the derivation of all components of IVPPs of any rational maps. It is based on the study of two examples of a 2 dimensional map and a 3 dimensional map. In particular, all components of IVPPs of the 2 dimensional example are completely determined by means of the cyclotomic polynomials.