Monomial Dynamical Systems of Dimension One over Finite Fields

TL;DR

This paper analyzes monomial dynamical systems over finite fields, providing formulas for periodic points and cycles, and exploring their distributions using arithmetic, graph theory, and function field techniques.

Contribution

It introduces formulas and distribution computations for periodic points and cycles in monomial dynamical systems over finite fields, linking them with function fields.

Findings

Formulas for the number of periodic points and cycles of given period

Distribution patterns of periodic points and cycles identified

Connection established between dynamical systems and function fields

Abstract

In this paper we study the monomial dynamical systems of dimension one over finite fields from the viewpoints of arithmetic and graph theory. We give formulas for the number of periodic points with period r and cycles with length r. Then we compute the natural distributions of periodic points and cycles. We also define and compute the Dirichlet distributions of periodic points and cycles. Especially, we associate the monomial dynamical systems with function fields to compute distributions.