Periodicity and Dynamical Systems of Dickson Polynomials in Finite Fields

This paper investigates the dynamical properties of Dickson polynomials over finite fields, focusing on the periodicity and structural behavior of their iterated sequences. We introduce and analyze the sequence $[D_n(x, \alpha) \mod (x^q - x)]_n$, where $D_n(x, \alpha)$ denotes a Dickson polynomial of the first kind, and explore its periodic nature when reduced modulo $x^q - x$. We derive explicit formulas for the period of these sequences, particularly in the case when $n$ is coprime to $q^2 - 1$. In addition, we identify a symmetric property of the polynomial coefficients that plays a crucial role in the analysis of these sequences. Using tools from combinatorics, elementary number theory, and finite fields, we present algorithms to compute the exact period and investigate the dynamical structure of these polynomials. We also highlight open problems in cases where the degree $n$ is not coprime to $q^2 - 1$. Our results offer deep insights into the algebraic structure of Dickson polynomials and their role in dynamical systems over finite fields.