Periodicity and Dynamical Systems of Dickson Polynomials in Finite Fields

Abstract

This paper investigates the dynamical properties of the Dickson polynomials $D_n(x,\hat{I}\pm )$ of the first kind over finite fields, with an emphasis on the periodicity and algebraic structure of their iterated sequences. We consider the sequence $[D_n(x,\hat{I}\pm )(\mathrm{mod}{x}^q-x){]}_{n\ge 1}$, and determine the exact period of this sequence. We then use the classical functional equation for the Dickson polynomials to relate the dynamics of $D_n(x,\hat{I}\pm )$ to the power map $u\mapsto u^n$ on a suitable subset of ${\mathbb{F}}_{q^2}^{\times }$. In the permutation case $\gcd (n,q^2-1)=1$, this gives an explicit description of the group of Dickson polynomial functions under composition. We also obtain partial structural result in the non-permutation case. As further applications, we derive several new identities for the Dickson polynomials. Finally, we identify and prove a symmetry property of the Dickson polynomial family.

18 pages, under revision