Periodicity and Dynamical Systems of Dickson Polynomials in Finite Fields

This paper investigates the dynamical properties of the Dickson polynomials $D_n(x, \alpha)$ of the first kind over finite fields, with an emphasis on the periodicity and algebraic structure of their iterated sequences. We consider the sequence $[D_n(x, \alpha) \pmod{x^q - x}]_{n\geq1}$, and determine the exact period of this sequence. We then use the classical functional equation for the Dickson polynomials to relate the dynamics of $D_n(x,\alpha)$ 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.