标题： 有限域中Dickson多项式的周期性和动力系统 显示英文标题

标题： Periodicity and Dynamical Systems of Dickson Polynomials in Finite Fields

摘要： 本文研究了有限域上Dickson多项式的动力学性质，重点关注其迭代序列的周期性和结构行为。 我们引入并分析了序列\( [D_n(x, \alpha) \mod (x^q - x)]_n \)，其中\( D_n(x, \alpha) \)表示第一类Dickson多项式，并探讨了在模\( x^q - x \)下其周期性。 我们推导了这些序列周期的显式公式，特别是在\( n \)与\( q^2 - 1 \)互质的情况下。 此外，我们识别了多项式系数的一个对称性质，该性质在这些序列的分析中起着关键作用。 利用组合数学、初等数论和有限域的工具，我们提出了计算精确周期并研究这些多项式动力学结构的算法。 我们还强调了当次数\( n \)不与\( q^2 - 1 \)互质时的开放问题。 我们的结果为Dickson多项式的代数结构及其在有限域上的动力系统中的作用提供了深入的见解。 显示英文摘要

摘要： 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.