Periods of Maps on Irreducible Polynomials over Finite Fields

The dynamics of permutations on irreducible polynomials

“…Proposition 5. 10. Suppose that f ∈ I k and L g is a permutation over F q k , where L g is the q-associate of g ∈ F q [x].…”

“…In [4,5], the authors explored this dynamics over I for some special classes of linearized polynomials and, in particular, they show the existence of infinitely many fixed points, i.e., orbits of length one. This study was extended in [10], where Morton showed that, for F (x) = x q − ax, a ∈ F * q , the dynamics induced by F on I yields infinitely many periodic points with period π F (f ) = n, for any positive integer n. In [7], Cohen and Hachenberger extended this last result to any linearized polynomial F (x) = s i=0 a i x q i not of the form ax q s for any a ∈ F * q .…”

The dynamics of linearized polynomials

“…In the latter three papers, the main emphasis is laid on the case in which G = x q -ax (a G F). The main result in [9] (Theorem 1) states that, for every n ^ 0, there exist infinitely many /z G IF, which are purely periodic and have primitive period n with respect to G (where G = x q -ax, a / 0 ) . In the present paper, we shall extend and generalize this result considerably.…”

“…The study of the dynamics of a mapping G on Ip was initiated by Vivaldi [13], Batra and Morton [1,2] and Morton [9]. In the latter three papers, the main emphasis is laid on the case in which G = x q -ax (a G F).…”

“…Some remarks are in order on the relationship of this work to Morton [9], which was our starting point. In his paper, Morton similarly divides the investigation into three parts, which correspond loosely to our § § 2-4, respectively.…”