Algebraic dynamics of polynomial maps on the algebraic closure of a finite field. I
If \(\sigma\) is a polynomial over a finite field \(F\), then it induces a map \(\widehat{\sigma}\) on the set of all irreducible monic polynomials over \(F\) by putting \(\widehat {\sigma}(g)=f\) if for every root \(\alpha\) of \(g\) in the algebraic closure \(\widehat{F}\) of \(D\) one has \(f(\sigma (\alpha))=0\). The authors prove that if \(\sigma\) is non-constant, then \(\widehat {\sigma}(x)\) has infinitely many fixed-points. Then the case \(\sigma(x)= x^ q+ ax\) (where \(q\) is the cardinality of \(F\) and \(a\neq 0\)) is considered and periodic points of the map \(\sigma: \widehat{F}\to \widehat{F}\) are studied, using the polynomial \(\Phi_{m\sigma} (x)= \prod_{d\mid m} (\sigma^ d (x)-x)^{\mu (m/d)}\), (with \(\sigma^ d\) denoting the \(d\)-th iterate of \(\sigma\), which has already shown its utility in such questions in earlier papers of the second author. In particular a formula for the number of periodic points of given order and degree (over \(F\)) is given and an interesting connection with Mersenne primes is established. Remarks: In a recent paper [Manuscr. Math. 83, 279-289 (1994; Zbl 0804.11059)] \textit{T. Pezda} determined the orders of periodic points, up to finitely many, of all polynomials over an algebraically closed field of positive characteristic. [Part II, cf. ibid. 24, No. 3, 905-932 (1994)].
scientific article; zbMATH DE number 645213
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