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The dynamics of linearized polynomials

Published online by Cambridge University Press:  20 January 2009

Stephen D. Cohen  and
Dirk Hachenberger
Stephen D. Cohen
Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, UK ([email protected])
Dirk Hachenberger
Affiliation:
Institut für Mathematik, Universität Augsburg, 86159 Augsburg, Germany ([email protected])
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Abstract

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Let F = GF(q). To any polynomial G ∈ F[x] there is associated a mapping Ĝ on the set IF of monic irreducible polynomials over F. We present a natural and effective theory of the dynamics of Ĝ for the case in which G is a monic q-linearized polynomial. The main outcome is the following theorem.

Assume that G is not of the form , where l ≥ 0 (in which event the dynamics is trivial). Then, for every integer n ≥ 1 and for every integer k ≥ 0, there exist infinitely many μ ∈ IF. having preperiod k and primitive period n with respect to Ĝ.

Previously, Morton, by somewhat different means, had studied the primitive periods of Ĝ when G = xq – ax, α a non-zero element of F. Our theorem extends and generalizes Morton's result. Moreover, it establishes a conjecture of Morton for the class of q-linearized polynomials.

Keywords

finite fieldpolynomial dynamicslinearized polynomialperiod
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 1 , February 2000 , pp. 113 - 128
Copyright
Copyright © Edinburgh Mathematical Society 2000

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