Dynamics of polynomial maps over finite fields

Abstract

Let ${\mathbb{F}}_q$ be a finite field with $q$ elements and let $n$ be a positive integer. In this paper, we study the digraph associated to the map $x\mapsto x^{n}h(x^{\frac{q-1}{m}})$, where $h(x)\in {\mathbb{F}}_q[x].$ We completely determine the associated functional graph of maps that satisfy a certain condition of regularity. In particular, we provide the functional graphs associated to monomial maps. As a consequence of our results, the number of connected components, length of the cycles and number of fixed points of these class of maps are provided.

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