Dynamics of polynomial maps over finite fields

Let 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 ↦ x n h ( x q - 1 m ) over F q , where h ( x ) ∈ 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, one have the number of connected components, length of the cycles and number of fixed points of these class of maps.