Function digraphs of quadratic maps modulo p

In this paper we will consider geometric representations of the iteration of quadratic polynomials modulo p. This is a discrete analogue of the classical quadratic Julia sets which have been the subject of much study [3,4]. In particular, let fdm(u(x)) denote the function digraph which has Zm as vertices and edges of the form (x,u(x)) where x is an element of Zm. This digraph geometrically represents the function u(x) and paths correspond to iteration of u(x). The function digraphs resulting from squaring mod m, fdm(x), have been studied when m is prime or has a primitive root [1,2,5,10]. In particular, the cycle and tree structures have been classified. In [8] these results were generalized from fdp(x) to fdp(x) and a correspondence between geometric subsets of the function digraph and subgroups of the group of units was established. Subsequently, most of the results were generalized to general moduli in [12]. The aim of our paper is to explore these same ideas for the iteration of general quadratic functions instead of powers. In other words, we will consider fdp(a0 + a1x + a2x) where a a p 0 1 , ∈Z and