Computational Aspects of Monomial Dynamical Systems

We consider the dynamics of x & x(n), where n >= 2 is an integer, over the multiplicative group modulo p(k), where k is a positive integer and p an odd prime. This paper is a review of earlier results by the author, but new results are also contained. Possible applications to pseudorandom number generation will be discussed. The main results are a description of the preperiodic points and an algorithm to find the longest possible cycle. The preperiodic points form trees, all isomorphic as graphs to the preperiodic points of the fixed point 1. When n is a prime, different from p, we can describe the tree structure completely. A formula for the length of the longest cycle is presented. We can find one of the longest cycles of the monomial system using a primitive root modulo p(k) as an initial value.