  In this paper we study the monomial dynamical systems of dimension one over
finite fields from the viewpoints of arithmetic and graph theory. We give
formulas for the number of periodic points with period r and cycles with length
r. Then we compute the natural distributions of periodic points and cycles. We
also define and compute the Dirichlet distributions of periodic points and
cycles. Especially, we associate the monomial dynamical systems with function
fields to compute distributions.
