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In this paper we study the monomial dynamical systems of dimension one overfinite fields from the viewpoints of arithmetic and graph theory. We giveformulas for the number of periodic points with period r and cycles with lengthr. Then we compute the natural distributions of periodic points and cycles. Wealso define and compute the Dirichlet distributions of periodic points andcycles. Especially, we associate the monomial dynamical systems with functionfields to compute distributions.

Monomial Dynamical Systems of Dimension One over Finite Fields