A dynamical system consists of a set V and a map f : V → V . The primary goal is to characterize points in V according to their limiting behaviors under iteration of the map f . Especially understanding dynamics of nonlinear maps is an important but difficult problem, and there are not many methods available. This work concentrates on dynamics of certain nonlinear maps over finite fields. First we study monomial dynamics over finite fields. We show that determining the number of fixed points of a boolean monomial dynamics is #P–complete problem and consider various cases in which the dynamics can be explained efficiently. We also extend the result to the monomial dynamics over general finite fields. Then we study the dynamics of a simple nonlinear map, f(x) = x + x-1, on fields of characteristic two. The main idea is to lift the map f to a proper finite covering map whose dynamics is easier to understand. We lift the map of f to an isogeny g on an elliptic curve where the dynamics of g can be further reduced to that of a linear map on Z –module. As an application of finite covering, we construct a new family of permutation maps over finite fields from the known permutation maps
