On dynamics of quadratic maps over finite fields of characteristic two

Abstract

Quadratic functions defined over finite fields of characteristic two are studied as discrete dynamical systems, especially in terms of classification, nonperiodic points, periodic points, and conjugacy. These functions are organized according to linearity their distinguished elements are counted, and their similarities are considered. By conjugacy and linearity, quadratic functions are categorized into three one-parameter families. One consists of linear transformations, another of affine ones, and the last a single map. The linear transformations are studied by constructing a basis for the underlying field, allowing their non-periodic and periodic points to be described. The affine transformations are associated with linear ones, characterizing their distinguished points. The special points of the remaining map are shown to depend solely on the size of the finite field. Finally, equivalence between maps in each family is given.