Regular Mappings of Sequence Spaces over Finite Fields

Let K be a finite field and let X be the set of doubly infinite sequences over K, that is, X = K where Z denotes the set of integers. In this paper I shall study some problems about mappings from X to itself which are related to earlier results in topological dynamics and cryptography. The second section includes a characterization of some algebra homomorphisms in terms of ultrafilters. Elements of X will be denoted by lower case bold letters x or by sequences of lower case italic letters (x,). The set X can be given the product topology derived from the discrete topology on K. Then X is a Cantor space. In topological dynamics, continuous mappings from X to itself which commute with the shift map a: X -* X, X, /»((*.•)) = {yil where yt = f{xh xi + 1,..., x . + ^ J . It is proved in [5] that fm is an endomorphism of the shift dynamical system over K, and moreover that all endomorphisms can be generated in this way. If F : X -> X is an endomorphism then there exist integers m and n and a function / : K" -> K such that F = fx o a m