The categories of flows of Set and Top

Following John Kennison, a flow (or discrete dynamical system) in a category C is a couple (X, f), where X is an object of C and f: X→ X is a morphism, called the iterator. If (A, f) and (B, g) are flows in C, then h: A→ B is a morphism of flows from (A, f) to (B, g) if h{ring operator} f=g{ring operator}. h. We let Flow(C) denote the resulting category of flows in C.This paper deals with Flow(Set) and Flow(Top), where Set and Top denote respectively the categories of sets and topological spaces.By a Gottschalk flow, we mean a flow (X, f) in Top satisfying the following conditions:. (i)If x∈X is any almost periodic point of f, then the closure Of(x)̄ is a minimal set of f;(ii)All points in any minimal set of f are almost periodic points.As proven by Gottschalk, if X is a compact Hausdorff space and f:. X→ X is a continuous function, then (X, f) is a Gottschalk flow.In this paper, we prove that for any flow (X, f) of Set, there is a topology P(f) on X for which ((X,P(f)),f) is a Gottschalk flow in Top. This, actually, defines a covariant functor P from Flow(Set) into Flow(Top).The main result of this paper provides a characterization of spaces in the image of the functor P in order-theoretical terms.Some categorical properties of Flow(Set) and Flow(Top) are also given.