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 ∘ f = g ∘ h . We let Flow ( C ) denote the resulting category of flows in C. aper deals with Flow ( Set ) and Flow ( Top ) , where Set and Top denote respectively the categories of sets and topological spaces. ottschalk flow, we mean a flow ( X , f ) in Top satisfying the following conditions:(i) X is any almost periodic point of f, then the closure O f ( x ) ¯ is a minimal set of f; ints in any minimal set of f are almost periodic points. ven by Gottschalk, if X is a compact Hausdorff space and f : X → X is a continuous function, then ( X , f ) is a Gottschalk flow. s 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 ) . in result of this paper provides a characterization of spaces in the image of the functor P in order-theoretical terms. ategorical properties of Flow ( Set ) and Flow ( Top ) are also given.