Nonmetrizable topological dynamical characterization of central sets

Without the restriction of metrizability, topological dynamical systems (X, 〈Ts〉s∈G) are defined and uniform recurrence and proximality are studied. Some well known results are generalized and some new results are obtained. In particular, a topological dynamical characterization of central sets in an arbitrary semigroup (G,+) is given and shown to be equivalent to the usual algebraic characterization. 0. Introduction. A topological dynamical system is usually defined to consist of a compact metric space X together with a semigroup (or group) acting on X by continuous transformations. (See [11, p. 19] or [2, Definition 6.1].) We generalize this notion by dropping the “metric” requirement and study in Section 1 natural generalizations of uniform recurrence and proximality. These generalizations turn out to be very useful, because they enable us to establish the equivalence of dynamical and algebraic characterizations of central sets. The notion of “central subset” was first developed by Furstenberg [11] in the semigroup (N,+) of natural numbers. Later Bergelson and Hindman [2] defined the notion of a central set in an arbitrary semigroup (G,+) in terms of the algebra of βG, the Stone–Čech compactification of G. They also defined a notion of *-central subset, using the natural extension of Furstenberg’s definition, and pointed out that any *-central subset of (G,+) is central in (G,+). Moreover, a result of Weiss (see [2, Theorem 6.11]) guarantees that in a countable semigroup (G,+), a subset of G is central if and only if it is *-central. In Section 2 of this paper, we show (inspired by Weiss) that if **-central is defined as the natural extension of the notion of *-central using the more general definition of topological dynamical system, 1991 Mathematics Subject Classification: 54H20, 22A15.