ABSTRACT ω-LIMIT SETS

The shift map sigma on omega* is the continuous self-map of omega* induced by the function n -> n +1 on omega. Given a compact Hausdorff space X and a continuous function f : X -> X, we say that (X, f) is a quotient of (omega*, sigma) whenever there is a continuous surjection Q : omega* -> X such that Q circle sigma = sigma circle f. Our main theorem states that if the weight of X is at most & alefsym;(1), then (X, f) is a quotient of (omega*, sigma) if and only if f is weakly incompressible (which means that no nontrivial open U subset of X has f(<(U)overbar>) subset of U). Under CH, this gives a complete characterization of the quotients of (omega*, sigma) and implies, for example, that (omega*, sigma(-1)) is a quotient of (omega*, sigma). In the language of topological dynamics, our theorem states that a dynamical system of weight aleph(1) is an abstract omega-limit set if and only if it is weakly incompressible. We complement these results by proving (1) our main theorem remains true when aleph(1) is replaced by any k < p, (2) consistently, the theorem becomes false if we replace aleph(1) by aleph(2), and (3) OCA + MA implies that (omega*, sigma(-1)) is not a quotient of (omega*, sigma).