ABSTRACT ω-LIMIT SETS

Abstract

The shift map σ on ω* is the continuous self-map of ω* induced by the function n ↦ n + 1 on ω. Given a compact Hausdorff space X and a continuous function f : X → X, we say that (X, f) is a quotient of (ω*, σ) whenever there is a continuous surjection Q : ω*→ X such that Q ○ σ = σ ○ f.Our main theorem states that if the weight of X is at most ℵ1, then (X, f) is a quotient of (ω*, σ), if and only if f is weakly incompressible (which means that no nontrivial open U ⊆ X has $f\left( {\bar{U}} \right) \subseteq U$). Under CH, this gives a complete characterization of the quotients of (ω*, σ) and implies, for example, that (ω*, σ−1) is a quotient of (ω*, σ).In the language of topological dynamics, our theorem states that a dynamical system of weight ℵ1 is an abstract ω-limit set if and only if it is weakly incompressible.We complement these results by proving (1) our main theorem remains true when ℵ1 is replaced by any κ < p, (2) consistently, the theorem becomes false if we replace ℵ1 by ℵ2, and (3) OCA + MA implies that (ω*, σ−1) is not a quotient of (ω*, σ).

Universal flows and automorphisms of P(ω)/fin

“…This corollary was first observed in Section 5 of [6], where it was also proved that the assumption of CH cannot be dropped: under OCA + MA, the shift map is not a quotient of its inverse (see Theorem 5.7 in [6] For more partial progress on this question, see [10]. In this subsection, we will see that Corollary 3.13 extends to every trivial map under CH, and that the answer to Question 3.14 may tell us something about all of them.…”

“…(2) r ↑ is universal for chain recurrent automorphisms of P(ω)/fin. Theorem 3.7(1) is the main result of [6]. The (straightforward) proof in this section therefore shows that the results in [6] are a special case of Theorem 2.4.…”

“…What is called the "Main Theorem" above is in fact just a corollary to a stronger result, which we call the "Main Lemma," but its statement is more technical and will be postponed until later. Roughly, the main lemma characterizes the weight-ℵ 1 quotients of any trivial S-flow on ω * for any countable discrete semigroup S. This improves on some former work of the author in [6], which gives a topological characterization of the weight-ℵ 1 quotients of a particular (N, +)-flow on ω * (the one generated by the shift map).…”

“…However, we claim that neither one is a quotient of the other, so neither one is universal. That OCA + MA implies (s ↑ ) −1 is not a quotient of s ↑ was proved (via Theorem 3.25) as Theorem 5.7 in [6]. The proof given there is easily adapted to show that s ↑ is not a quotient of (s ↑ ) −1 .…”

“…In proving our Main Lemma, the first part of B laszczyk and Szymański's proof goes through: every G-flow of weight ℵ 1 is a length-ω 1 inverse limit of metrizable G-flows. However, we run into trouble with the analogue of Lemma 2.7: the analogous lemma for flows is false (see Example 3.4 in [6]).…”