2016/05/04 by Will Brian, Brian, Will
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #General Topology (math.GN) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1605.01385
openalex publication_date 2016/05/04 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
The shift map \σ on \ω^* is the continuous self-map of \ω^*\ninduced by the function n \↦ n+1 on \ω. Given a compact Hausdorff\nspace X and a continuous function f: X \→ X, we say that (X,f)\nis a quotient of (\ω^*,\σ) whenever there is a continuous surjection\nQ: \ω^* \→ X such that Q \∘ \σ = f \∘ Q.\n Our main theorem states that if the weight of X is at most \ℵ1, then\n(X,f) is a quotient of (\ω^*,\σ) if and only if f is weakly\nincompressible (which means that no nontrivial open U \⊆ X has\nf(\U) \⊆ U). Under CH, this gives a complete characterization of\nthe quotients of (\ω^*,\σ) and implies, for example, that\n(\ω^*,\σ-1) is a quotient of (\ω^*,\σ).\n In the language of topological dynamics, our theorem states that a dynamical\nsystem of weight \ℵ1 is an abstract \ω-limit set if and only if it\nis weakly incompressible.\n We complement these results by proving (1) our main theorem remains true\nwhen \ℵ1 is replaced by any \κ < mathfrakp, (2) consistently,\nthe theorem becomes false if we replace \ℵ1 by \ℵ2, and (3)\nOCA+MA implies that (\ω^*,\σ-1) is not a quotient of\n(\ω^*,\σ).\n