Quotients of ℕ^*, ω-limit sets, and chain transitivity

2014/12/31 by Brian, William R.
#Dynamical Systems (math.DS) #FOS: Mathematics #General Topology (math.GN)

paper · doi:10.48550/arxiv.1501.00157

Abstract

ℕ^* = βℕ ∖ ℕ has a canonical dynamical structure provided by the shift map, the unique continuous extension to βℕ of the map n ↦ n+1 on ℕ. Here we investigate the question of what dynamical systems can be written as quotients of ℕ^*. We prove that a dynamical system is a quotient of ℕ^* if and only if it is isomorphic to the ω-limit set of some point in some larger system. This provides a full external characterization of the quotients of ℕ^*. We also prove, assuming MA_σ-centered(κ), that a dynamical system of weight κ is a quotient of ℕ^* if and only if it is chain transitive. This provides a consistent partial internal characterization of the quotients of ℕ^*, and a full internal characterization for metrizable systems.

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