Quotients of \(\mathbb{N}^\), \(\omega\)-limit sets, and chain transitivity

\(\mathbb{N}^* = \beta\mathbb{N} \setminus \mathbb{N}\) has a canonical dynamical structure provided by the shift map, the unique continuous extension to \(\beta\mathbb{N}\) of the map \(n \mapsto n+1\) on \(\mathbb{N}\). Here we investigate the question of what dynamical systems can be written as q...

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Bibliographic Details
Published in:arXiv.org 2019-04
Main Author: Brian, William R
Format: Article
Language:English
Subjects:
Chains
Dynamical systems
Quotients
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Online Access:Get full text
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Summary:\(\mathbb{N}^* = \beta\mathbb{N} \setminus \mathbb{N}\) has a canonical dynamical structure provided by the shift map, the unique continuous extension to \(\beta\mathbb{N}\) of the map \(n \mapsto n+1\) on \(\mathbb{N}\). Here we investigate the question of what dynamical systems can be written as quotients of \(\mathbb{N}^*\). We prove that a dynamical system is a quotient of \(\mathbb{N}^*\) if and only if it is isomorphic to the \(\omega\)-limit set of some point in some larger system. This provides a full external characterization of the quotients of \(\mathbb{N}^*\). We also prove, assuming MA\(_{\sigma\text{-centered}}(\kappa)\), that a dynamical system of weight \(\kappa\) is a quotient of \(\mathbb{N}^*\) if and only if it is chain transitive. This provides a consistent partial internal characterization of the quotients of \(\mathbb{N}^*\), and a full internal characterization for metrizable systems.
ISSN:2331-8422