### Abstract

A set Λ is internally chain transitive if for any x,yΛ and >0 there is an -pseudo-orbit in Λ between x and y. In this paper we characterize all ω-limit sets in shifts of finite type by showing that, if Λ is a closed, strongly shift-invariant subset of a shift of finite type, X, then there is a point zX with ω(z)=Λ if and only if Λ is internally chain transitive. It follows immediately that any closed, strongly shift-invariant, internally chain transitive subset of a shift space over some alphabet is the ω-limit set of some point in the full shift space over . We use similar techniques to prove that, for a tent map f, a closed, strongly f-invariant, internally chain transitive subset of the interval is the ω-limit set of a point provided it does not contain the image of the critical point. We give an example of a sofic shift space ZG (a factor of a shift space of finite type) that is not of finite type that has an internally chain transitive subset that is not the ω-limit set of any point in ZG.

Translated title of the contribution | A characterization of omega-limit sets in shift spaces |
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Original language | English |

Pages (from-to) | 21 - 31 |

Number of pages | 11 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 30 |

Issue number | 1 |

Early online date | 26 Feb 2009 |

DOIs | |

Publication status | Published - 1 Feb 2010 |

### Bibliographical note

Publisher: Cambridge University Press## Fingerprint Dive into the research topics of 'A characterization of ω-limit sets in shift spaces'. Together they form a unique fingerprint.

## Cite this

Barwell, AD., Good, C., Knight, R., & Raines, BE. (2010). A characterization of ω-limit sets in shift spaces.

*Ergodic Theory and Dynamical Systems*,*30*(1), 21 - 31. https://doi.org/10.1017/S0143385708001089