## Abstract

Let f:X→X be a continuous map on a compact metric space X and let α
_{f}, ω
_{f} and ICT
_{f} denote the set of α-limit sets, ω-limit sets and nonempty closed internally chain transitive sets respectively. We show that if the map f has shadowing then every element of ICT
_{f} can be approximated (to any prescribed accuracy) by both the α-limit set and the ω-limit set of a full-trajectory. Furthermore, if f is additionally expansive then every element of ICT
_{f} is equal to both the α-limit set and the ω-limit set of a full-trajectory. In particular this means that shadowing guarantees that α
_{f}‾=ω
_{f}‾=ICT
_{f} (where the closures are taken with respect to the Hausdorff topology on the space of compact sets), whilst the addition of expansivity entails α
_{f}=ω
_{f}=ICT
_{f}. We progress by introducing novel variants of shadowing which we use to characterise both maps for which α
_{f}‾=ICT
_{f} and maps for which α
_{f}=ICT
_{f}.

Original language | English |
---|---|

Article number | 124291 |

Number of pages | 19 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 491 |

Issue number | 1 |

Early online date | 3 Jun 2020 |

DOIs | |

Publication status | Published - 1 Nov 2020 |

## Keywords

- shadowing
- α-limit set
- ω-limit set
- Internally chain transitive
- Expansive
- pseudo-orbit
- Shadowing
- Pseudo-orbit

## ASJC Scopus subject areas

- General Mathematics
- Analysis
- Applied Mathematics