### Abstract

Given a map Φ defined on bounded subsets of the (base) metric space X and with bounded sets as its values, one can follow the orbits A, Φ (A), Φ^{2}(A), …, of nonempty, closed, and bounded sets A in X. This is the system (Φ, X). On the other hand, the same orbits can be viewed as trajectories of points in the hyperspace X^{♯} of nonempty, closed, and bounded subsets of X. This is the system (Φ, X^{♯}). We study the existence and properties of global attractors for both (Φ, X) and (Φ, X^{♯}). We give very basic conditions on Φ, stated at the level of the base space X, that are necessary and suffcient for the existence of a global attractor for (Φ, X). Continuity is not among those conditions, but if Φ is continuous in a certain sense then the attractor and the ω-limit sets are Φ-invariant. If (Φ, X) has a global attractor, then (Φ, X^{♯}) has a global attractor as well. Every point of the global attractor of (Φ, X^{♯}) is a compact set in X, and the union of all the points of that attractor is the global attractor of (Φ, X).

Original language | English (US) |
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Pages (from-to) | 199-227 |

Number of pages | 29 |

Journal | Topological Methods in Nonlinear Analysis |

Volume | 44 |

Issue number | 1 |

State | Published - Sep 1 2014 |

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### Keywords

- Dynamical systems
- Global attractors
- Hyperspace
- Iterated functions systems

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Topological Methods in Nonlinear Analysis*,

*44*(1), 199-227.