Attractors in hyperspace

Lev Kapitanski, Sanja Živanović Gonzalez

Research output: Contribution to journalArticle

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), Φ2(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 languageEnglish (US)
Pages (from-to)199-227
Number of pages29
JournalTopological Methods in Nonlinear Analysis
Volume44
Issue number1
StatePublished - Sep 1 2014

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Keywords

  • Dynamical systems
  • Global attractors
  • Hyperspace
  • Iterated functions systems

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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