Attractors in hyperspace
KeywordsHyperspace, global attractors, dynamical systems, iterated functions systems
AbstractGiven a map $\Phi$ defined on bounded subsets of the (base) metric space $X$ and with bounded sets as its values, one can follow the orbits $A$, $\Phi(A)$, $\Phi^2(A)$, $\ldots$, of nonempty, closed, and bounded sets $A$ in $X$. This is the system $(\Phi, X)$. On the other hand, the same orbits can be viewed as trajectories of points in the hyperspace $X^\sharp$ of nonempty, closed, and bounded subsets of $X$. This is the system $(\Phi, X^\sharp)$. We study the existence and properties of global attractors for both $(\Phi, X)$ and $(\Phi, X^\sharp)$. We give very basic conditions on $\Phi$, stated at the level of the base space $X$, that are necessary and sufficient for the existence of a global attractor for $(\Phi, X)$. Continuity is not among those conditions, but if $\Phi$ is continuous in a certain sense then the attractor and the $\omega$-limit sets are $\Phi$-invariant. If $(\Phi, X)$ has a global attractor, then $(\Phi, X^\sharp)$ has a global attractor as well. Every point of the global attractor of $(\Phi, X^\sharp)$ is a compact set in $X$, and the union of all the points of that attractor is the global attractor of $(\Phi, X)$.
How to Cite
KAPITANSKI, Lev & ŽIVANOVIĆ GONZALEZ, Sanja. Attractors in hyperspace. Topological Methods in Nonlinear Analysis [online]. 12 April 2016, T. 44, nr 1, s. 199–227. [accessed 25.3.2023].
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