On the Continuity of the Hutchinson Operator

Abstract

We investigate when the Hutchinson operator associated with an iterated function system is continuous. The continuity with respect to both the Hausdorff metric and Vietoris topology is carefully considered. An example showing that the Hutchinson operator on the hyperspace of nonempty closed bounded sets need not be Hausdorff continuous is given. Infinite systems are also discussed. The work clarifies and generalizes several partial results scattered across the literature.

Keywords

• iterated function system
• Hutchinson operator
• multifunction
• boundedly uniformly continuous map
• upper semicontinuity
• strict attractor
• Vietoris continuity
• compact-open topology