On the Continuity of the Hutchinson Operator

What is it about?

The Hutchinson operator provides a way to associate the dynamics with the iterated function system, that is the collection of maps acting on a space. In the article it is answered when the Hutchinson operator induced by a collection of continuous maps is continuous. This is important for proving various properties of iterated function systems. In particular, for the validity of the chaos game algorithm for generating attractors/fractals. It turns out that the Hutchinson operator is always continuous when the hyperspace of sets is equipped with the Vietoris convergence. In the case of the continuity of the Hutchinson operator with respect to the Hausdorff metric one needs some amount of uniform continuity in maps, namely local uniform continuity. Otherwise, as suitable construction in an infinite dimensional Banach space shows, even a single continuous map may not lift up to a continuous Hutchinson operator on the hyperspace of nonempty closed bounded sets equipped with the Husdorff metric. The mentioned construction is built on ideas around local uniform continuity of nonlinear functionals on a Banach space, as studied by Izzo (A. J. Izzo "Locally uniformly continuous functions" Proc. Am. Math. Soc. 1994, vol. 122, 1095–1100).

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