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Symmetry 2015, 7(4), 1831-1840; https://doi.org/10.3390/sym7041831

On the Continuity of the Hutchinson Operator

1
Mathematical Sciences Institute, The Australian National University, Union Lane, Canberra, ACT 2601, Australia
2
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toru´n, ul. Chopina 12/18, Toru´n 87-100, Poland
*
Author to whom correspondence should be addressed.
Academic Editor: Rémi Léandre
Received: 17 August 2015 / Revised: 16 September 2015 / Accepted: 8 October 2015 / Published: 15 October 2015
(This article belongs to the Special Issue Symmetry and Fractals)
Full-Text   |   PDF [209 KB, uploaded 15 October 2015]

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. View Full-Text
Keywords: iterated function system; Hutchinson operator; multifunction; boundedly uniformly continuous map; upper semicontinuity; strict attractor; Vietoris continuity; compact-open topology iterated function system; Hutchinson operator; multifunction; boundedly uniformly continuous map; upper semicontinuity; strict attractor; Vietoris continuity; compact-open topology
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Barnsley, M.F.; Leśniak, K. On the Continuity of the Hutchinson Operator. Symmetry 2015, 7, 1831-1840.

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