J Austral Math Soc Ser B 36 pp175--193, 1994.

Continuity properties of attractors for treated fuzzy set systems

B. Forte, M. Lo Schiavo and E. R. Vrscay

Abstract

An N-map Iterated Fuzzy Set System (IFZS), introduced in [4] and to be denoted as (w, PHI), is a system of N contraction maps w_i : X -> X over a compact metric space (X, d ), with associated "grey level" maps phi_i : [0, 1] -> [0, 1]. Associated with an IFZS (w, PHI) is a fixed point u /belongs F^*(X), the class of normalized fuzzy sets on X, u : X -> [0, 1]. We are concerned with the continuity properties of u with respect to changes in the w_i, and the phi_i. Establishing continuity for the fixed points of IFZS is more complicated than for traditional Iterated Function Systems (IFS) with probabilities since a composition of functions is involved. Continuity at each specific attractor u may be established over a suitably restricted domain of phi_i maps. Two applications are (i) animation of images and (ii) the inverse problem of fractal construction.

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Authors

B. Forte

E. R. Vrscay

Department of Applied Mathematics, University of Waterloo, Waterloo, Canada N2L 3G1.

M. Lo Schiavo

Metodi e Mod. Mat. Sc. Appl., Universita' di Roma "La Sapienza", 00161 Rome, Italy.