CONTINUITY PROPERTIES OF ATTRACTORS FOR ITERATED FUZZY SET SYSTEMS

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 is-an-element-of 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.