Continuity properties of attractors for iterated fuzzy set systems

Abstract

Abstract An N -map Iterated Fuzzy Set System (IFZS), introduced in [4] and to be denoted as (w, Φ), is a system of N contraction maps w i : X → X over a compact metric space ( X, d) , with associated “grey level” maps ø i : [0, 1] → [0, 1]. Associated with an IFZS (w, Φ) is a fixed point u ∈ 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 φ 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 φ i maps. Two applications are (i) animation of images and (ii) the inverse problem of fractal construction.