Iterated Function Systems on Multifunctions

Dedicated to the memory of Professor Bruno Forte Summary. We introduce a method of iterated function systems (IFS) over the space of set-valued mappings (multifunctions). This is done by first considering a couple of useful metrics over the space of multifunctions F(X,Y). Some appropri-ate IFS-type fractal transform operators T: F(X,Y) → F(X,Y) are then defined which combine spatially-contracted and range-modified copies of a multifunction u to produce a new multifunction v = Tu. Under suitable conditions, the fractal trans-form T is contractive, implying the existence of a fixed-point set-valued mapping ū. Some simple examples are then presented. We then consider the inverse problem of approximation of set-valued mappings by fixed points of fractal transform operators T and present some preliminary results.