"Chaos Games" for Iterated Function Systems with Grey Level Maps

Abstract

Two random iteration algorithms, or games, for iterated function systems (IFS) on function spaces, namely IFS with grey level maps (IFSM), are described. The first algorithm can be interpreted as a game in code space and is guaranteed to work only in the case of nonoverlapping IFS maps. In the second algorithm, applicable to IFSM with overlapping IFS maps but affine grey level maps, the (normalized) IFSM attractor function $\bar u$ serves as the density for an invariant measure $\bar \mu$ of an IFS with probabilities with condensation measure. As such, approximations to the attractor function of the IFSM are yielded by visitation histograms, as in the case of IFS with probabilities on measure spaces. Some computer results illustrating the convergence of this chaos game for a simple overlapping IFSM on [0,1] are also presented.