APPROXIMATION BY ABSOLUTELY CONTINUOUS INVARIANT MEASURES OF ITERATED FUNCTION SYSTEMS WITH PLACE-DEPENDENT PROBABILITIES

Let [Formula: see text] be the attractor (fractal) of a contractive iterated function system (IFS) with place-dependent probabilities. An IFS with place-dependent probabilities is a random map [Formula: see text] where the probabilities [Formula: see text] of switching from one transformation to another are functions of positions, that is, at each step, the random map [Formula: see text] moves the point [Formula: see text] to [Formula: see text] with probability [Formula: see text]. If the random map [Formula: see text] has a unique invariant measure [Formula: see text], then the support of [Formula: see text] is the attractor [Formula: see text]. For a bounded region [Formula: see text], we prove the existence of a sequence [Formula: see text] of IFSs with place-dependent probabilities whose invariant measures [Formula: see text] are absolutely continuous with respect to Lebesgue measure. Moreover, if [Formula: see text] is a compact metric space, we prove that [Formula: see text] converges weakly to [Formula: see text] as [Formula: see text] We present examples with computations.