In this work we present iterated function systems with general measures (IFSm) formed by a set of maps _ acting over a compact space X, for a compact space of indices,. The Markov process Zₖ associated to the IFS iteration is defined using a general family of probabilities measures qₓ on, where x X: Z₊+₁ is given by _ (Zₖ), with randomly chosen according to qₓ. We prove the existence of the topological attractor and the existence of the invariant attracting measure for the Markov Process. We also prove that the support of the invariant measure is given by the attractor and results on the stochastic stability of the invariant measures, with respect to changes in the family qₓ.
Oliveira et al. (Wed,) studied this question.