Quotient Space Theory about An Irreducible Iterated Function System

In this paper, we use the relations of quotient space theory and martingale theory to research the iterated function system that is fractal geometry images, and propose these conclusions: Given an irreducible iterated function system {X, w_i, p_ij; i, j = 1, 2, ..., n} , then exists a corresponding chain of quotient space {W_k = (X_k, mu_k, F_k); k = 1, 2, ...} and a martingale {(mu_k, F_k); k = 1, 2, ...} on the chain, therefore there are: 1) Assume P_k is a invariant subsets of W_k, P is a invariant subsets of W, then exists lim_krarrinfin P_k = P and the convergence is according to Hausdorff distance. 2) Assume mu_k is a invariant measure of F_k, mu is a invariant measure of F, then exists lim_krarrinfin mu_k = mu. 3) P_k is a support set of mu_k, P is a support set of mu. Namely we present the quotient approximation theorem about fractal geometry images, and build relations among chain of quotient space, martingale , fractal geometry images and Markovian process.