The model of quotient fractal based on the theory of quotient space

In this paper, the relationship between theory of fractal geometry and theory of quotient space is discussed and a new model which combines the character of granularity and fractal is put forward. Furthermore, approaching to fractal graph with quotient granularity is investigated. Some conclusions are proved separately. 1) Given a function iterative system{X,w_i,s_i,i=1,2,hellip,n}, a hierarchical structure {X_k,k=1,2,hellip} can be proved, and the distance is induced on the quotient chain {X_k,k=1,2,hellip} . 2) Given quotient mapping W_k and quotient sets P_k on X_k, then P_k are proved to be invariable sets on X_k along with W_k. 3) The model of quotient-fractal {(X_k,W_k,P_k),k=1,2,hellip} is constructed. 4) Lay X_k into primary space, {P_k} are proved that converge to P with Housdoff distance, 5) a sufficient and necessary condition which performed from a function iterative system to fractal graph is proposed.