Attractors of trees of maps and of sequences of maps between spaces with applications to subdivision

An extension of the Banach fixed-point theorem for a sequence of maps on a complete metric space (X, d) has been presented in a previous paper. It has been shown that backward trajectories of maps X→ X converge under mild conditions and that they can generate new types of attractors such as scale-dependent fractals. Here we present two generalizations of this result and some potential applications. First, we study the structure of an infinite tree of maps X→ X and discuss convergence to a unique “attractor” of the tree. We also consider “staircase” sequences of maps, that is, we consider a countable sequence of metric spaces { (X_i, d_i) } and an associated countable sequence of maps { T_i} , T_i: X_i→ X_{i - 1}. We examine conditions for the convergence of backward trajectories of the { T_i} to a unique attractor. An example of such trees of maps are trees of function systems leading to the construction of fractals which are both scale-dependent and location dependent. The staircase structure facilitates linking all types of linear subdivision schemes to attractors of function systems.