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

Abstract

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 $X\rightarrow X$ 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 $X\rightarrow X$ 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 $\{T_i\}$, $T_i:X_{i}\rightarrow X_{i-1}$. 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.