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→ 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 { (Xi, di) } and an associated countable sequence of maps { Ti} , Ti: Xi→ Xi - 1. We examine conditions for the convergence of backward trajectories of the { Ti} 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.
Original language | English |
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Article number | 14 |
Journal | Journal of Fixed Point Theory and Applications |
Volume | 22 |
Issue number | 1 |
DOIs | |
State | Published - 1 Mar 2020 |
Keywords
- Fractals
- attractors
- fixed points
- function systems
- subdivision schemes