## 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 { (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.

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