# Attractors of Trees of Maps and of Sequences of Maps between Spaces with Application to Subdivision

## TL;DR

This paper generalizes fixed point theorems for sequences and trees of maps in metric spaces, demonstrating convergence to attractors, including fractals, with applications to subdivision schemes and complex structures.

## Contribution

It introduces new convergence results for infinite trees and staircase sequences of maps, extending the theory of attractors and fractals in metric spaces.

## Key findings

- Backward trajectories converge to unique attractors under new conditions.
- Trees of maps can generate scale and location dependent fractals.
- Staircase sequences link subdivision schemes to attractors of function systems.

## Abstract

In a previous paper we considered a sequence of maps on a complete metric space $(X,d)$ and derived an extension of the Banach fixed point theorem. We showed that backward trajectories of maps $X\to X$ converge under mild conditions and that they can generate new types of attractors such as scale dependent fractals. Here we present two generalisations of this result and some potential applications. First, we study the structure of an infinite tree of maps $X\to X$ 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 $\{T_i\}$, $T_i:X_{i}\to X_{i-1}$. 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.