Hypercomplex Iterated Function Systems

TL;DR

This paper introduces hypercomplex iterated function systems (IFS) on metric spaces, defining their attractors and exploring how backward trajectories can produce fractal shapes with diverse local structures.

Contribution

It presents the novel concept of hypercomplex IFSs, expanding fractal theory into hypercomplex spaces and analyzing their attractors and backward trajectories.

Findings

Hypercomplex IFSs have well-defined attractors.

Backward trajectories can generate diverse fractal shapes.

Attractors exhibit different local (fractal) structures.

Abstract

We introduce the novel concept of hypercomplex iterated function system (IFS) on the complete metric space $(\mathbb{A}_{n+1}^k,d)$ and define its hypercomplex attractor. Systems of hypercomplex function systems arising from hypercomplex IFSs and their backward trajectories are also introduced and it is shown that the attractors of such backward trajectories possess different local (fractal) shapes.