Hutchinson's Theorem in Semimetric Spaces

TL;DR

This paper extends Hutchinson's theorem from complete metric spaces to semimetric spaces, establishing existence, uniqueness, and stability of fractals in this broader setting.

Contribution

It generalizes Hutchinson's theorem to semimetric spaces and investigates properties of fractal spaces within this new framework.

Findings

Extended Hausdorff's theorem for compactness

Extended Blaschke's theorem for completeness

Proved an analogue of Hutchinson's theorem in semimetric spaces

Abstract

One of the important consequences of the Banach Fixed Point Theorem is Hutchinson's theorem which states the existence and uniqueness of fractals in complete metric spaces. The aim of this paper is to extend this theorem for semimetric spaces using the results of Bessenyei and P\'ales published in 2017. In doing so, some properties of semimetric spaces as well as of the fractal space are investigated. We extend Hausdorff's theorem to characterize compactness and Blaschke's theorems to characterize the completeness of the fractal space. Based on these preliminaries, an analogue of Hutchinson's Theorem in the setting of semimetric spaces is proved and finally, error estimates and stability of fractals are established as well.