Hutchinson's Theorem in Semimetric Spaces
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Publication:6384396
DOI10.1007/S11784-022-00991-4zbMATH Open1508.54013arXiv2111.15553WikidataQ115230390 ScholiaQ115230390MaRDI QIDQ6384396
Publication date: 30 November 2021
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.
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