Realization of a snowflaked interval as a Euclidean self-similar set

Fatma Diğdem Koparal, Yunus Özdemir, Derya Çelik and Şahin Koçak

Chaos, Solitons & Fractals, 2020, vol. 139, issue C

Abstract: The metric space ([0, 1], dα) with 0 < α < 1 is called a snowflaked version of the interval [0,1] with the standard metric d. Assouad has shown in 1983 that such a snowflaked interval can be embedded bi-Lipschitzly into RNwhere N=[[1α]]+1. We give an alternative proof of this nice theorem in terms of iterated function systems (IFS). We construct three similitudes on RNsuch that the image of the snowflaked interval under our bi-Lipschitz embedding becomes the attractor of the IFS consisting of these three similitudes. In this way the image of the bi-Lipschitz embedding becomes a self-similar subset of RNwith Hausdorff dimension 1α.

Keywords: Snowflake metric space; Assouad’s theorem; Bi-Lipschitz embedding; Iterated function system; Self-similar set (search for similar items in EconPapers)
Date: 2020
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:139:y:2020:i:c:s096007792030583x

DOI: 10.1016/j.chaos.2020.110187

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