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Title: Realization of a snowflaked interval as a Euclidean self-similar set
Authors: Koparal, Fatma Diğdem
Özdemir, Yunus
Çelik, Derya
Koçak, Şahin
Keywords: Snowflake metric space
Assouad's theorem
Bi-Lipschitz embedding
Iterated function system
Self-similar set
Issue Date: 2020
Publisher: Pergamon-Elsevier Science Ltd
Abstract: The metric space ([0, 1], d(alpha)) with 0 < alpha < 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 R-N where N = [[1/alpha]] + 1. We give an alternative proof of this nice theorem in terms of iterated function systems (IFS). We construct three similitudes on R-N such 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 R-N with Hausdorffdimension 1/alpha. (C) 2020 Elsevier Ltd. All rights reserved.
ISSN: 0960-0779
Appears in Collections:Scopus İndeksli Yayınlar Koleksiyonu
WoS İndeksli Yayınlar Koleksiyonu

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