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

Koparal F. D., Ozdemir Y., Celik D., Kocak S.

CHAOS SOLITONS & FRACTALS, vol.139, 2020 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 139
  • Publication Date: 2020
  • Doi Number: 10.1016/j.chaos.2020.110187
  • Journal Name: CHAOS SOLITONS & FRACTALS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Compendex, INSPEC, zbMATH
  • Keywords: Snowflake metric space, Assouad's theorem, Bi-Lipschitz embedding, Iterated function system, Self-similar set
  • Anadolu University Affiliated: Yes

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.