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

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.