Hausdorff dimensions of self-similar sets and shortest path metrics
type: Item
instance of: scholarly article
title: Hausdorff dimensions of self-similar sets and shortest path metrics
zbMATH Open document ID: 0851.28002
author: Jun Kigami
published in: Journal of the Mathematical Society of Japan
publication date: 1996-11-18T00:00:00Z
review text: This very interesting paper contains two main subjects. First, it studies the Hausdorff dimension of self-similar sets, where the open set condition cannot be applied. Second, it deals with post critically finite self-similar sets (p.c.f. sets) using the ``shortest path metric''. The paper also contains a great variety of examples and remarkable observations. Recall the well-known result of P. Moran [see for example \textit{K. Falconer}: ``The geometry of fractals'' (1985; Zbl 0587.28004)]: Let \(f_i\) be the \(r_i\)-similitude of \(\mathbb{R}^n\) for \(i = 1, 2, \dots, N\); that is for all \(x,y \in \mathbb{R}^n\), \(d(f_i (x), f_i(y)) = r_i d(x,y)\), where \(d\) is the Euclidean metric on \(\mathbb{R}^n\). We suppose that the open set condition is satisfied, i.e., there exists an open set \(O \subset \mathbb{R}^n\) such that \(\cup^N_{i = 1} f_i (O) \subset O\) and \(f_i (O) \cap f_j (O) = \emptyset\) for \(i \neq j\), then the Hausdorff dimension of \(K\) such that \(K = \cup_{i = 1}^N f_i (K)\) is the unique number \(\alpha\) that satisfies \(\sum^N_{i = 1} r^\alpha_i = 1\). The author generalizes this result using instead of the ordinary metric \(d\) the effective resistance metric introduced in his previous paper [Math. Proc. Camb. Philos. Soc. 115, No. 2, 291-303 (1994; Zbl 0803.60074)]. He is able to apply his Theorem to estimate the dimension of certain fractals as for example the Levy's curve. In the second part of this work it is given a necessary and sufficient condition for the existence of shortest path metric on p.c.f. sets (note that for any two points \(p\) and \(q\) on this metric space there exists a geodesic between \(p\) and \(q)\). The notion p.c.f. sets, introduced by the author [Trans. Am. Math. Soc. 355, No. 2, 721-755 (1993; Zbl 0773.31009)], is a mathematical justification of the ``finitely ramified'' self-similar sets. Roughly speaking, a set \(K\) as above is finitely ramified if \(\# (\cup_{i \not \equiv j} (f_i (K) \cap f_j (K)))\) is …
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 800604
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: self-similar sets
zbMATH Keywords: resistance metric
zbMATH Keywords: shortest path metric
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04730381
OpenAlex ID: W1995020887
DOI: 10.2969/JMSJ/04730381
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description: scientific article; zbMATH DE number 800604