Hausdorff dimensions of self-similar sets and shortest path metrics
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 finite. Note: Recently, similar problems were examined with the ``interior metric''; see \textit{C. Bandt} and \textit{T. Kuschel} [Suppl. Rend. Circ. Mat. Palermo, II. Ser. 28, 307-317 (1992; following review)].
scientific article; zbMATH DE number 800604