Dimension of uniformly random self-similar fractals. (English) Zbl 1303.28013

The author proposes a model of a random, statistically self-similar fractal using similarity transformations whose translation components are fixed but whose scaling, rotation, and reflection components are chosen uniformly and independently.
Fix \(m \geq 2\) as the number of similarities to be used in the construction and \(a_1,\ldots, a_m \in \mathbb{R}^{d}\) as set of translations. Pick \((\sigma_{-},\sigma_{+} ) \subset (0,1)\) and denote \( \mathcal{O}(d)\) the orthogonal group on \(\mathbb{R}^{d}\). Both of these sets support a uniform distribution so \(T_i = a_i + r_iQ_i\) can be chosen uniformly from \((\sigma_{-},\sigma_{+}) \times \mathcal{O}(d)\). Let \(\mathbb{J}\) be the \(m-\)tree whose edges are labeled with maps \(T\) where each is chosen randomly as above. Then each path from the origin to the leaves of this tree gives a countable sequence of similarities to compose. The image of \(0\) under this infinite composition of similarities is well-defined and the fractal is the union of the image of \(0\) under the transformations given by all the paths in the tree. The fractal so produced has no strict spatial homogeneity but does have a statistical homogeneity in the same sense that it has a statistical self-similarity.
Replacing the usual Moran formula for the dimension, \(s,\) of a self-similar fractal satisfying the open set condition, \((\sum_{i=1}^{m} r_i^{s} = 1\)). An expected version is used; let \(s\) be the solution to \(\mathbb{E}(\sum_{i=1}^{m} r_i^{s}) = 1\). The existence of such an \(s\) is proved in Lemma 2.1. The main result of the paper is then:
{ Theorem}: The Hausdorff dimension of the fractal is almost surely \(\min\{s,d\}\). If \(s > d\) then the fractal almost surely has positive \(d-\) dimensional Lebesgue measure.
The paper also includes a brief, but representative, set of references for the dimensionality of other randomly constructed fractals.


28A80 Fractals
28A78 Hausdorff and packing measures
60D05 Geometric probability and stochastic geometry
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