## 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.

### MSC:

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