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A subset E of ℝⁿ is called self-affine with respect to a collection ϕ₁,...,ϕₜ of affinities if E is the union of the sets ϕ₁(E),...,ϕₜ(E). For S ⊂ ℝⁿ let . If Φ(S) ⊂ S let denote . For given Φ consisting of contracting “pseudo-dilations” (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets of each Hausdorff dimension between zero and a positive number depending on Φ. We also investigate...
We obtain a metrical property on the asymptotic behaviour of the maximal run-length function in the Lüroth expansion. We also determine the Hausdorff dimension of a class of exceptional sets of points whose maximal run-length function has sub-linear growth rate.
The inner knowledge of volumes from images
is an ancient problem. This question becomes complicated when it
concerns quantization, as the case of any measurement and in
particular the calculation of fractal dimensions. Trabecular bone
tissues have, like many natural elements, an architecture which
shows a fractal aspect. Many studies have already been developed
according to this approach. The question which arises however is to
know to which extent it is possible to get an exact determination of
the...
In this paper, we study optimal transportation problems for multifractal random measures. Since these measures are much less regular than optimal transportation theory requires, we introduce a new notion of transportation which is intuitively some kind of multistep transportation. Applications are given for construction of multifractal random changes of times and to the existence of random metrics, the volume forms of which coincide with the multifractal random measures.
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