Journal ArticleDOI
Computability of the Hausdorff and packing measures on self-similar sets and the self-similar tiling principle
TLDR
In this article, the authors show that any open subset of a self-similar set with open set condition may be tiled without loss of measure by copies under similitudes of any closed subset with positive measure.Abstract:
We state a self-similar tiling principle which shows that any open subset of a self-similar set with open set condition may be tiled without loss of measure by copies under similitudes of any closed subset with positive measure. We use this method to get the optimal coverings and packings which give the exact value of the Hausdorff-type and packing measures. In particular, we show that the exact value of these measures coincides with the supremum or with the infimum of the inverse of the density of the natural probability measure on suitable classes of sets. This gives criteria for the numerical analysis of the measures, and allows us to compare their complexity in terms of computability.read more
Citations
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Diffusion relaxation limit of a bipolar hydrodynamic model for semiconductors
TL;DR: In this paper, the authors discuss the relaxation limit of a bipolar isentropic hydrodynamical model for semiconductors with small momentum relaxation time and prove that periodic initial-value problems of a scaled bipolar hydrodynamic model have unique smooth solutions.
Journal ArticleDOI
Weakly controlled Moran constructions and iterated functions systems in metric spaces
Tapio Rajala,Markku Vilppolainen +1 more
TL;DR: In this article, the authors studied the Hausdorff measures of limit sets of weakly controlled Moran constructions in metric spaces and investigated different separation conditions for semiconformal iterated function systems.
Journal ArticleDOI
Self-similar sets with optimal coverings and packings
Marta Llorente,Manuel Morán +1 more
TL;DR: In this paper, it was shown that if a self-similar set E in Rn with Hausdorff dimension s satisfies the strong separation condition, then the maximal values of the Hs-density on the class of arbitrary subsets of Rn and on the classes of Euclidean balls are attained, and the inverses of these values give the exact values of both the spherical Hs and Hs measures of E. The authors also showed that a ball of minimal density exists, and that the inverse density of this ball gives the exact packing measure of E
Posted Content
Weakly controlled Moran constructions and iterated functions systems in metric spaces
Tapio Rajala,Markku Vilppolainen +1 more
TL;DR: In this article, the authors studied the Hausdorff measures of limit sets of weakly controlled Moran constructions in metric spaces and investigated different separation conditions for semiconformal iterated function systems.
Journal ArticleDOI
Hausdorff measure of cartesian product of the ternary cantor set
Juan Deng,Hui Rao,Zhi-Ying Wen +2 more
TL;DR: In this article, the authors studied the optimal sets of C × C: their diameters, measures, symmetries, and shapes, and showed that the diameter of the optimal set B is between 1.2993 and 1.3082.
References
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Journal ArticleDOI
Separation properties for self-similar sets
TL;DR: In this article, it was shown that the strong open set condition and the strong closed set condition are both equivalent to Ha (K) > 0, where a is the similarity dimension of K and Ha denotes the Hausdorff measure of this dimension.
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Packing measure, and its evaluation for a Brownian path
S. James Taylor,Claude Tricot +1 more
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An optimized box-assisted algorithm for fractal dimensions
TL;DR: An optimized algorithm for estimating the correlation dimension of an attractor based on very long time sequences is presented, using a mesh in order to count only near neighbors in the correlation sum using linked lists.
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Exact Hausdorff measure and intervals of maximum density for Cantor sets
TL;DR: In this article, the authors give an algorithm for computing 7-t (K) exactly as the maximum of a finite set of elementary functions of the parameters of the i.i.f. parameters.
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Twelve open problems on the exact value of the Hausdorff measure and on topological entropy: a brief survey of recent results*
Zuoling Zhou,Li Feng +1 more
TL;DR: In this article, the authors give a brief survey of the recent research results related to these open problems and conjecture, and present a conjecture on the Hausdorff measure of self-similar sets.