Cut-out sets and the Zipf law for fractal voids

PHYSICA D-NONLINEAR PHENOMENA

卷:223

Article

Gaite, Jose

关键词: fractals;    Zipf's law;   

DOI  :  10.1016/j.physd.2006.09.021

来源: Elsevier

【 摘 要 】

Cut-out sets are fractals that can be obtained by removing a sequence of disjoint regions from an initial region of d-dimensional euclidean space. Conversely, a description of some fractals in terms of their void complementary set is possible. The essential property of a sequence of fractal voids is that their sizes decrease as a power law, that is, they follow Zipf's law. We prove the relation between the box dimension of the fractal set (for d <= 3) and the exponent of the Zipf law for convex voids; namely, if the Zipf law exponent e is such that 1 < e < d/(d-1) and, in addition, we forbid the appearance of degenerate void shapes, we prove that the corresponding cut-out set has box dimension d/e (such that d-1 < d/e < d). We explore various physical applications of this result, in particular, the application to the description of the cosmic structure using cosmic foam models. (c) 2006 Elsevier B.V. All rights reserved.

【 授权许可】

Free