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The Hausdorff dimension of a class of random self-similar fractal trees

Part of: Geometric probability and stochastic geometry Markov processes Classical measure theory

Published online by Cambridge University Press:  01 July 2016

D. A. Croydon
D. A. Croydon*
Affiliation:
University of Warwick
*
∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: d.a.croydon@warwick.ac.uk
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Abstract

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In this article a collection of random self-similar fractal dendrites is constructed, and their Hausdorff dimension is calculated. Previous results determining this quantity for random self-similar structures have relied on geometrical properties of an underlying metric space or the scaling factors being bounded uniformly away from 0. However, using a percolative argument, and taking advantage of the tree-like structure of the sets considered here, it is shown that conditions such as these are not necessary. The scaling factors of the recursively defined structures in consideration form what is known as a multiplicative cascade, and results about the height of this random object are also obtained.

Keywords

Hausdorff dimensionmultiplicative cascaderandom treeself-similar fractal

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 28A80: Fractals
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 3 , September 2007 , pp. 708 - 730
Copyright
Copyright © Applied Probability Trust 2007 

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