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

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

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: [email protected]
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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 

References

Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (London Math. Soc. Lecture Notes Ser. 167), Cambridge University Press, pp. 23–70.CrossRefGoogle Scholar
Alon, N. and Spencer, J. H. (2000). The Probabilistic Method, 2nd edn. John Wiley, New York.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
Barlow, M. T. (1998). Diffusions on fractals. In Lectures on Probability Theory and Statistics (Lecture Notes Math. 1690), Springer, Berlin, pp. 1–121.CrossRefGoogle Scholar
Barlow, M. T. and Bass, R. F. (1989). The construction of Brownian motion on the Sierpiński carpet. Ann. Inst. H. Poincaré Prob. Statist. 25, 225–257.Google Scholar
Croydon, D. A. (2006). Random fractal dendrites. , University of Oxford.Google Scholar
Croydon, D. A. and Hambly, B. M. (2007). Self-similarity and spectral asymptotics for the continuum random tree. To appear in Stoch. Process. Appl. Google Scholar
Falconer, K. J. (1986). Random fractals. Math. Proc. Camb. Philos. Soc. 100, 559–582.CrossRefGoogle Scholar
Falconer, K. J. (1987). Cut-set sums and tree processes. Proc. Amer. Math. Soc. 101, 337–346.CrossRefGoogle Scholar
Falconer, K. J. (1990). Fractal Geometry. John Wiley, Chichester.Google Scholar
Hambly, B. M. (1997). Brownian motion on a random recursive Sierpiński gasket. Ann. Prob. 25, 1059–1102.CrossRefGoogle Scholar
Hambly, B. M. and Jones, O. D. (2003). Thick and thin points for random recursive fractals. Adv. Appl. Prob. 35, 251–277.CrossRefGoogle Scholar
Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.Google Scholar
Karp, R. M. (1990). The transitive closure of a random digraph. Random Structures Algorithms 1, 73–93.CrossRefGoogle Scholar
Kigami, J. (1995). Harmonic calculus on limits of networks and its application to dendrites. J. Funct. Anal. 128, 48–86.CrossRefGoogle Scholar
Kigami, J. (2001). Analysis on Fractals (Camb. Tracts Math. 143). Cambridge University Press.CrossRefGoogle Scholar
Liu, Q. (1996). The growth of an entire characteristic function and the tail probabilities of the limit of a tree martingale. In Trees (Progress Prob. 40), Birkhäuser, Basel, pp. 51–80.CrossRefGoogle Scholar
Liu, Q. and Rouault, A. (2000). Limit theorems for Mandelbrot's multiplicative cascades. Ann. Appl. Prob. 10, 218–239.CrossRefGoogle Scholar
Mandelbrot, B. (1974). Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–353.CrossRefGoogle Scholar
Mauldin, R. D. and Williams, S. C. (1986). Random recursive constructions: asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295, 325–346.CrossRefGoogle Scholar