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Thick and thin points for random recursive fractals

Part of: Markov processes Stochastic processes Classical measure theory

Published online by Cambridge University Press:  01 July 2016

B. M. Hambly  and
O. D. Jones
B. M. Hambly*
Affiliation:
University of Oxford
O. D. Jones*
Affiliation:
University of Southampton
*
∗ Postal address: Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK. Email address: [email protected]
∗∗ Postal address: Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, UK.
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Abstract

We consider random recursive fractals and prove fine results about their local behaviour. We show that for a class of random recursive fractals the usual multifractal spectrum is trivial in that all points have the same local dimension. However, by examining the local behaviour of the measure at typical points in the set, we establish the size of fine fluctuations in the measure. The results are proved using a large deviation principle for a class of general branching processes which extends the known large deviation estimates for the supercritical Galton-Watson process.

Keywords

General branching processlarge deviationslocal dimension

MSC classification

Primary: 60G57: Random measures
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 28A80: Fractals
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 1 , March 2003 , pp. 251 - 277
Copyright
Copyright © Applied Probability Trust 2003 

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