The Mathematical Institute, University of Oxford, Eprints Archive

Random fractal strings: their zeta functions, complex dimensions and spectral asymptotics

Hambly, B. M. and Lapidus, M. L. (2003) Random fractal strings: their zeta functions, complex dimensions and spectral asymptotics. Transactions of the American Mathematical Society . (In Press)

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Abstract

In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that using a random recursive self-similar construction it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary.

Item Type:Article
Subjects:O - Z > Probability theory and stochastic processes
Research Groups:Stochastic Analysis Group
Oxford Centre for Industrial and Applied Mathematics
ID Code:113
Deposited By:Ben Hambly
Deposited On:04 Aug 2004
Last Modified:20 Jul 2009 14:18

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