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ISSN 1088-6850(e) ISSN 0002-9947(p)

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

Authors: B. M. Hambly and Michel L. Lapidus
Journal: Trans. Amer. Math. Soc. 358 (2006), 285-314
MSC (2000): Primary 28A80, 60D05; Secondary 11M41, 58J50, 60J80
Posted: February 18, 2005
MathSciNet review: 2171234
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Abstract | References | Similar Articles | Additional Information

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 by 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.

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