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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics
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by B. M. Hambly and Michel L. Lapidus PDF
Trans. Amer. Math. Soc. 358 (2006), 285-314 Request permission


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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Additional Information
  • B. M. Hambly
  • Affiliation: Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, United Kingdom
  • MR Author ID: 325616
  • Email:
  • Michel L. Lapidus
  • Affiliation: Department of Mathematics, University of California, Riverside, California 92521-0135
  • Email:
  • Received by editor(s): October 21, 2003
  • Received by editor(s) in revised form: February 17, 2004
  • Published electronically: February 18, 2005
  • Additional Notes: The second author was supported in part by the U.S. National Science Foundation under grants DMS-9623002, DMS-0070497.
  • © Copyright 2005 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 285-314
  • MSC (2000): Primary 28A80, 60D05; Secondary 11M41, 58J50, 60J80
  • DOI:
  • MathSciNet review: 2171234