## 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

## 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.## References

- M. T. Barlow and B. M. Hambly,
*Transition density estimates for Brownian motion on scale irregular Sierpinski gaskets*, Ann. Inst. H. Poincaré Probab. Statist.**33**(1997), no. 5, 531–557 (English, with English and French summaries). MR**1473565**, DOI 10.1016/S0246-0203(97)80104-5 - M. V. Berry,
*Distribution of modes in fractal resonators*, Structural stability in physics (Proc. Internat. Symposia Appl. Catastrophe Theory and Topological Concepts in Phys., Inst. Inform. Sci., Univ. Tübingen, Tübingen, 1978) Springer Ser. Synergetics, vol. 4, Springer, Berlin, 1979, pp. 51–53. MR**556688**, DOI 10.1007/978-3-642-67363-4_{7} - N. H. Bingham and R. A. Doney,
*Asymptotic properties of supercritical branching processes. II. Crump-Mode and Jirina processes*, Advances in Appl. Probability**7**(1975), 66–82. MR**378125**, DOI 10.2307/1425854 - Jean Brossard and René Carmona,
*Can one hear the dimension of a fractal?*, Comm. Math. Phys.**104**(1986), no. 1, 103–122. MR**834484** - W. D. Evans and D. J. Harris,
*Fractals, trees and the Neumann Laplacian*, Math. Ann.**296**(1993), no. 3, 493–527. MR**1225988**, DOI 10.1007/BF01445117 - K. J. Falconer,
*Random fractals*, Math. Proc. Cambridge Philos. Soc.**100**(1986), no. 3, 559–582. MR**857731**, DOI 10.1017/S0305004100066299 - Dimitris Gatzouras,
*Lacunarity of self-similar and stochastically self-similar sets*, Trans. Amer. Math. Soc.**352**(2000), no. 5, 1953–1983. MR**1694290**, DOI 10.1090/S0002-9947-99-02539-8 - B. M. Hambly,
*Brownian motion on a random recursive Sierpinski gasket*, Ann. Probab.**25**(1997), no. 3, 1059–1102. MR**1457612**, DOI 10.1214/aop/1024404506 - B. M. Hambly,
*On the asymptotics of the eigenvalue counting function for random recursive Sierpinski gaskets*, Probab. Theory Related Fields**117**(2000), no. 2, 221–247. MR**1771662**, DOI 10.1007/s004400050005 - Jun Kigami,
*Analysis on fractals*, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR**1840042**, DOI 10.1017/CBO9780511470943 - Michel L. Lapidus,
*Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture*, Trans. Amer. Math. Soc.**325**(1991), no. 2, 465–529. MR**994168**, DOI 10.1090/S0002-9947-1991-0994168-5 - Michel L. Lapidus and Carl Pomerance,
*The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums*, Proc. London Math. Soc. (3)**66**(1993), no. 1, 41–69. MR**1189091**, DOI 10.1112/plms/s3-66.1.41 - Michel L. Lapidus and Carl Pomerance,
*Counterexamples to the modified Weyl-Berry conjecture on fractal drums*, Math. Proc. Cambridge Philos. Soc.**119**(1996), no. 1, 167–178. MR**1356166**, DOI 10.1017/S0305004100074053 - Michel L. Lapidus and Machiel van Frankenhuysen,
*Fractal geometry and number theory*, Birkhäuser Boston, Inc., Boston, MA, 2000. Complex dimensions of fractal strings and zeros of zeta functions. MR**1726744**, DOI 10.1007/978-1-4612-5314-3 - M.L. Lapidus and M. van Frankenhuysen, Fractality, self-similarity and complex dimensions. in
*Fractal Geometry and Applications: a Jubilee of Benoit Mandelbrot*, Proc. Sympos. Pure Math., vol. 72, Part 1, Amer. Math. Soc., Providence, R.I., 2004, pp. 349–372. - Torgny Lindvall,
*Lectures on the coupling method*, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1992. A Wiley-Interscience Publication. MR**1180522** - R. Daniel Mauldin and S. C. Williams,
*Random recursive constructions: asymptotic geometric and topological properties*, Trans. Amer. Math. Soc.**295**(1986), no. 1, 325–346. MR**831202**, DOI 10.1090/S0002-9947-1986-0831202-5 - Olle Nerman,
*On the convergence of supercritical general (C-M-J) branching processes*, Z. Wahrsch. Verw. Gebiete**57**(1981), no. 3, 365–395. MR**629532**, DOI 10.1007/BF00534830 - Jim Pitman and Marc Yor,
*The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator*, Ann. Probab.**25**(1997), no. 2, 855–900. MR**1434129**, DOI 10.1214/aop/1024404422 - Charles Stone,
*On moment generating functions and renewal theory*, Ann. Math. Statist.**36**(1965), 1298–1301. MR**179857**, DOI 10.1214/aoms/1177700003

## 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: hambly@maths.ox.ac.uk
**Michel L. Lapidus**- Affiliation: Department of Mathematics, University of California, Riverside, California 92521-0135
- Email: lapidus@math.ucr.edu
- 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: https://doi.org/10.1090/S0002-9947-05-03646-9
- MathSciNet review: 2171234