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

Hambly, B.; Lapidus, Michel

doi:10.1090/S0002-9947-05-03646-9

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