Lacunarity of self-similar and stochastically self-similar sets

Transactions of the American Mathematical Society

Volumn 352, Issue 5, 2000, Pages 1953-1983

  Gatzouras, Dimitris ^a,b  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^b UNIVERSITY OF CRETE   (Greece)

Author keywords

Branching random walk; Cantor set; E neighborhood; Minkowski content; Renewal theorem; Stable process

Indexed keywords

EID: 23044520951     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-99-02539-8     Document Type: Article

References (26)

1. T. Bedford and A. Fisher, Analogues of the Lebesgue density theorem for fractal sets of reals and integers, Proc. London Math. Soc. 64 (1992), 95-124. MR 92j:58058
2. J. Bertoin, Levy Processes, Cambridge Tracts in Math. 121, Camb. Univ. Press, 1996. MR 98e:60117
3. J. Biggins, Martingale convergence in the branching random walk, 3, Appl. Probab. 14 (1977), 25-37. MR 55:6592
4. P. Billingsley, Probability and Measure, 3rd ed., Wiley, New York, 1995. MR 95k:60001
5. R.M. Blumenthal and R.K. Getoor, The dimension of the set of zeros and the graph of a symmetric stable process, Illinois J. Math. 6 (1962), 308-316. MR 25:1581
6. K. Falconer, Random Fractals, Math. Proc. Camb. Phil. Soc. 100 (1986), 559-582. MR 88e:28005
7. _, On the Minkowski measurability of fractals, Proc. Amer. Math. Soc. 123 (1995), 1115-1124. MR 95e:28003
8. W. Feller, An Introduction to Probability Theory and its Applications, Vol. 2, 2nd ed., Wiley, New York, 1971. MR 42:5292
9. B. Fristedt and S.J. Taylor, Constructions of local time for a Markov process, Z. Wahrsch. verw. Gebiete 62 (1983), 73-112. MR 85c:60121
10. J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 82h:49026
11. K. Ito and H. McKean, Diffusion processes and their sample paths, Springer Verlag, BerlinHeidelberg, 1965. MR 33:8031
12. S. Lallcy, The packing and covering functions of some self-similar fractals, Indiana Univ. Math. J. 37 (1988), 699-709. MR 89k:28013
13. _, A renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations, and their fractal limits, Acta Math. 163 (1989), 1-55. MR 91c:58112
14. M. Lapidus, FYactal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc. 325 (1991), 465-529. MR 91j:58163
15. M. Lapidus and C. Pomerance, The Riemann zeta-function and the one-dimensional WeylBerry conjecture for fractal drums, Proc. London Math. Soc. (3) 66 (1993), 41-69. MR 93k:58217
16. _, Counterexamples to the modified Weyl-Berry conjecture on fractal drums, Math. Proc. Camb. Phil. Soc. 119 (1996), 167-178. MR 96h:58175
17. R. Lyons, A simple path to Biggins' martingale convergence for branching random walk, in Classical and Modern Branching Processes (K. Athreya, P. Jagers, eds.), IMA Vol. Math. Appl., Vol. 84, Springer Verlag, New York, 1997, pp. 217-221. CMP 98:08
18. R. Lyons, R. Pemantle and Y. Pères, Conceptual proofs of L log L criteria for mean behavior of branching processes, Ann. Probab. 23 (1995), 1125-1138. MR 96m:60194
19. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1983. MR 84h:00021
20. D. Mauldin and S. Williams, Random recursive constructions: Asymptotic geometric and topological properties, Trans. Amer. Math. Soc. 295 (1986), 325-346. MR 87j:60027
21. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Camb. Phil. Soc. 42 (1946), 15-23. MR 7:278f
22. O. Herman, On the convergence of supercritical general (C-M-J) branching processes, Z. Wahrsch. verw. Gebiete 57 (1981), 365-395. MR 82m:60104
23. A. Schief, Separation properties of self-similar sets, Proc. Amer. Math. Soc. 122 (1994), 111-115. MR 94k:28012
24. S.J. Taylor and J.G. Wendel, The exact Hausdorff measure of the zero set of a stable process, Z. Wahrsch. verw. Gebiete 6 (1966), 170-180. MR 35:1090
25. M. Lapidus and M. van Frankenhuysen, Complex dimensions of fractal strings and oscillatory phenomena in fractal geometry and arithmetic, Spectral Problems in Geometry and Arithmetic (T. Branson, ed.), Contemporary Mathematics, Vol. 237, Amer. Math. Soc., 1999.
26. M. Lapidus, Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl-Berry conjecture, in Ordinary and Partial Differential Equations, Vol. IV, Proc. 12th International Conference on the Theory of Ordinary and Partial Differential Equations (Dundee, Scotland), Pitman Research Notes in Mathematics Series 289 (B. D. Sleeman and R. J. Jarvis Eds.), Longman Scientific and Technical, London, 1993, pp. 126-209. MR 95g:58247