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Hausdorff dimension of random fractals with overlaps

Published online by Cambridge University Press:  17 April 2009

Z. G. Yu ,
V. V. Anh  and
K. S. Lau
Z. G. Yu
Affiliation:
Centre in Statistical Science and Industrial Mathematics, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia Department of Mathematics, Xiangtan University, Hunan 411105, Peoples Republic ov China
V. V. Anh
Affiliation:
Centre in Statistical Science and Industrial Mathematics, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia
K. S. Lau
Affiliation:
Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong
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Abstract

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This paper considers random fractals generated by contractive transformations which satisfy a separation condition weaker than the open set condition. This condition allows overlaps in the iterations. Estimates of the Hausdorff dimension of this kind of random fractals are obtained.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 2 , April 2002 , pp. 315 - 328
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Arbeiter, M., ‘Random recursive construction of self-similar measures: the non-compact case’ Probab. Theory Related Fields 88 (1991), 497–520.CrossRefGoogle Scholar
[2]Barnsley, M.F., Fractal everywhere (Academic Press, New York, 1988).Google Scholar
[3]Billingsley, P., Ergodic theory and informations (John Wiley and Sons, New York, 1965).Google Scholar
[4]Erdös, P., ‘On the smoothness properties of a family of symmetric Bernoulli convolutions’, Amer. J. Math. 62 (1940), 180–186.CrossRefGoogle Scholar
[5]Falconer, K.J., ‘Random fractals’, Math. Proc. Cambridge Philos. Soc. 100 (1986), 559–582.CrossRefGoogle Scholar
[6]Falconer, K.J., Fractal geomentry, mathematical Foundations and Applications (John Wiley and Sons Ltd., 1990).Google Scholar
[7]Falconer, K.J., Techniques in fractal geometry (John Wiley and Sons, Chichester, 1997).Google Scholar
[8]Garsia, A.M., ‘Arithmetic properties of Bernoulli convolutions’, Trans. Amer. Math. Soc. 102 (1962), 409–432.CrossRefGoogle Scholar
[9]Giona, M. and Roman, H.E., ‘Fractional diffusion equation for transport phenomena in random media‘, Physica A 185 (1992), 97–97.CrossRefGoogle Scholar
[10]Graf, S., ‘Statistically self-similar fractals’, Probab. Theory Related Fields 74 (1987), 357–394.CrossRefGoogle Scholar
[11]Harris, T.E., The theory of branching processes, Die Grundlehren der Mathematischen Wisenschaften Bd 119 (Springer-Verlag, Berlin, 1963).CrossRefGoogle Scholar
[12]Hutchinson, J.E., ‘Fractals and self similarity’, Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[13]Hutchinson, J.E. and Rüschendorf, L., ‘Random fractal measures via the contraction method’, Indiana Univ. Math. J. 47 (1998), 471–487.CrossRefGoogle Scholar
[14]Kenyon, R., ‘Projecting the one-dimensional Sierpinski Gasket’, Israel J. Math. 97 (1997), 221–238.CrossRefGoogle Scholar
[15]Lau, K.S., ‘Iterated functions systems with overlaps and multifractal structure’, in Trends in Probability and Related Analysis, (Kono, K. and Shieh, N-R., Editors) (World Scientific, River Edge, N.J., 1999), PP.35–76.Google Scholar
[16]Lau, K.S. and Ngai, S.M., ‘Multifractal measures and a weak separation condition’, Adv. Math. 141 (1999), 45–96.CrossRefGoogle Scholar
[17]Lau, K.S., Ngai, S.M. and Rao, H., ‘Iterated function systems with overlaps and self-similar measures’, J. London Math. Soc. (to appear).Google Scholar
[18]Liang, J-R. and Ren, F-Y., ‘Hausdorff dimensions of random net fractals’, Stochastic Process. Appl. 74 (1998), 235–250.CrossRefGoogle Scholar
[19]Mandelbrot, B.B., ‘The fractal geometry of nature’, in Bifurcation theory and applications in Scientific disciplines, Ann. New York Acad. Sci. 316 (W.C. Freeman and Co., New York, 1982), pp. 463–464.Google Scholar
[20]Mauldin, R.D. and Williams, S.C., ‘Random recursive constructions: asympotic geometric and topological properties’, Trans. Amer. Math. Soc. 295 (1986), 325–346.CrossRefGoogle Scholar
[21]Olsen, L., Random geometrically graph directed self-similar multifractals, Pitman Research Notes in Math. Series 307 (Longman, Harlow).Google Scholar
[22]Peres, Y. and Solomyak, B., ‘Absolute continuity of Bernoulli convolutions, a simpler proof’, Math. Research Letter 3 (1996), 231–239.CrossRefGoogle Scholar
[23]Peres, Y. and Solomyak, B., ‘Self-similar measures and intersection of Cantor sets’, Trans. Amer. Math. Soc. (to appear).Google Scholar
[24]Peres, Y. and Schlag, W., ‘Smoothness of projections, Bernoulli convolutions and the dimension of exceptions’, (preprint).Google Scholar
[25]Pesin, Y. and Weiss, H., ‘On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture’, Comm. Math. Phys. 182 (1996), 105–153.CrossRefGoogle Scholar
[26]Rao, H. and Wen, Z.Y., ‘Some studies of a class of self-similar fractals with overlap structure’, Adv. in Appl. Math. 20 (1998), 50–72.CrossRefGoogle Scholar
[27]Solomyak, B., ‘On the random series Σ±λn (an Erdös problem)’, Ann. of Math. 142 (1995), 611–625.CrossRefGoogle Scholar
[28]yu, Z-G., Ren, F-Y. and Zhou, J., ‘Fractional integral associated to generalized coolie-cutter set and its physical interpretation’, J. Phys. A. 30 (1997), 5569–5577.CrossRefGoogle Scholar