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RANDOM SUBSETS OF SELF-AFFINE FRACTALS

Part of: Geometric probability and stochastic geometry Classical measure theory

Published online by Cambridge University Press:  10 December 2009

Kenneth Falconer  and
Jun Miao
Kenneth Falconer
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, U.K. (email: kjf@st-andrews.ac.uk)
Jun Miao
Affiliation:
Department of Mathematics, East China Normal University, 3663 Zhong Shan North Road, Shanghai 200062, China (email: jjmiao@math.ecnu.edu.cn)
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Abstract

We find the almost-sure Hausdorff and box-counting dimensions of random subsets of self-affine fractals obtained by selecting subsets at each stage of the hierarchical construction in a statistically self-similar manner.

MSC classification

Secondary: 28A80: Fractals 60D05: Geometric probability and stochastic geometry
Type
Research Article
Information
Mathematika , Volume 56 , Issue 1 , January 2010 , pp. 61 - 76
Copyright
Copyright © University College London 2010

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References

[1]Athreya, K. B. and Ney, P. E., Branching Processes, Springer (Berlin, 1972).CrossRefGoogle Scholar
[2]Barański, K., Hausdorff dimension of the limit sets of some planar geometric constructions. Adv. Math. 210 (2007), 215–245.CrossRefGoogle Scholar
[3]Dekking, F. M. and Meester, R. W. J., On the structure of Mandelbrot’s percolation process and other random Cantor sets. J. Stat. Phys. 58 (1990), 1109–1126.CrossRefGoogle Scholar
[4]Edgar, G. A., Fractal dimension of self-affine sets: some examples. Rend. Circ. Mat. Palermo (2) Suppl. 28 (1992), 341–358.Google Scholar
[5]Falconer, K. J., The Geometry of Fractal Sets, Cambridge University Press (Cambridge, 1985).CrossRefGoogle Scholar
[6]Falconer, K. J., Random fractals. Math. Proc. Cambridge Philos. Soc. 100 (1986), 559–582.CrossRefGoogle Scholar
[7]Falconer, K. J., The Hausdorff dimension of self-affine fractals. Math. Proc. Cambridge Philos. Soc. 103 (1988), 169–179.CrossRefGoogle Scholar
[8]Falconer, K. J., The dimension of self-affine fractals II. Math. Proc. Cambridge Philos. Soc. 111 (1992), 339–350.CrossRefGoogle Scholar
[9]Falconer, K. J., Fractal Geometry: Mathematical Foundations and Applications, 2nd edn., John Wiley (Chichester, 2003).CrossRefGoogle Scholar
[10]Falconer, K. J. and Miao, J., Dimensions of self-affine fractals and multifractals generated by upper-triangular matrices. Fractals 15 (2007), 289–299.CrossRefGoogle Scholar
[11]Gatzouras, D. and Lalley, S. P., Statistically self-affine sets: Hausdorff and box dimensions. J. Theoret. Probab. 7 (1994), 437–468.CrossRefGoogle Scholar
[12]Graf, S., Statistically self-similar fractals. Probab. Theor. Related Fields 74 (1987), 357–392.CrossRefGoogle Scholar
[13]Hutchinson, J. E., Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[14]Jordan, T., Pollicott, M. and Simon, K., Hausdorff dimension for randomly perturbed self affine attractors. Comm. Math. Phys. 207 (2007), 519–544.CrossRefGoogle Scholar
[15]Käenmäki, A. and Shmerkin, P., Overlapping self-affine sets of Kakeya type. Ergod. Th & Dynam. Sys. 29 (2009), 941–965.CrossRefGoogle Scholar
[16]Mandelbrot, B. B., The Fractal Geometry of Nature, W. H. Freeman (New York, 1982).Google Scholar
[17]Mauldin, R. D. and Williams, S. C., Random recursive constructions: asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295 (1986), 325–346.CrossRefGoogle Scholar
[18]Miao, J. J., The geometry of self-affine fractals. PhD Thesis, University of St Andrews, 2008.Google Scholar
[19]Przytycki, F. and Urbanski, M., On the Hausdorff dimension of some fractals. Studia Math. 93 (1989), 155–186.CrossRefGoogle Scholar
[20]Rogers, C. A., Hausdorff Measures, Cambridge University Press (Cambridge, 1998).Google Scholar
[21]Solomyak, B., Measure and dimensions for some fractal families. Math. Proc. Cambridge Philos. Soc. 124 (1998), 531–546.CrossRefGoogle Scholar
[22]Williams, D., Probability with Martingales, 2nd edn., Cambridge University Press (Cambridge, 1991).CrossRefGoogle Scholar
[23]Yu, J. and Ding, L., Hausdorff dimensions of generalized statistically self-affine fractals. Acta Math. Sci. Ser. B. Engl. Ed. 24 (2004), 421–433.CrossRefGoogle Scholar