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Towards numerically estimating Hausdorff dimensions

Published online by Cambridge University Press:  17 February 2009

David E. Stewart
David E. Stewart
Affiliation:
Mathematics Department, University of Iowa, Iowa City, IA 52242, USA.
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Abstract

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This paper gives a numerical method for estimating the Hausdorff-Besicovitch dimension where this differs from the fractal (or capacity or box-counting) dimension. The method has been implemented, and numerical results obtained for the set {1/n | n ∈ N} and the Cantor set. Comments about the practical use of the estimation algorithms are made.

Type
Research Article
Information
The ANZIAM Journal , Volume 42 , Issue 4 , April 2001 , pp. 451 - 461
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Falconer, K. J., The geometry of fractal sets (Cambridge Univ. Press, Cambridge, 1985).Google Scholar
[2]Hall, P. and Wood, A., “On the performance of box-counting estimators of fractal dimension”, Biometrica 80 (1993) 246–252.Google Scholar
[3]Hunt, F., “Error analysis and convergence of capacity dimension algorithms”, SIAM J. Appl. Math. 50 (1990) 307–321.Google Scholar
[4]Hunt, F. and Sullivan, F., “Efficient algorithms for computing fractal dimensions”, in Dimensions and entropies in chaotic systems: Quantification of complex behaviour (ed. Mayer-Kress, G.), Springer Series in Synergetics 32, (Springer, Berlin, 1986) 74–81.CrossRefGoogle Scholar
[5]Hutchinson, J., “Fractals and self-similarity”, Indiana University J. Math. 30 (1981) 713–747.Google Scholar
[6]Kernighan, B. W. and Ritchie, D., The C programming language, 2nd ed. (Prentice Hall, 1989).Google Scholar
[7]Mandelbrot, B., The fractal geometry of nature (W. H. Freeman, San Francisco, 1982).Google Scholar
[8]Molteno, T. C. A., “Fast O(N) box-counting algorithm of estimating dimensions”, Phys. Rev. E 48 (1993) R3263–R3266.Google Scholar
[9]Parker, T. S. and Chua, L. O., Practical numerical algorithms for chaotic systems (Springer, New York, 1989).Google Scholar
[10]Temam, R., Infinite-dimensional dynamical systems in mechanics and physics (Springer, Berlin, 1988).Google Scholar