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Methods of computing fractal dimensions

Part of the Lecture Notes in Mathematics book series (LNM,volume 1394)

Abstract

We discuss two frequently calculated fractal dimensions, the capacity and information dimension and present efficient methods for their computation for sets embedded in ℝn. In particular we show how Monte Carlo calculation of areas and volumes can be used to compute the capacity using fewer box counting operations than straightforward box counting, and we discuss an efficient implementation of the method using a very fast one-dimensional sorting algorithm. Sets embedded in ℝ2 and ℝ3 are mapped to [0,1] using a folding map φ with the property that n·dim(φ(X)) = dim(X) where X c ℝn and dim is either the Hausdorff or the capacity dimension. Thus the problem of calculating the capacity dimension or the information dimension (in the case it coincides with the Hausdorff dimension) is reduced to computing these quantities for φ(X) in [0,1].

Keywords

  • Hausdorff Dimension
  • Chaotic Attractor
  • Information Dimension
  • Lorenz Attractor
  • Capacity Dimension

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Partially supported by NSF grant DMS-8603703.

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© 1989 Springer-Verlag

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Hunt, F., Sullivan, F. (1989). Methods of computing fractal dimensions. In: Gill, T.L., Zachary, W.W. (eds) Nonlinear Semigroups, Partial Differential Equations and Attractors. Lecture Notes in Mathematics, vol 1394. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086754

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  • DOI: https://doi.org/10.1007/BFb0086754

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51594-4

  • Online ISBN: 978-3-540-46679-6

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