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Dimensions and Entropies in Chaotic Systems pp 42–53Cite as

Hausdorff Dimensions for Sets with Broken Scaling Symmetry

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Part of the book series: Springer Series in Synergetics ((SSSYN,volume 32))

Abstract

Based on Hausdorff’s original approach to fractional dimensions, we study systems which are not sufficiently characterized by their “fractal” or scaling dimension. We construct informative examples of such sets and relate them to sets observed in the context of dynamical systems.

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References

  1. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman and Co., San Francisco, 1982).

    Google Scholar 

  2. S. J. Taylor Pr. of the Cambridge Philosophical Society 60, 253 (1964)

    Article  ADS  MATH  Google Scholar 

  3. Felix Hausdorff, Math. Ann. 79, 157 (1919)

    Article  MathSciNet  Google Scholar 

  4. D.K. Umberger and J.D. Farmer, Phys. Rev. Lett. 55, 661 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  5. J. D. Farmer, Phys. Rev. Lett. 55, 351 (1985).

    Article  MathSciNet  ADS  Google Scholar 

  6. C. Grebogi, S. W. McDonald, E. Ott, and J. A. Yorke, Phys. Lett. 110A, 1 (1985).

    MathSciNet  ADS  Google Scholar 

  7. R. Ecke, J. D. Farmer, and D. K. Umberger, unpublished (1985).

    Google Scholar 

  8. J. D. Farmer, E. Ott, and J. A. Yorke, Physica 7D, 153 (1983).

    MathSciNet  ADS  Google Scholar 

  9. The original definition of the Hausdorff measure utilizes arbitrary covers. We have chosen the formulation in terms of D dimensional squares for simplicity.

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  10. The definition of the Hausdorff measure which we choose guarantees this, however in the general case (see previous reference) the Lebesgue measure and the Hausdorff measure differ by a nonzero factor.

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  11. G. H. Hardy, Orders of Infinity, Cambridge Tracts in Mathematics and Mathematical Physics no. 12, Cambridge University Press (1954).

    Google Scholar 

  12. S. Takasue, Ph.D. dissertation, Tokyo University, 1983

    Google Scholar 

  13. S. W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, to appear in Physica D

    Google Scholar 

  14. B. Mandelbrot: this volume

    Google Scholar 

  15. D. A. Russell, J. D. Hanson, and E. Ott, Phys. Rev. Lett. 45, 1175 (1980).

    Article  MathSciNet  ADS  Google Scholar 

  16. D. K. Umberger and R. Eykholt, unpublished (1985).

    Google Scholar 

  17. J. D. Farmer and D.K. Umberger, to appear Phys. Rev. Lett. (1985).

    Google Scholar 

  18. R. Eykholt and D.K. Umberger, unpublished (1985).

    Google Scholar 

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Author information

Authors and Affiliations

  1. Center for Nonlinear Studies, MS B258, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    D. K. Umberger, G. Mayer-Kress & E. Jen

  2. Dept. of Physics, Univ. of Arkansas, Fayetteville, AR, 72701, USA

    D. K. Umberger

Authors
  1. D. K. Umberger
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  2. G. Mayer-Kress
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  3. E. Jen
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Editor information

Editors and Affiliations

  1. Center for Nonlinear Studies, Los Alamos National Laboratory, Mail Stop: B258, Los Alamos, NM, 87545, USA

    Gottfried Mayer-Kress

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© 1986 Springer-Verlag Berlin Heidelberg

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Umberger, D.K., Mayer-Kress, G., Jen, E. (1986). Hausdorff Dimensions for Sets with Broken Scaling Symmetry. In: Mayer-Kress, G. (eds) Dimensions and Entropies in Chaotic Systems. Springer Series in Synergetics, vol 32. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-71001-8_6

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  • DOI: https://doi.org/10.1007/978-3-642-71001-8_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-71003-2

  • Online ISBN: 978-3-642-71001-8

  • eBook Packages: Springer Book Archive

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