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Irregularity Index and Spherical Densities of the Penta-Sierpinski Gasket

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Abstract

We compute the centred Hausdorff measure, \(C^{s}(\textbf{P})\sim 2.44\), and the packing measure, \(P^{s}(\textbf{P})\sim 6.77\), of the penta-Sierpinski gasket, \(\textbf{P}\), with explicit error bounds. We also compute the full spectra of asymptotic spherical densities of these measures in \(\textbf{P}\), which, in contrast with that of the Sierpinski gasket, consists of a unique interval. These results allow us to compute the irregularity index of \(\textbf{P}\), \(\mathcal {I}(\textbf{P})\sim 0.6398\), which we define for any self-similar set E with open set condition as \(\mathcal {I}(E)=1-\frac{C^{s}(E)}{P^{s}(E)}\).

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References

  1. Barnsley, M.F.: Fractals Everywhere. Courier Corporation, Chelmsford (2012)

    MATH  Google Scholar 

  2. Berry, M.V., Lewis, Z.V., Nye, J.F.: On the Weierstrass-Mandelbrot fractal function. Proc. R. Soc. Lond. Ser. A 370, 459–484 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications. Wiley, Chichester (2014)

    MATH  Google Scholar 

  4. Flake, G.W.: The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation. MIT Press, Cambridge (2000)

    MATH  Google Scholar 

  5. Guariglia, E.: Harmonic Sierpinski gasket and applications. Entropy 20(9), 714 (2018)

    Article  MathSciNet  Google Scholar 

  6. Guariglia, E.: Primality, fractality and image analysis. Entropy 21(3), 304 (2019)

    Article  MathSciNet  Google Scholar 

  7. Guariglia, E., Silvestrov, S.: Fractional-wavelet analysis of positive definite distributions and wavelets on D’(C). In: Silvestrov, S., Rancic, M. (eds.) Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization, pp. 337–353. Springer, Cham (2016)

    Chapter  MATH  Google Scholar 

  8. Hausdorff, F.: Dimension und äusseres. Math. Ann. 79, 157–179 (1919)

    Article  MATH  Google Scholar 

  9. Hentschel, H.G.E., Procaccia, I.: The infinite number of generalized dimensions of fractals and strange attractors. Physica D 8(3), 435–444 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hutchinson, J.E.: Fractals and self-similarity. Ind. J. Math. 30, 713–747 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kubacki, R., Czyżewski, M., Laskowski, D.: Minkowski island and crossbar fractal microstrip antennas for broadband applications. Appl. Sci. 8, 334 (2018)

    Article  Google Scholar 

  12. Lalley, S.P.: The packing and covering functions of some self-similar fractals. Indiana Univ. Math. J. 37(3), 699–710 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Llorente, M., Morán, M.: Self-similar sets with optimal coverings and packings. J. Math. Anal. Appl. 334(2), 1088–1095 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Llorente, M., Morán, M.: Advantages of the centered Hausdorff measure from the computability point of view. Math. Scand. 107(1), 103–122 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Llorente, M., Morán, M.: An algorithm for computing the centered Hausdorff measures of self-similar sets. Chaos Solitons Fractals 45(3), 246–255 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Llorente, M., Morán, M.: Computability of the packing measure of totally disconnected self-similar sets. Ergod. Theory Dyn. Syst. 36(5), 1534–1556 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Llorente, M., Mera, M.E., Morán, M.: Rate of convergence: the packing and centered Hausdorff measures of totally disconnected self-similar sets. Chaos Solitons Fractals 98, 220–232 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Llorente, M., Mera, M.E., Morán, M.: On the packing measure of the Sierpinski gasket. Nonlinearity 31, 2571–2589 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Llorente, M., Mera, M.E., Morán, M.: On the centered Hausdorff measure of the Sierpinski gasket. Fractals (2023). https://doi.org/10.1142/S0218348X23501074

    Article  MATH  Google Scholar 

  20. Mandelbrot, B.: The Fractal Geometry of Nature. Times Books, New York (1982)

    MATH  Google Scholar 

  21. Marstrand, J.M.: The \((\phi, s)\) regular subsets of \(n\) space. Trans. Am. Math. Soc. 113, 369–392 (1964)

    MathSciNet  MATH  Google Scholar 

  22. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  23. Morán, M.: Dynamical boundary of a self-similar set. Fundam. Math. 160(1), 1–14 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Morán, M.: Computability of the Hausdorff and packing measures on self-similar sets and the self-similar tiling principle. Nonlinearity 18(2), 559–570 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Morán, M., Llorente, M., Mera, M.E.: Local geometry of self-similar sets: typical balls, tangent measures and asymptotic spectra. Fractals 31(5), 2350059 (2023)

    Article  MATH  Google Scholar 

  26. Moran, P.: Additive functions of intervals and Hausdorff measure. Math. Proc. Camb. Philos. Soc. 42(1), 15–23 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  27. Saint Raymond, X., Tricot, C.: Packing regularity of sets in \( n\)-space. Math. Proc. Camb. Philos. Soc. 103(1), 133–145 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  28. Salli, A.: Upper Density Properties of Hausdorff Measures on Fractals. Volume 55 of Annales Academiae Scientiarum Fennicae: Mathematica. Suomalainen Tiedeakatemia, Helsinki (1985)

    Google Scholar 

  29. Schief, A.: Separation properties for self-similar sets. Proc. Am. Math. Soc. 122(1), 111–115 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tricot, C.: Rectifiable and fractal sets. In: Bélair, J., Dubuc, S. (eds.) Fractal Geometry and Analysis. NATO ASI Series, vol. 346, pp. 364–403. Kluber, Dordrecht (1991)

    Google Scholar 

  31. Yu, B., Li, J.: Some fractal characters of porous media. Fractals 9(3), 365–372 (2001)

    Article  Google Scholar 

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Acknowledgements

This work was supported by the Universidad Complutense de Madrid and the Banco de Santander (PR108/20-14).

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Authors and Affiliations

  1. Dpto. Análisis Económico y Economía Cuantitativa, Universidad Complutense de Madrid, Campus de Somosaguas, 28223, Madrid, Spain

    María Eugenia Mera & Manuel Morán

  2. IMI-Institute of Interdisciplinary Mathematics, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040, Madrid, Spain

    Manuel Morán

Authors
  1. María Eugenia Mera
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  2. Manuel Morán
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All the authors wrote the main manuscript, designed the codes, prepared figures and tables and reviewed the manuscript.

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Correspondence to Manuel Morán.

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Mera, M.E., Morán, M. Irregularity Index and Spherical Densities of the Penta-Sierpinski Gasket. Mediterr. J. Math. 20, 322 (2023). https://doi.org/10.1007/s00009-023-02528-6

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