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Exact Computation and Approximation of Stochastic and Analytic Parameters of Generalized Sierpinski Gaskets

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Abstract

The interplay of fractal geometry, analysis and stochastics on the one-parameter sequence of self-similar generalized Sierpinski gaskets is studied. An improved algorithm for the exact computation of mean crossing times through the generating graphs SG(m) of generalized Sierpinski gaskets sg(m) for m up to 37 is presented and numerical approximations up to m = 100 are shown. Moreover, an alternative method for the approximation of the mean crossing times, the walk and the spectral dimensions of these fractal sets based on quasi-random so-called rotor walks is developed, confidence bounds are calculated and numerical results are shown and compared with exact values (if available) and with known asymptotic formulas.

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Author notes

  1. Christoph Thäle

    Present address: Institute of Mathematics, University of Osnabrück, Albrechtstr. 28a, 49076, Osnabrück, Germany

Authors and Affiliations

  1. University of Siegen, Emmy-Noether-Campus, Walter-Flex-Str. 3, 57068, Siegen, Germany

    Uta Freiberg

  2. Department of Mathematics, University of Fribourg, Chemin du musée 23, 1700, Fribourg, Switzerland

    Christoph Thäle

Authors
  1. Uta Freiberg
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  2. Christoph Thäle
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Correspondence to Christoph Thäle.

Additional information

The second author was supported by the Swiss National Science Foundation Grant SNF PP002-114715/1.

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Freiberg, U., Thäle, C. Exact Computation and Approximation of Stochastic and Analytic Parameters of Generalized Sierpinski Gaskets. Methodol Comput Appl Probab 15, 485–509 (2013). https://doi.org/10.1007/s11009-011-9254-7

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  • DOI: https://doi.org/10.1007/s11009-011-9254-7

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