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Generic Hölder level sets and fractal conductivity

Author

Listed:
  • Buczolich, Zoltán
  • Maga, Balázs
  • Vértesy, Gáspár

Abstract

Hausdorff dimensions of level sets of generic continuous functions defined on fractals can give information about the “thickness/narrow cross-sections” of a “network” corresponding to a fractal set, F. This lead to the definition of the topological Hausdorff dimension of fractals. In this paper we continue our study of the level sets of generic 1-Hölder-α functions. While in a previous paper we gave the initial definitions and established some properties of these generic level sets, in this paper we provide numerical estimates in the case of the Sierpiński triangle. These calculations give better insight and illustrate why can one think of these generic 1-Hölder-α level sets as something measuring “thickness/narrow cross-sections/conductivity” of a fractal “network”.

Suggested Citation

  • Buczolich, Zoltán & Maga, Balázs & Vértesy, Gáspár, 2022. "Generic Hölder level sets and fractal conductivity," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    • Handle: RePEc:eee:chsofr:v:164:y:2022:i:c:s096007792200875x
    DOI: 10.1016/j.chaos.2022.112696
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    References listed on IDEAS

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    1. Balankin, Alexander S., 2020. "Fractional space approach to studies of physical phenomena on fractals and in confined low-dimensional systems," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    2. Balka, Richárd & Buczolich, Zoltán & Elekes, Márton, 2012. "Topological Hausdorff dimension and level sets of generic continuous functions on fractals," Chaos, Solitons & Fractals, Elsevier, vol. 45(12), pages 1579-1589.
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