A method for evaluating the fractal dimension in the plane, using coverings with crosses

Volume 172 / 2002

Claude Tricot Fundamenta Mathematicae 172 (2002), 181-199 MSC: 28A75, 28A80. DOI: 10.4064/fm172-2-5

Abstract

Various methods may be used to define the Minkowski–Bouligand dimension of a compact subset $E$ in the plane. The best known is the box method. After introducing the notion of $\varepsilon $-connected set $E_{\varepsilon }$, we consider a new method based upon coverings of $E_{\varepsilon }$ with crosses of diameter $2{\varepsilon }$. To prove that this cross method gives the fractal dimension for all $E$, the main argument consists in constructing a special pavement of the complementary set with squares. This method gives rise to a dimension formula using integrals, which generalizes the well known variation method for graphs of continuous functions.

Authors

  • Claude TricotLaboratoire de Mathématiques Pures
    Université Blaise Pascal
    63177 Aubière Cedex, France
    e-mail

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