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

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Tricot C.

Fundamenta Mathematicae

2002

tom 172

nr 2

181-199

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 ε-connected set $E_{ε}$, we consider a new method based upon coverings of $E_{ε}$ with crosses of diameter 2ε. 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.

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1 Fundamenta Mathematicae

1 Tricot C.

1 2002

Wyniki wyszukiwania

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