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

• Autorzy: Claude Tricot
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 28A75: Length, area, volume, other geometric measure theory
• 28A80: Fractals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2002
• Tom: 172
• Numer: 2
• Strony: 181-199

Daty

wydano

2002

Twórcy

autor

Claude Tricot

• Laboratoire de Mathématiques Pures, Université Blaise Pascal, 63177 Aubière Cedex, France

Identyfikatory

DOI

10.4064/fm172-2-5