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

Claude Tricot¹

¹ Laboratoire de Mathématiques Pures Université Blaise Pascal 63177 Aubière Cedex, France

Fundamenta Mathematicae, Tome 172 (2002) no. 2, pp. 181-199

Résumé

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.

Détail

DOI : 10.4064/fm172-2-5

Keywords: various methods may define minkowski bouligand dimension compact subset plane best known box method after introducing notion varepsilon connected set varepsilon consider method based coverings varepsilon crosses diameter varepsilon prove cross method gives fractal dimension main argument consists constructing special pavement complementary set squares method gives rise dimension formula using integrals which generalizes known variation method graphs continuous functions