Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

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

Tom 172 / 2002

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

Streszczenie

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.

Autorzy

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

Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Odśwież obrazek