zbMATH — the first resource for mathematics

Dimensions - their determination and properties. (English) Zbl 0734.28008
Fractal geometry and analysis, Proc. NATO ASI Sémin. Math. Supér., Montréal/Can. 1989, NATO ASI Ser., Ser. C 346, 221-254 (1991).
[For the entire collection see Zbl 0728.00016.]
This paper gives six lectures concerned with Hausdorff measure and dimension as well as box-counting dimension, fractal constructions and calculation of dimension by mass distribution principle. It demonstrates the interaction of dimension and geometrical properties of fractals (projection properties, Vitushkin’s conjecture in complex variable theory, intersection of fractals). Self-affine sets, their dimension properties and the Lipschitz equivalence of certain Julia sets are also discussed. The most material is taken from the author’s book “Fractal geometry: mathematical foundations and applications” (1990; Zbl 0689.28003).

28A78 Hausdorff and packing measures
28A80 Fractals
28A75 Length, area, volume, other geometric measure theory