Density theorems for Hausdorff and packing measures of self-similar sets

• Autores: Lars Olsen
• Localización: Aequationes mathematicae, ISSN 0001-9054, Vol. 75, Nº. 3, 2008, págs. 208-225
• Idioma: inglés
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• Resumen

  • We analyze the local behaviour of the Hausdorff measure and the packing measure of self-similar sets. In particular, if K is a self-similar set whose Hausdorff dimension and packing dimension equal s, a special case of our main results says that if K satisfies the Open Set Condition, then there exists a number r 0 such that 1 and 2 for all x ? K and all 0 < r < r 0, where denotes the s-dimensional Hausdorff measure and denotes the s-dimensional packing measure. Inequality (1) and inequality (2) are used to obtain a number of very precise density theorems for Hausdorff and packing measures of self-similar sets. These density theorems can be applied to compute the exact value of the s-dimensional Hausdorff measure and the exact value of the s-dimensional packing measure of self-similar sets K.