The pointwise dimensions of Moran measures
The calculation of the Hausdorff dimension of a fractal set is in many cases very difficult to execute. Much of the theory of Hausdorff and packing dimension of sets depends on local properties of a suitably defined measure. The lower pointwise dimension of the measure is related to the Hausdorff dimension and the upper pointwise dimension is related to the packing dimension. So it is important to consider the pointwise dimension of a measure in the study of fractals. In this paper the authors extend the known pointwise dimension results of some self-similar measures and Moran measures. The main results are contained in Theorems 2.7, 2.8 and 2.9, where formulas of upper (lower) pointwise dimensions of some Moran measures on Moran sets in \(\mathbb R^d\) are obtained under the strong separation condition. Also, formulas for the dimension of the Moran measures are derived. Some examples are given at the end of the paper. The paper is very well written and the subject is interesting and up to date.
scientific article; zbMATH DE number 5774123