Dimension of homogeneous rational self-similar measures with overlaps
The self-similar measures play an important role in the studies of fractal geometry, the case having overlaps is still a difficult question. The author consider a class of self-similar measures with overlaps which are obtained from systems of homotheties with centres in a lattice where the contraction ratios are all equal to the inverse of a natural number \(L\). In this case, the author gets the local dimension of the measure as the Shannon entropy (up to a constant), which shows also some relations with the properties of the absolute continuity or singularity of the measure.
scientific article; zbMATH DE number 5549434
scientific article