In this note we present some one-parameter families of homogeneous self-similar measures on the line such that • the similarity dimension is greater than 1 for all parameters and • the singularity of some of the self-similar measures from this family is not caused by exact overlaps between the cylinders. We can obtain such a family as the angle-α projections of the natural measure of the Sierpiński carpet. We present more general one-parameter families of self-similar measures vα, such that the set of parameters α for which vα is singular is a dense G δ set but this “exceptional” set of parameters of singularity has zero Hausdorff dimension.
ASJC Scopus subject areas
- Applied Mathematics