Singularity versus exact overlaps for self-similar measures

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 Sierpi'nski carpet. We present more general one-parameter families of self-similar measures $ν_{α}$ , such that the set of parameters $α$ for which $ν_{α}$ is singular is a dense $G_{δ}$ set but this “exceptional” set of parameters of singularity has zero Hausdorff dimension.

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 Sierpi'nski carpet. We present more general one-parameter families of self-similar measures $ν_{α}$ , such that the set of parameters $α$ for which $ν_{α}$ is singular is a dense $G_{δ}$ set but this “exceptional” set of parameters of singularity has zero Hausdorff dimension.

文献摘要