Singularity versus exact overlaps for self-similar measures

Simon, Károly; Vágó, Lajos

doi:10.1090/proc/14042

Singularity versus exact overlaps for self-similar measures
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by Károly Simon and Lajos Vágó PDF

Proc. Amer. Math. Soc. 147 (2019), 1971-1986 Request permission

Abstract:

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-$\alpha$ projections of the natural measure of the Sierpiński carpet. We present more general one-parameter families of self-similar measures $\nu _\alpha$, such that the set of parameters $\alpha$ for which $\nu _\alpha$ is singular is a dense $G_\delta$ set but this “exceptional” set of parameters of singularity has zero Hausdorff dimension.

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