Singularity versus exact overlaps for self-similar measures

2017/02/22 by Károly Simon, Simon, Károly, Lajos Vágó +1
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Primary: 28A80 #Secondary: 28A99 #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1702.06785

openalex publication_date 2017/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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-α projections of the natural measure of the Sierpiński 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.

Related

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• A self-similar measure with dense rotations, singular projections and discrete slices
• Self-similar sets and measures on the line
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