Absolute continuity in families of parametrised non-homogeneous self-similar measures

Journal of fractal geometry: mathematics of fractals and related topics

article

Antti Käenmäki¹  Tuomas Orponen² 

[1] University of Eastern Finland;University of Helsinki

关键词: Self-similar measure;    absolute continuity;    convolution;    projection;   

DOI  :  10.4171/jfg/127

来源: European Mathematical Society

【 摘 要 】

Let μ\muμ be a planar self-similar measure with similarity dimension exceeding 111, satisfying a mild separation condition, and such that the fixed points of the associated similitudes do not share a common line. Then, we prove that the orthogonal projections πe♯(μ)\pi_{e\sharp}(\mu)πe♯​(μ) are absolutely continuous for all e∈S1∖Ee \in S^{1} \setminus Ee∈S1∖E, where the exceptional set EEE has zero Hausdorff dimension. The result is obtained from a more general framework which applies to certain parametrised families of self-similar measures on the real line. Our results extend the previous work of Shmerkin and Solomyak from 2016, where it was assumed that the similitudes associated with μ\muμ have a common contraction ratio.

【 授权许可】

Unknown