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

2018/12/12 by Käenmäki, Antti, Orponen, Tuomas
#42A38 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Primary 28A80 #Secondary 28A78

paper · doi:10.48550/arxiv.1812.05006

Abstract

Let μ be a planar self-similar measure with similarity dimension exceeding 1, 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 π_e\sharp(μ) are absolutely continuous for all e ∈ S¹ ∖ E, where the exceptional set E 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 previous work of Shmerkin and Solomyak from 2016, where it was assumed that the similitudes associated with μ have a common contraction ratio.

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