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

Abstract

Let \mu 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 \pi_{e\sharp}(\mu) are absolutely continuous for all e \in S^{1} \setminus 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 the previous work of Shmerkin and Solomyak from 2016, where it was assumed that the similitudes associated with \mu have a common contraction ratio.