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

## TL;DR

This paper proves that for a broad class of planar self-similar measures with dimension exceeding one, most orthogonal projections are absolutely continuous, extending previous results to more general parametrised families.

## Contribution

It extends prior work by removing the common contraction ratio assumption, showing absolute continuity for a wider class of self-similar measures and their projections.

## Key findings

- Projections are absolutely continuous outside a zero Hausdorff dimension set.
- Results apply to parametrised families of self-similar measures on the real line.
- General framework broadens applicability beyond previous assumptions.

## 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 previous work of Shmerkin and Solomyak from 2016, where it was assumed that the similitudes associated with $\mu$ have a common contraction ratio.