On self-similar measures with absolutely continuous projections and dimension conservation in each direction

Abstract

Relying on results due to Shmerkin and Solomyak, we show that outside a $0$-dimensional set of parameters, for every planar homogeneous self-similar measure $\nu$, with strong separation, dense rotations and dimension greater than $1$, there exists $q>1$ such that $\{P_z\nu {\}}_{z\in S}\subset L^q(\mathbb{R})$. Here $S$ is the unit circle and $P_{z}w=⟨z,w⟩$ for $w\in {\mathbb{R}}^2$. We then study such measures. For instance, we show that $\nu$ is dimension conserving in each direction and that the map $z\to P_z\nu$ is continuous with respect to the weak topology of $L^q(\mathbb{R})$.