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=\left\langle z,w\right\rangle $ 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 $z\rightarrow P_{z}\nu$ is continuous with respect to the weak topology of $L^{q}(\mathbb{R})$.