Absolute continuity of self-similar measures on the plane

Abstract

Consider an iterated function system consisting of similarities on the complex plane of the form $g_i(z)={\lambda }_{i}z+t_i,{\lambda }_i,t_i\in \mathbb{C},|{\lambda }_i|<1,i=1,\dots ,k$. We prove that for almost every choice of $({\lambda }_1,\dots ,{\lambda }_k)$ in the super-critical region (with fixed translations and probabilities), the corresponding self-similar measure is absolutely continuous. This extends results of Shmerkin-Solomyak (in the homogenous case) and Saglietti-Shmerkin-Solomyak (in the one-dimensional non-homogeneous case). As the main steps of the proof, we obtain results on the dimension and power Fourier decay of random self-similar measures on the plane, which may be of independent interest.

Final authors version to appear in Indiana Univ. Math. J