Absolute Continuity of Self-similar Measures on the Plane

Consider an iterated function system consisting of similarities on the complex plane of the form g_i(z) = λ_iz+t_i, λ_i, t_i ∈ C, |λ_i | < 1, i = 1,…,k. We prove that for almost every choice of (λ₁,…,λ_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.