Absolute continuity of self-similar measures on the plane

TL;DR

This paper proves that for most parameter choices, self-similar measures generated by planar similarities are absolutely continuous, extending previous results and providing new insights into their dimension and Fourier decay properties.

Contribution

It extends the absolute continuity results of self-similar measures on the plane to a broader parameter range, including the super-critical region, and introduces new findings on their dimension and Fourier decay.

Findings

Almost every parameter choice yields absolutely continuous measures.

Established new bounds on the dimension of random self-similar measures.

Proved power Fourier decay properties of these measures.

Abstract

Consider an iterated function system consisting of similarities on the complex plane of the form $g_{i}(z) = \lambda_i z + t_i,\ \lambda_i, t_i \in \mathbb{C},\ |\lambda_i|<1, i=1,\ldots, k$. We prove that for almost every choice of $(\lambda_1, \ldots, \lambda_k)$ 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.