On absolute continuity of inhomogeneous and contracting on average self-similar measures

Abstract

We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In fact, for $d\ge 1$ and any given rotations in $O(d)$ acting irreducibly on ${\mathbb{R}}^d$ as well as any distinct translations, all having algebraic coefficients, we construct absolutely continuous self-similar measures with the given rotations and translations. We furthermore strengthen Varjú's result for Bernoulli convolutions, treat complex Bernoulli convolutions and in dimension $\ge 3$ improve the condition on absolute continuity by Lindenstrauss-Varjú. Moreover, self-similar measures of contracting on average measures are studied, which may include expanding similarities in their support.

73 pages. Strengthening of the main theorem to include perturbations of rotations and added additional corollaries