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

TL;DR

This paper establishes a new condition for the absolute continuity of self-similar measures across dimensions, providing explicit examples and extending previous results to more general settings including inhomogeneous and contracting-on-average measures.

Contribution

It introduces a novel criterion for absolute continuity of self-similar measures, constructs explicit examples in low dimensions, and extends existing results to complex and higher-dimensional cases.

Findings

Constructed explicit absolutely continuous inhomogeneous self-similar measures in dimensions one and two.

Extended Varjú's results for Bernoulli convolutions and treated complex cases.

Improved conditions for absolute continuity in dimensions three and above.

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 $d\geq 1$ 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 Varj\'u's result for Bernoulli convolutions, treat complex Bernoulli convolutions and in dimension $\geq 3$ improve the condition on absolute continuity by Lindenstrauss-Varj\'u. Moreover, self-similar measures of contracting on average measures are studied, which may include expanding similarities in their support.