Exceptions in the domain of generic absolute continuity of non-homogeneous self-similar measures

TL;DR

This paper constructs specific examples of singular non-homogeneous self-similar measures within a domain where such measures are typically absolutely continuous, challenging previous generic results.

Contribution

It introduces algebraic curves to explicitly construct exceptional singular measures in the domain where absolute continuity is expected.

Findings

Constructed explicit singular measures using algebraic curves.

Identified exceptions within the parameter domain of absolute continuity.

Challenged the generic absolute continuity results for non-homogeneous self-similar measures.

Abstract

Non-homogeneous self-similar measures are generically absolute continuous in the domain of parameters for which the similarity dimension is larger than one, see \cite{[SSS]}. Using certain algebraic curves we construct here exceptional singular non-homogeneous self-similar measures in this domain.