A construction of singular overlapping asymmetric self-similar measures

TL;DR

This paper constructs specific singular measures within a class of overlapping asymmetric self-similar measures, expanding understanding of their measure-theoretic properties beyond the typical absolutely continuous cases.

Contribution

It introduces a method to explicitly construct singular measures in a previously studied class of overlapping asymmetric self-similar measures.

Findings

Identified conditions under which measures are singular.

Demonstrated existence of singular measures in the class.

Extended the classification of measure types within self-similar systems.

Abstract

In [8] we found a class of overlapping asymmetric self-similar measures on the real line, which are generically absolutely continuous with respect to the Lebesgue measure. Here we construct exceptional measures in this class being singular.