Singularity results for functional equations driven by linear fractional transformations

TL;DR

This paper investigates the singularity of measures arising from solutions to functional equations driven by linear fractional transformations, establishing conditions for singularity and exploring their connection to stationary measures.

Contribution

It provides a necessary and sufficient condition for the singularity of measures associated with these functional equations and links them to stationary measures.

Findings

Established a criterion for measure singularity.

Connected solutions to stationary measure properties.

Analyzed Hausdorff dimension implications.

Abstract

We consider functional equations driven by linear fractional transformations, which are special cases of de Rham's functional equations. We consider Hausdorff dimension of the measure whose distribution function is the solution. We give a necessary and sufficient condition for singularity. We also show that they have a relationship with stationary measures.