Quantitative estimates for singularity for conjugate equations driven by linear fractional transformations

TL;DR

This paper provides quantitative criteria for the singularity of solutions to conjugate equations driven by families of maps, including linear fractional transformations, extending de Rham's functional equations.

Contribution

It introduces sufficient conditions for solution singularity in conjugate equations with non-affine maps and linear fractional transformations, with quantitative estimates.

Findings

Identifies conditions leading to solution singularity.

Provides quantitative estimates for singular solutions.

Extends de Rham's functional equations framework.

Abstract

We consider the conjugate equation driven by two families of finite maps on the unit interval satisfying a compatibility condition. This framework contains de Rham's functional equations. We give sufficient conditions for singularity of the solution with quantitative estimates in the case where the equation is driven by a family of non-affine maps and a family of linear fractional transformations.