Regularity and singularity for conjugate equations driven by linear fractional transformations

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 consider some real analytic properties of the solution in the case that the equation is driven by non-affine maps, in particular, linear fractional transformations. We give sufficient conditions for the regularity in the sense of Ullman-Stahl-Totik and for the singularity of the solution.