  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.
