Cantor sets in the line: scaling functions and the smoothness of the shift-map

Consider d disjoint closed subintervals of the unit interval and consider an orientation preserving expanding map which maps each of these subintervals to the whole unit interval. The set of points where all iterates of this expanding map are defined is a Cantor set. Associated with the construction of this Cantor set is the scaling function which records the infinitely deep geometry of this Cantor set. This scaling function is an invariant of conjugation. Dennis Sullivan posed the inverse problem: given a scaling function, determine the maximal possible smoothness of any expanding map which produces it. We solve this problem in the case of finite smoothness and in the real-analytic case.