Conformal measures and density estimation

The notion of conformal measures or densities goes back to Patterson's work [8]. By a theorem of Milnor and Thurston [7] a piecewise monotone and continuous map of the interval is semiconjugate to one with constant slopes. The semiconjugacies can be defined by distribution functions of conformal measures as shown in [2]. In this note we show that for some transformations the conjugacies are estimable functions and can be used to improve estimation procedures, in particular density estimations.