Conjugations between circle maps with a single break point

Consider two circle homeomorphisms f(i) is an element of C2+alpha (S\{b(i)}), alpha > 0. i = 1, 2 with a single break point b(i) i.e. a discontinuity in the derivative Df(i), and identical irrational rotation number rho. Suppose the jump ratios sigma(1) = Df(1)(b(1)-0)/Df(1)(b(1)+0) and sigma(2) = Df(1)(b(2)-0)/Df(2)(b(2)+0) do not coincide. Then the map psi conjugating f(1) and f(2) is a singular function i.e. it is continuous on S-1 and D psi(x) = 0 a.e. with respect to LebeSgUe measure. (C) 2009 Elsevier Inc. All rights reserved.