On conjugations of circle homeomorphisms with two break points

Let f(i) is an element of C2+alpha (S-1\{a(i), b(i)}), alpha > 0, i = 1, 2, be circle homeomorphisms with two break points a(i), b(i), that is, discontinuities in the derivative Df(i), with identical irrational rotation number rho and mu(1)([a(1), b(1)]) = mu(2)([a(2), b(2)]), where mu(i) are the invariant measures of f(i), i = 1, 2. Suppose that the products of the jump ratios of Df(1) and Df(2)do not coincide, that is, Df(1)(a(1) - 0)/Df(1)(a(1) + 0). Df(1)(b(1) - 0)/Df(1)(b(1) + 0) not equal Df(2)(a(2) - 0)/Df(2)(a(2) + 0). Df(2)(b(2) - 0)/Df(2)(b(2) + 0). Then the map psi conjugating f(1) and f(2) is a singular function, that is, it is continuous on S-1, but D psi(x) = 0 almost everywhere with respect to Lebesgue measure.