On conjugations of circle homeomorphisms with two break points

Let $f_i\in C^{2+\alpha }(S^1\setminus \{a_i,b_i\}),\alpha >0,i=1,2$ be circle homeomorphisms with two break points $a_i,b_i$, i.e. discontinuities in the derivative $f_i$, with identical irrational rotation number $rho$ and ${\mu }_1([a_1,b_1])={\mu }_2([a_2,b_2])$, where ${\mu }_i$ are invariant measures of $f_i$. Suppose the products of the jump ratios of $D{f}_1$ and $D{f}_2$ do not coincide, i.e. $\frac{Df_1(a_1-0)}{Df_1(a_1+0)}\times \frac{Df_1(b_1-0)}{Df_1(b_1+0)}\neq \frac{Df_2(a_2-0)}{Df_2(a_2+0)}\times \frac{Df_2(b_2-0)}{Df_2(b_2+0)}$\mathrm{.}Thenthemap$\psi$conjugating$f_1$and$f_2$isasingularfunction,i\mathrm{.}e\mathrm{.}itiscontinuouson$S^1$,but$D\psi = 0$ a.e. with respect to Lebesgue measure