Let $f_{ i},$ $i=1,2$ be piecewise-smooth $C^{ 1}$ circle homeomorphisms with two break points, $\log Df_{ i},$ $i=1,2$ are absolutely continuous on each continuity intervals of $Df_{ i}$ and $D\log Df_{ i}\in L^{ p}$ for some $p>1.$ Suppose, the jump ratios of $f_{ 1} $ and $f_{ 2} $ at their break points do not coincide but have the same total jumps (i.e. the product of jump ratios) and identical irrational rotation number of bounded type. Then the conjugation $h$ between $f_{ 1} $ and $f_{ 2} $ is a singular function, i.e. it is continuous on $S^1,$ but $Dh(x)=0$ a.e. with respect to Lebesgue measure.
Categories: math.DS