Nonrigidity of piecewise-smooth circle maps

Abstract

Let $f_i,$ $i=1,2$ be piecewise-smooth $C^1$ circle homeomorphisms with two break points, $\log D{f}_i,$ $i=1,2$ are absolutely continuous on each continuity intervals of $D{f}_i$ and $D\log D{f}_i\in L^p$ 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.

21 pages 2 figures