Conjugacies between P-homeomorphisms with several breaks

Abstract

Let $f_i,i=1,2$ be orientation preserving circle homeomorphisms with a finite number of break points, at which the first derivatives $D{f}_i$ have jumps, and with identical irrational rotation number $\rho ={\rho }_{f_1}={\rho }_{f_2}.$ The jump ratio of $f_i$ at the break point $b$ is denoted by ${\sigma }_{f_i}(b)$, i.e. ${\sigma }_{f_i}(b):=\frac{D{f}_i(b-0)}{D{f}_i(b+0)}$. Denote by ${\sigma }_{f_i},i=1,2,$ the total jump ratio given by the product over all break points $b$ of the jump ratios ${\sigma }_{f_i}(b)$ of $f_i$. We prove, that for circle homeomorphisms $f_i,i=1,2$, which are $C^{2+\epsilon },\epsilon >0$, on each interval of continuity of $D{f}_i$ and whose total jump ratios ${\sigma }_{f_1}$ and ${\sigma }_{f_2}$ do not coincide, the congugacy between $f_1$ and $f_2$ is a singular function.

22 pages