Conjugacies between P-homeomorphisms with several breaks

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

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