A note on the conjugacy between two critical circle maps

We study a conjugacy between two critical circle homeomorphisms with irrational rotation number. Let $f_{i}, i=1,2$ be a $C^{3}$ circle homeomorphisms with critical point $x_{cr}^{(i)}$ of the order $2m_{i}+1$. We prove that if $2m_{1}+1 \neq 2m_{2}+1$, then conjugating between $f_{1}$ and $f_{2}$ is a singular function.