On the Singularity of the Conjugation Between Piecewise-Smooth Circle Homeomorphisms

Abstract

Let $$f_1$$ and $$f_2$$ be two orientation-preserving circle homeomorphisms with the same irrational rotation number $$\rho $$ and each with a single break point $$b_1$$ and $$b_2$$ , respectively. Suppose that the derivatives $$Df_1$$ and $$Df_2$$ satisfy a certain Zygmund condition except break points and the jumps $$\sigma (b_1)=\frac{Df_1(b_1-0)}{Df_1(b_1+0)}$$ , $$\sigma (b_2)=\frac{Df_2(b_2-0)}{Df_1(b_2+0)}$$ do not coincide. Then the map $$\psi $$ conjugating $$f_1$$ and $$f_2$$ is singular.