On conjugations of circle homeomorphisms with two break points

Abstract

Let \(f_i\in C^{2+\alpha}(S^1\setminus \{a_i,b_i\}), \alpha >0, i=1,2\) be circle homeomorphisms with two break points \(a_i,b_i\), i.e. discontinuities in the derivative \(f_i\), with identical irrational rotation number \(rho\) and \(\mu_1([a_1,b_1])= \mu_2([a_2,b_2])\), where \(\mu_i\) are invariant measures of \(f_i\). Suppose the products of the jump ratios of \(Df_1\) and \(Df_2\) do not coincide, i.e. \(\frac{Df_1(a_1-0)}{Df_1(a_1+0)}\times \frac{Df_1(b_1-0)}{Df_1(b_1+0)}\neq \frac{Df_2(a_2-0)}{Df_2(a_2+0)}\times \frac{Df_2(b_2-0)}{Df_2(b_2+0)}\). Then the map \(\psi\) conjugating \(f_1\) and \(f_2\) is a singular function, i.e. it is continuous on \(S^1\), but \(D\psi = 0\) a.e. with respect to Lebesgue measure