Singular measures of circle homeomorphisms with two break points

Abstract

Let $T_{f}$ be a circle homeomorphism with two break points $a_{b},c_{b}$ and irrational rotation number $\varrho_{f}$. Suppose that the derivative $Df$ of its lift $f$ is absolutely continuous on every connected interval of the set $S^{1}\backslash\{a_{b},c_{b}\}$, that $DlogDf \in L^{1}$ and the product of the jump ratios of $ Df $ at the break points is nontrivial, i.e. $\frac{Df_{-}(a_{b})}{Df_{+}(a_{b})}\frac{Df_{-}(c_{b})}{Df_{+}(c_{b})}\neq1$. We prove that the unique $T_{f}$- invariant probability measure $\mu_{f}$ is then singular with respect to Lebesgue measure $l$ on $S^{1}$.