Invariant measure of circle maps with mixed type of singularities

In this paper we consider the critical circle homeomorphisms with several break points. It is well known that a circle homeomorphisms $f$ with irrational rotation number $\rho$ is strictly ergodic, i.e. it has a unique $f$ –invariant probability measure $\mu$. We prove that invariant measure of critical circle homeomorphisms with finite number of break points is singular w.r.t Lebegue measure.