Invariant Measure of a Circle Map with a Mixed Type of Singularities

In this work, a critical circle homeomorphism with several break points is considered. The circle homeomorphism with the irrational rotation number is well known to be strictly ergodic; i.e., it has the unique -invariant probability measure . The invariant measure of critical circle homeomorphisms with a finite number of break points is proved to be singular with respect to the Lebesgue measure.