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 $$f$$ with the irrational rotation number $$\rho $$ is well known to be strictly ergodic; i.e., it has the unique $$f$$ -invariant probability measure $$\mu $$ . 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.

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