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M. Herman showed that the invariant measure μh of a piecewise linear (PL) circle homeomorphism h with two break points and an irrational rotation number ρh is absolutely continuous iff the two break points belong to the same orbit. We extend Herman’s result to the class P of piecewise C2+ε-circle maps f with an irrational rotation number ρf and two break points a0, c0, which do not lie on the same orbit and whose total jump ratio is σf = 1, as follows: if μf denotes the invariant measure of the P -homeomorphism f , then for Lebesgue almost all values of μf ([a0, c0]) the measure μf is singular with respect to Lebesgue measure.

An Extention of Herman’s Theorem for Nonlinear Circle Maps with Two Breaks