Singular measures of piecewise smooth circle homeomorphisms with two break points

Let Tf : S1 ! S1 be a circle homeomorphism with two break points ab,cb, that means the derivative Df of its lift f : R ! R has discon- tinuities at the pointsab, ˜cb which are the representative points of ab, cb in the interval (0,1), and irrational rotation number ‰f. Suppose that Df is ab- solutely continuous on every connected interval of the set (0,1)\{˜ab, ˜cb}, that DlogDf 2 L1((0,1)) and the product of the jump ratios of Df at the break points is nontrivial, i.e. Df ¡(˜ ab) Df+(˜ ab) Df¡(˜ cb) Df+(˜ cb) 6= 1. We prove that the unique Tf- invariant probability measure µf is then singular with respect to Lebesgue measure on S1.