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

showed that the invariant measure h of a piecewise linear (PL) circle homeomorphism h with two break points and an irrational rotation number h is absolutely continuous iff the two break points belong to the same orbit. We extend Herman's result to the class P of piecewise C 2+ -circle maps f with an irrational rotation number f and two break points a 0 , c 0 , which do not lie on the same orbit and whose total jump ratio is f = 1, as follows: if f denotes the invariant measure of the P -homeomorphism f , then for Lebesgue almost all values of f ([a 0 , c 0 ]) the measure f is singular with respect to Lebesgue measure.

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