Invariant Measures of Piecewise Smooth Circle Maps with Breaks

In this paper we consider general orientation preserving circle homeomorphisms f ∈ C2+ε(S1∈{a(0), c(0)}), ε \u3e 0, with an irrational rotation number ρf and two break points a(0), c(0). Denote by (:)()()bfbbDfxxDfxσ+−=, xb = a(0), c(0), the jump ratios of f at the two break points and by σf:= σf (a(0)) ∈ σf(c(0)) its total jump ratio. Let h be a piecewise-linear (PL) circle homeomorphism with two break points a0, c0, irrational rotation number ρh and total jump ratio σh = 1. M. Herman’s showed, that the invariant measure μh is absolutely continuous if the two break points belong to the same orbit. We extend Herman’s result for the above class of piecewise C2+ε-circle maps f with irrational rotation number ρf and two break points b(1), b(2) not lying on the same orbit with total jump ratio σf = 1 as follows: if μf denotes the invariant measure of the P-homeomorphism f and ()()12([]),1fbbβμβ=+, then for almost all β the measure μf is singular with respect to Lebesgue measure