Singular measures for class<i>P</i>-circle homeomorphisms with several break points

Abstract Let f be a class P -homeomorphism of the circle with break point singularities, that is, differentiable except at some singular points where the derivative has a jump. Let f have irrational rotation number and Df be absolutely continuous on every continuity interval of Df . We prove that if the product of the f -jumps along any subset of break points is distinct from 1 then the invariant measure μ f is singular with respect to the Haar measure. This result generalizes previous results obtained by Dzhalilov and Khanin, Dzhalilov, Akhadkulov, Dzhalilov–Liousse and Mayer. Moreover, we prove that if the rotation number ρ ( f ) is irrational of bounded type then (a) if the product of the f -jumps on some orbit is distinct from 1 then the invariant measure μ f is singular with respect to the Haar measure m , and (b) if the product of the f -jumps on each orbit is equal to 1 and D 2 f ∈ L p ( S 1 ) for some p &gt;1 then μ f is equivalent to the Haar measure.

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