Nonrigidity of piecewise-smooth circle maps

Original · EN

Let fᵢ, i=1,2 be piecewise-smooth C¹ circle homeomorphisms with two break points, Dfᵢ, i=1,2 are absolutely continuous on each continuity intervals of Dfᵢ and D Dfᵢ∈ Lᵖ for some p>1. Suppose, the jump ratios of f₁ and f₂ at their break points do not coincide but have the same total jumps (i.e. the product of jump ratios) and identical irrational rotation number of bounded type. Then the conjugation h between f₁ and f₂ is a singular function, i.e. it is continuous on S¹, but Dh(x)=0 a.e. with respect to Lebesgue measure.