On conjugations of circle homeomorphisms with two break points

Abstract Let f i ∈ C 2+ α ( S 1 ∖{ a i , b i }), α >0, i =1,2, be circle homeomorphisms with two break points a i , b i , that is, discontinuities in the derivative Df i , with identical irrational rotation number ρ and μ 1 ([ a 1 , b 1 ])= μ 2 ([ a 2 , b 2 ]), where μ i are the invariant measures of f i , i =1,2. Suppose that the products of the jump ratios of Df 1 and Df 2 do not coincide, that is, Df 1 ( a 1 −0)/ Df 1 ( a 1 +0)⋅ Df 1 ( b 1 −0)/ Df 1 ( b 1 +0)≠ Df 2 ( a 2 −0)/ Df 2 ( a 2 +0)⋅ Df 2 ( b 2 −0)/ Df 2 ( b 2 +0) . Then the map ψ conjugating f 1 and f 2 is a singular function, that is, it is continuous on S 1 , but Dψ ( x )=0 almost everywhere with respect to Lebesgue measure.

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