On conjugations of circle homeomorphisms with two break points
Habibulla AkhadkulovSchool of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan, 43600 UKM Bangi, Selangor Darul Ehsan, MalaysiaAkhtam DzhalilovFaculty of Mathematics and Mechanics, Samarkand State University, Boulevard st. 15, 703004 Samarkand, UzbekistanDieter MayerInstitut für Theoretische Physik, TU Clausthal, Leibnizstraße 10, D-38678 Clausthal-Zellerfeld, Germany
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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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