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On conjugations of circle homeomorphisms with two break points

Published online by Cambridge University Press:  30 November 2012

HABIBULLA AKHADKULOV ,
AKHTAM DZHALILOV  and
DIETER MAYER
HABIBULLA AKHADKULOV
Affiliation:
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia (email: [email protected])
AKHTAM DZHALILOV
Affiliation:
Faculty of Mathematics and Mechanics, Samarkand State University, Boulevard st. 15, 703004 Samarkand, Uzbekistan (email: [email protected])
DIETER MAYER
Affiliation:
Institut für Theoretische Physik, TU Clausthal, Leibnizstraße 10, D-38678 Clausthal-Zellerfeld, Germany (email: [email protected])
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Abstract

Let fi∈C2+α(S1∖{ai,bi}), α>0,i=1,2, be circle homeomorphisms with two break points ai,bi, that is, discontinuities in the derivative Dfi, with identical irrational rotation number ρ and μ1([a1,b1])=μ2([a2,b2]), where μi are the invariant measures of fi,i=1,2. Suppose that the products of the jump ratios of Df1 and Df2do not coincide, that is, Df1 (a1 −0)/Df1 (a1 +0)⋅Df1 (b1 −0)/Df1 (b1 +0)≠Df2(a2−0)/Df2(a2+0)⋅Df2(b2−0)/Df2(b2+0) . Then the map ψ conjugating f1 and f2 is a singular function, that is, it is continuous on S1, but Dψ(x)=0 almost everywhere with respect to Lebesgue measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 3 , June 2014 , pp. 725 - 741
Copyright
©2012 Cambridge University Press 

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