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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: akhadkulov@yahoo.com)
AKHTAM DZHALILOV
Affiliation:
Faculty of Mathematics and Mechanics, Samarkand State University, Boulevard st. 15, 703004 Samarkand, Uzbekistan (email: a_dzhalilov@yahoo.com)
DIETER MAYER
Affiliation:
Institut für Theoretische Physik, TU Clausthal, Leibnizstraße 10, D-38678 Clausthal-Zellerfeld, Germany (email: dieter.mayer@tu-clausthal.de)
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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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References

[1]Arnol’d, V. I.. Small denominators: I. Mappings from the circle onto itself. Izv. Akad. Nauk SSSR, Ser. Mat. 25 (1961), 21–86.Google Scholar
[2]Denjoy, A.. Sur les courbes définies par les équations différentielles à la surface du tore. J. Math. Pures Appl. 11 (1932), 333–375.Google Scholar
[3]Dzhalilov, A. A. and Khanin, K. M.. On invariant measure for homeomorphisms of a circle with a point of break. Funct. Anal. Appl. 32(3) (1998), 153–161.Google Scholar
[4]Dzhalilov, A. A. and Liousse, I.. Circle homeomorphisms with two break points. Nonlinearity 19 (2006), 1951–1968.Google Scholar
[5]Dzhalilov, A. A., Liousse, I. and Mayer, D.. Singular measures of piecewise smooth circle homeomorphisms with two break points. Discrete Contin. Dyn. Syst. 24(2) (2009), 381–403.Google Scholar
[6]Dzhalilov, A. A., Akin, H. and Temir, S.. Conjugations between circle maps with a single break point. J. Math. Anal. Appl. 366 (2010), 1–10.CrossRefGoogle Scholar
[7]Cornfeld, I. P., Fomin, S. V. and Sinai, Ya. G.. Ergodic Theory. Springer, Berlin, 1982.CrossRefGoogle Scholar
[8]Herman, M.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. Inst. Hautes Études Sci. 49 (1979), 225–234.CrossRefGoogle Scholar
[9]Katznelson, Y. and Ornstein, D.. The absolute continuity of the conjugation of certain diffeomorphisms of the circle. Ergod. Th. & Dynam. Sys. 9 (1989), 681–690.CrossRefGoogle Scholar
[10]Khanin, K. M. and Sinai, Ya. G.. Smoothness of conjugacies of diffeomorphisms of the circle with rotations. Russian Math. Surveys 44 (1989), 69–99, translation of Uspekhi. Mat. Nauk 44, 57–82 (1989).Google Scholar
[11]Khanin, K. M. and Vul, E. B.. Circle homeomorphisms with weak discontinuities. Adv. Sov. Math. 3 (1993), 57–98.Google Scholar
[12]Khanin, K. M. and Khmelev, D.. Renormalizations and rigidity theory for circle homeomorphisms with singularities of the break type. Comm. Math. Phys. 235 (2003), 69–124.Google Scholar
[13]Liousse, I.. Nombre de rotation, mesures invariantes et ratio set des homéomorphisms affines par morceaux du cercle. Ann. Inst. Fourier 55 (2005), 431–482.Google Scholar
[14]Moser, J.. A rapid convergent iteration method and non-linear differential equations. II. Ann. Sc. Norm. Super. Pisa 20(3) (1966), 499–535.Google Scholar
[15]de Melo, W. and van Strien, S.. One Dimensional Dynamics. Springer, Berlin, 1993, pp. 3–25.CrossRefGoogle Scholar
[16]Stein, M.. Groups of piecewise linear homeomorphisms. Trans. Amer. Math. Soc. 32 (1992), 477–514.Google Scholar
[17]Teplinskii, A. Yu. and Khanin, K. M.. Rigidity for circle diffeomorphisms with simgularities. Russian Math. Surveys 759(2) (2004), 329–353, translation of Uspekhi. Mat. Nauk 59(2), 137–160 (2004).CrossRefGoogle Scholar
[18]Yoccoz, J. C.. Il n’y a pas de contre-exemple de Denjoy analytique. C. R. Acad. Sci. Paris 298(7) (1984), 141–144.Google Scholar