Acta of Turin Polytechnic University in Tashkent


In this paper we consider general orientation preserving circle homeomorphisms f ∈ C2+ε(S1∈{a(0), c(0)}), ε > 0, with an irrational rotation number ρf and two break points a(0), c(0). Denote by (:)()()bfbbDfxxDfxσ+−=, xb = a(0), c(0), the jump ratios of f at the two break points and by σf:= σf (a(0)) ∈ σf(c(0)) its total jump ratio. Let h be a piecewise-linear (PL) circle homeomorphism with two break points a0, c0, irrational rotation number ρh and total jump ratio σh = 1. M. Herman’s showed, that the invariant measure μh is absolutely continuous if the two break points belong to the same orbit. We extend Herman’s result for the above class of piecewise C2+ε-circle maps f with irrational rotation number ρf and two break points b(1), b(2) not lying on the same orbit with total jump ratio σf = 1 as follows: if μf denotes the invariant measure of the P-homeomorphism f and ()()12([]),1fbbβμβ=+, then for almost all β the measure μf is singular with respect to Lebesgue measure.


1. V.I. Arnold, Small denominators I. Mapping the circle onto itself, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961) 21–86. 2. I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai, Ergodic Theory, Springer Verlag, Berlin (1982). 3. A.A. Dzhalilov and K.M. Khanin, On invariant measure for homeomorphisms of a circle with a point of break, Functional Anal. i Prilozhen. 32 (3) (1998) 11–21, translation in Funct. Anal. Appl. 32 (3) p.153-161, (1998) . 4. A.A. Dzhalilov and I. Liousse, Circle homeomorphisms with two break points, Nonlinearity, 19 1951–1968,(2006). 5. A.A. Dzhalilov, I. Liousse and D. Mayer, Singular measures of piecewise smooth circle homeomorphisms with two break points, Discrete Contin. Dyn. Syst. 24 (2) p. 381–403,(2009) . 6. A.A. Dzhalilov, D. Mayer and U.A. Safarov, Piecwise-smooth circle homeomorphisms with several break points, Izv. Ross. Akad. Nauk Ser. Mat. 76 (1) (2012) 101–120, translation in Izv. Math. 76 (1) p.94–112, (2012). 7. M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Etudes Sci. Publ. Math. 49 (1979) 5–233. 8. Y. Katznelson and D. Ornstein, The absolute continuity of the conjugation of certain diffeomorphisms of the circle, Ergodic Theory Dynamic. Systems 9 , p.681–690, (1989). 9. K.M. Khanin and Ya.G. Sinai, Smoothness of conjugacies of diffeomorphisms of the circle with rotations, Russ. Math. Surveys 44 (1) (1989) 69–99, translation of Uspekhi Mat. Nauk 44 (1) (1989) 57–82.