An Extention of Herman’s Theorem for Nonlinear Circle Maps with Two Breaks

    Received 15 November 2018; accepted 15 November 2018

    2018, Vol. 14, no. 4, pp.  553-577

    Author(s): Dzhalilov A., Mayer D., Djalilov S., Aliyev A.

    M. Herman showed that the invariant measure $\mu_h$ of a piecewise linear (PL) circle homeomorphism $h$ with two break points and an irrational rotation number $\rho_{h}$ is absolutely continuous iff the two break points belong to the same orbit. We extend Herman's result to the class P of piecewise $ C^{2+\varepsilon} $-circle maps $f$ with an irrational rotation number $\rho_f$ and two break points $ a_{0}, c_{0}$, which do not lie on the same orbit and whose total jump ratio is $\sigma_f=1$, as follows: if $\mu_f$ denotes the invariant measure of the $P$-homeomorphism $f$, then for Lebesgue almost all values of $\mu_f([a_0, c_{0}])$ the measure $\mu_f$ is singular with respect to Lebesgue measure.
    Keywords: piecewise-smooth circle homeomorphism, break point, rotation number, invariant measure
    Citation: Dzhalilov A., Mayer D., Djalilov S., Aliyev A., An Extention of Herman’s Theorem for Nonlinear Circle Maps with Two Breaks, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 4, pp.  553-577
    DOI:10.20537/nd180409


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    References
    [1] Adouani, A. and Marzougui, H., “Singular Measures for Class $P$-Circle Homeomorphisms with Several Break Points”, Ergodic Theory Dynam. Systems, 34:2 (2014), 423–456  crossref  mathscinet  zmath
    [2] Arnol'd, V. I., “Small Denominators: 1. Mapping the Circle onto Itself”, Izv. Akad. Nauk SSSR. Ser. Mat., 1:1 (1961), 21–86 (Russian)  mathnet  mathscinet
    [3] Coelho, Z., Lopes, A., and da Rocha, L. F., “Absolutely Continuous Invariant Measures for a Class of Affine Interval Exchange Maps”, Proc. Amer. Math. Soc., 123:11 (1995), 3533–3542  crossref  mathscinet  zmath
    [4] Cornfeld, I. P., Fomin, S. V., and Sinai, Ya. G., Ergodic Theory, Grundlehren Math. Wiss., 245, Springer, New York, 1982, x+486 pp.  crossref  mathscinet  zmath
    [5] Denjoy, A., “Sur les courbes définies par les équations différentielles à la surface du tore”, J. Math. Pures Appl. (9), 11 (1932), 333–375
    [6] Dzhalilov, A. A. and Khanin, K. M., “On an Invariant Measure for Homeomorphisms of a Circle with a Point of Break”, Funct. Anal. Appl., 32:3 (1998), 153–161  mathnet  crossref  mathscinet  zmath; Funktsional. Anal. i Prilozhen., 32:3 (1998), 11–21 (Russian)  crossref  mathscinet  zmath
    [7] Dzhalilov, A. A., “Piecewise Smoothness of Conjugate Homeomorphisms of a Circle with Corners”, Theoret. and Math. Phys., 120:2 (1999), 179–192  mathnet  crossref  mathscinet  zmath; Teoret. Mat. Fiz., 120:2 (1999), 961–972 (Russian)  crossref  mathscinet  zmath
    [8] Dzhalilov, A. A. and Liousse, I., “Circle Homeomorphisms with Two Break Points”, Nonlinearity, 19:8 (2006), 1951–1968  crossref  mathscinet  zmath  adsnasa
    [9] Dzhalilov, 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  crossref  mathscinet  zmath
    [10] Dzhalilov, A., Jalilov, A., and Mayer, D., “A Remark on Denjoy's Inequality for $PL$ Circle Homeomorphisms with Two Break Points”, J. Math. Anal. Appl., 458:1 (2018), 508–520  crossref  mathscinet  zmath
    [11] Dzhalilov, A. A., Maĭer, D., and Safarov, U. A., “Piecewise-Smooth Circle Homeomorphisms with Several Break Points”, Izv. Math., 76:1 (2012), 94–112  mathnet  crossref  mathscinet  zmath; Izv. Ross. Akad. Nauk Ser. Mat., 76:1 (2012), 101–120 (Russian)  crossref  mathscinet  zmath
    [12] Katznelson, Y. and Ornstein, D., “The Absolute Continuity of the Conjugation of Certain Diffeomorphisms of the Circle”, Ergodic Theory Dynam. Systems, 9:4 (1989), 681–690  mathscinet
    [13] Hawkins, J. and Schmidt, K., “On $C^{2}$-Diffeomorphisms of the Circle Which Are of Type ${\rm III}_{1}$”, Invent. Math., 66:3 (1982), 511–518  crossref  mathscinet  zmath  adsnasa
    [14] Herman, M., “Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations”, Inst. Hautes Études Sci. Publ. Math., 49 (1979), 5–233  crossref  mathscinet  zmath
    [15] Khanin, K. M. and Kocić, S., “Hausdorff Dimension of Invariant Measure of Circle Diffeomorphisms with Breaks”, Ergod. Theory Dyn. Syst., 63 (2017), 1–9  crossref
    [16] Sinai, Ya. G. and Khanin, K. M., “Smoothness of Conjugacies of Diffeomorphisms of the Circle with Rotations”, Russian Math. Surveys, 44:1 (1989), 69–99  mathnet  crossref  mathscinet  adsnasa; Uspekhi Mat. Nauk, 44:1(265) (1989), 57–82, 247  mathscinet
    [17] Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences, Pure Appl. Math., Wiley, New York, 1974, xiv+390 pp.  mathscinet  zmath
    [18] Larcher, G., “A Convergence Problem Connected with Continued Fractions”, Proc. Amer. Math. Soc., 103:3 (1988), 718–722  crossref  mathscinet  zmath
    [19] Liousse, I., “$PL$ Homeomorphisms of the Circle Which Are Piecewise $C^1$ conjugate to Irrational Rotations”, Bull. Braz. Math. Soc. (N. S.), 35:2 (2004), 269–280  crossref  mathscinet  zmath
    [20] Liousse, I., “Nombre de rotation, mesures invariantes et ratio set des homéomorphismes affines par morceaux du cercle”, Ann. Inst. Fourier (Grenoble), 55:2 (2005), 431–482  crossref  mathscinet  zmath
    [21] Nakada, H., “Piecewise Linear Homeomorphisms of Type ${\rm III}$ and the Ergodicity of Cylinder Flows”, Keio Math. Sem. Rep., 1982, no. 7, 29–40  mathscinet  zmath
    [22] Teplinsky, A., “A Circle Diffeomorphism with Breaks That Is Smoothly Linearizable”, Ergod. Theory Dyn. Syst., 38:1 (2018), 371–383  crossref  mathscinet  zmath
    [23] Yoccoz, J.-Ch., “Il n'y a pas de contre-exemple de Denjoy analytique”, C. R. Acad. Sci. Paris Sér. 1 Math., 298:7 (1984), 141–144  mathscinet  zmath



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