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A sufficient condition for \(C^1\)-smoothness of the conjugation between piecewise smooth circle homeomorphisms. (English) Zbl 1427.37031

Summary: Let \(T_1\) and \(T_2\) be two piecewise smooth circle homeomorphisms with countably many break points and identical irrational rotation number. We provide a sufficient condition for \(C^1\)-smoothness of the conjugation between \(T_1\) and \(T_2\).

MSC:

37E10 Dynamical systems involving maps of the circle
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
37E45 Rotation numbers and vectors
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[1] A. Adouani, Conjugation between circle maps with several break points, Ergodic Theory Dynam. Systems 36 (2016), no. 8, 2351-2383. · Zbl 1370.37080
[2] A. Adouani and H. Marzougui, Sur les homéomorphismes du cercle de classe \(PC^r\) par morceaux \((r\geq 1)\) qui sont conjugués \(C^r\) par morceaux aux rotations irrationnelles, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 3, 755-775. · Zbl 1147.37024
[3] A. Adouani and H. Marzougui, Singular measures for class \(P\)-circle homeomorphisms with several break points, Ergodic Theory Dynam. Systems 34 (2014), no. 2, 423-456. · Zbl 1300.37029
[4] H. Akhadkulov, Some circle homeomorphisms with break-type singularities (in Russian), Uspekhi Mat. Nauk 61 (2006), no. 5, 183-184; English translation in Russian Math. Surveys 61 (2006), no. 5, 981-983. · Zbl 1142.37340
[5] H. Akhadkulov, A. Dzhalilov, and K. Khanin, Notes on a theorem of Katznelson and Orstein, Discrete Contin. Dyn. Syst. 37 (2017), no. 9, 4587-4609. · Zbl 1368.37046
[6] H. Akhadkulov, A. Dzhalilov, and D. Mayer, On conjugations of circle homeomorphisms with two break points, Ergodic Theory Dynam. Systems 34 (2014), no. 3, 725-741. · Zbl 1350.37047
[7] H. Akhadkulov, A. Dzhalilov, and M. S. Md. Noorani, On conjugacies between piecewise-smooth circle maps, Nonlinear Anal. 99 (2014), 1-15. · Zbl 1347.37046
[8] H. Akhadkulov, M. S. Md. Noorani, and S. Akhatkulov, Renormalization of circle diffeomorphisms with a break-type singularity, Nonlinearity 30 (2017), no. 7, 2687-2717. · Zbl 1431.37039
[9] I. P. Cornfeld, S. V. Fomin, and Y. G. Sinaĭ, Ergodic Theory, Grundlehren Math. Wiss. 245, Springer, New York, 1982.
[10] A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl. (9) 11 (1932), 333-375. · JFM 58.1124.04
[11] A. Dzhalilov and K. Khanin, On an invariant measure for homeomorphisms of the circle with one break point (in Russian), Uspekhi Mat. Nauk 51 (1996), no. 6, 201-202; English translation in Russian Math. Surveys 51 (1996), no. 6, 1198-1199. · Zbl 0894.58033
[12] A. Dzhalilov, D. Mayer, and U. Safarov, Piecewise-smooth circle homeomorphisms with several break points (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 76 (2012), no. 1, 101-120; English translation in Izv. Math. 76 (2012), no. 1, 94-112. · Zbl 1242.37030
[13] A. Dzhalilov, D. Mayer, and U. Safarov, On the conjugation of piecewise smooth circle homeomorphisms with a finite number of break points, Nonlinearity 28 (2015), no. 7, 2441-2461. · Zbl 1359.37092
[14] M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5-233. · Zbl 0448.58019
[15] Y. Katznelson and D. Ornstein, The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergodic Theory Dynam. Systems 9 (1989), no. 4, 643-680. · Zbl 0819.58033
[16] Y. Katznelson and D. Ornstein. The absolute continuity of the conjugation of certain diffeomorphisms of the circle, Ergodic Theory Dynam. Systems 9 (1989), no. 4, 681-690.
[17] K. Khanin, S. Kocić, and E. Mazzeo, \(C^1\)-rigidity of circle maps with breaks for almost all rotation numbers, Ann. Sci. Éc. Norm. Supér. (4) 50 (2017), no. 5, 1163-1203. · Zbl 1388.37050
[18] A. Khinchin, Continued Fractions, University of Chicago Press, Chicago, 1964. · Zbl 0117.28601
[19] H. Poincaré, Mémoire sur les courbes définies par une équation différentielle (I), J. Math. Pures Appl. (9) 7 (1881), 375-422. · JFM 13.0591.01
[20] Y. Sinaĭ and K. Khanin, Smoothness of conjugacies of diffeomorphisms of the circle with rotations (in Russian), Uspekhi Mat. Nauk 44 (1989), no. 1, 57-82; English translation in Russian Math. Surveys 44 (1989), no. 1, 69-99. · Zbl 0701.58053
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