# American Institute of Mathematical Sciences

November  2014, 34(11): 4487-4513. doi: 10.3934/dcds.2014.34.4487

## Renormalizations of circle hoemomorphisms with a single break point

 1 Faculty of Mathematics and Mechanics. Samarkand State University, Boulevard Street 15, 140104 Samarkand, Uzbekistan 2 Turin Politechnic University in Tashkent, Kichik halqa yuli 17, 100095 Tashkent, Uzbekistan 3 Institut für Theoretische Physik, TU Clausthal, Leibnizstrasse 10, D-38678 Clausthal-Zellerfeld, Germany

Received  November 2013 Revised  March 2014 Published  May 2014

Let $f$ be an orientation preserving circle homeomorphism with a single break point $x_b,$ i.e. with a jump in the first derivative $f'$ at the point $x_b,$ and with irrational rotation number $\rho=\rho_{f}.$ Suppose that $f$ satisfies the Katznelson and Ornstein smoothness conditions, i.e. $f'$ is absolutely continuous on $[x_b,x_b+1]$ and $f''(x)\in \mathbb{L}^{p}([0,1), d\ell)$ for some $p>1$, where $\ell$ is Lebesque measure. We prove, that the renormalizations of $f$ are approximated by linear-fractional functions in $\mathbb{C}^{1+L^{1}}$, that means, $f$ is approximated in $C^{1}-$ norm and $f''$ is appoximated in $L^{1}-$ norm. Also it is shown, that renormalizations of circle diffeomorphisms with irrational rotation number satisfying the Katznelson and Ornstein smoothness conditions are close to linear functions in $\mathbb{C}^{1+L^{1}}$- norm.
Citation: Abdumajid Begmatov, Akhtam Dzhalilov, Dieter Mayer. Renormalizations of circle hoemomorphisms with a single break point. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4487-4513. doi: 10.3934/dcds.2014.34.4487
##### References:
 [1] V. I. Arnol'd, Small denominators: I. Mappings from the circle onto itself, Izv. Akad. Nauk SSSR, Ser. Mat., 25 (1961), 21-86.  Google Scholar [2] H. Akhadkulov, A. Dzhalilov and D. Mayer, On conjugations of circle homeomorphisms with two break points, Ergod. Theor. and Dynam. Syst., 34 (2014), 725-741. doi: 10.1017/etds.2012.159.  Google Scholar [3] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar [4] A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl., 11 (1932), 333-375. Google Scholar [5] A. A. Dzhalilov and K. M. Khanin, On invariant measure for homeomorphisms of a circle with a point of break, Funct. Anal. Appl., 32 (1998), 153-161. doi: 10.1007/BF02463336.  Google Scholar [6] A. A. Dzhalilov and I. Liousse, Circle homeomorphismswith two break points, Nonlinearity, 19 (2006), 1951-1968. doi: 10.1088/0951-7715/19/8/010.  Google Scholar [7] A. A. Dzhalilov, I. Liousse and D. Mayer, Singular measures of piecewise smooth circle homeomorphisms with two break points, Discrete and continuous dynamical systems, 24 (2009), 381-403. doi: 10.3934/dcds.2009.24.381.  Google Scholar [8] A. A. Dzhalilov, H. Akin and S. Temir, Conjugations between circle maps with a single break point, Journal of Mathematical Analysis and Applications, 366 (2010), 1-10. doi: 10.1016/j.jmaa.2009.12.050.  Google Scholar [9] A. A. Dzhalilov, D. Mayer and U. A. Safarov, Piecwise-smooth circle homeomorphisms with several break points, Izvestiya RAN: Ser. Mat., 76 (2012), 101-120, translation of Izvestiya: Mathematics, 76 (2012), 94-112.  Google Scholar [10] E. de Faria and W. de Melo, Rigidity of critical circle mappings, I. J. Eur. Math. Soc. (JEMS), 1 (1999), 339-392. doi: 10.1007/s100970050011.  Google Scholar [11] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Etudes Sci. Publ. Math., 49 (1979), 5-233. doi: 10.1007/BF02684798.  Google Scholar [12] Y. Katznelson and D. Ornstein, The differentability continuity of the conjugation of certain diffeomorphisms of the circle, Ergod. Theor. Dyn. Syst., 9 (1989), 643-680. doi: 10.1017/S0143385700005277.  Google Scholar [13] Y. Katznelson and D. Ornstein, The absolute continuity of the conjugation of certain diffeomorphisms of the circle, Ergod. Theor. Dyn. Syst., 9 (1989), 681-690 . doi: 10.1017/S0143385700005289.  Google Scholar [14] K. M. Khanin and Ya. G. Sinai, Smoothness of conjugacies of diffeomorphisms of the circle with rotations, Russ. Math. Surv., 44 (1989), 69-99, translation of Usp. Mat. Nauk, 44 (1989), 57-82. doi: 10.1070/RM1989v044n01ABEH002008.  Google Scholar [15] K. M. Khanin and E. B. Vul, Circle homeomorphisms with weak discontinuities, Advances in Soviet Mathematics, 3 (1991), 57-98.  Google Scholar [16] K. M. Khanin and D. Khmelev, Renormalizations and Rigidity Theory for Circle Homeomorphisms with Singularities of the Break Type, Commun. Math. Phys., 235 (2003), 69-124. doi: 10.1007/s00220-003-0809-5.  Google Scholar [17] I. Liousse, PL Homeomorphisms of the circle which are piecewise $C^1$ conjugate to irrational rotations, Bull. Braz. Math. Soc., 35 (2004), 269-280. doi: 10.1007/s00574-004-0014-y.  Google Scholar [18] J. Stark, Smooth conjugacy and renormalization for diffeomorfisms of the circle, Nonlinearity, 1 (1988), 541-575. doi: 10.1088/0951-7715/1/4/004.  Google Scholar [19] G. Swiatek, Rational rotation number for maps of the circle, Commun. Math. Phys., textbf{119} (1988), 109-128. doi: 10.1007/BF01218263.  Google Scholar [20] M. Stein, Groups of piecewise linear homeomorphisms, Trans. A.M.S., 332 (1992), 477-514. doi: 10.1090/S0002-9947-1992-1094555-4.  Google Scholar [21] A. Yu. Teplinskii and K. M. Khanin, Robust rigidity for circle diffeomorphisms with singularities, Inventiones mathematicae, 169 (2007), 193-218. doi: 10.1007/s00222-007-0047-0.  Google Scholar [22] J. C. Yoccoz, Il n'y a pas de contre-exemple de Denjoy analytique, C. R. Acad. Sci. Paris, 298 (1984), 141-144.  Google Scholar

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##### References:
 [1] V. I. Arnol'd, Small denominators: I. Mappings from the circle onto itself, Izv. Akad. Nauk SSSR, Ser. Mat., 25 (1961), 21-86.  Google Scholar [2] H. Akhadkulov, A. Dzhalilov and D. Mayer, On conjugations of circle homeomorphisms with two break points, Ergod. Theor. and Dynam. Syst., 34 (2014), 725-741. doi: 10.1017/etds.2012.159.  Google Scholar [3] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar [4] A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl., 11 (1932), 333-375. Google Scholar [5] A. A. Dzhalilov and K. M. Khanin, On invariant measure for homeomorphisms of a circle with a point of break, Funct. Anal. Appl., 32 (1998), 153-161. doi: 10.1007/BF02463336.  Google Scholar [6] A. A. Dzhalilov and I. Liousse, Circle homeomorphismswith two break points, Nonlinearity, 19 (2006), 1951-1968. doi: 10.1088/0951-7715/19/8/010.  Google Scholar [7] A. A. Dzhalilov, I. Liousse and D. Mayer, Singular measures of piecewise smooth circle homeomorphisms with two break points, Discrete and continuous dynamical systems, 24 (2009), 381-403. doi: 10.3934/dcds.2009.24.381.  Google Scholar [8] A. A. Dzhalilov, H. Akin and S. Temir, Conjugations between circle maps with a single break point, Journal of Mathematical Analysis and Applications, 366 (2010), 1-10. doi: 10.1016/j.jmaa.2009.12.050.  Google Scholar [9] A. A. Dzhalilov, D. Mayer and U. A. Safarov, Piecwise-smooth circle homeomorphisms with several break points, Izvestiya RAN: Ser. Mat., 76 (2012), 101-120, translation of Izvestiya: Mathematics, 76 (2012), 94-112.  Google Scholar [10] E. de Faria and W. de Melo, Rigidity of critical circle mappings, I. J. Eur. Math. Soc. (JEMS), 1 (1999), 339-392. doi: 10.1007/s100970050011.  Google Scholar [11] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Etudes Sci. Publ. Math., 49 (1979), 5-233. doi: 10.1007/BF02684798.  Google Scholar [12] Y. Katznelson and D. Ornstein, The differentability continuity of the conjugation of certain diffeomorphisms of the circle, Ergod. Theor. Dyn. Syst., 9 (1989), 643-680. doi: 10.1017/S0143385700005277.  Google Scholar [13] Y. Katznelson and D. Ornstein, The absolute continuity of the conjugation of certain diffeomorphisms of the circle, Ergod. Theor. Dyn. Syst., 9 (1989), 681-690 . doi: 10.1017/S0143385700005289.  Google Scholar [14] K. M. Khanin and Ya. G. Sinai, Smoothness of conjugacies of diffeomorphisms of the circle with rotations, Russ. Math. Surv., 44 (1989), 69-99, translation of Usp. Mat. Nauk, 44 (1989), 57-82. doi: 10.1070/RM1989v044n01ABEH002008.  Google Scholar [15] K. M. Khanin and E. B. Vul, Circle homeomorphisms with weak discontinuities, Advances in Soviet Mathematics, 3 (1991), 57-98.  Google Scholar [16] K. M. Khanin and D. Khmelev, Renormalizations and Rigidity Theory for Circle Homeomorphisms with Singularities of the Break Type, Commun. Math. Phys., 235 (2003), 69-124. doi: 10.1007/s00220-003-0809-5.  Google Scholar [17] I. Liousse, PL Homeomorphisms of the circle which are piecewise $C^1$ conjugate to irrational rotations, Bull. Braz. Math. Soc., 35 (2004), 269-280. doi: 10.1007/s00574-004-0014-y.  Google Scholar [18] J. Stark, Smooth conjugacy and renormalization for diffeomorfisms of the circle, Nonlinearity, 1 (1988), 541-575. doi: 10.1088/0951-7715/1/4/004.  Google Scholar [19] G. Swiatek, Rational rotation number for maps of the circle, Commun. Math. Phys., textbf{119} (1988), 109-128. doi: 10.1007/BF01218263.  Google Scholar [20] M. Stein, Groups of piecewise linear homeomorphisms, Trans. A.M.S., 332 (1992), 477-514. doi: 10.1090/S0002-9947-1992-1094555-4.  Google Scholar [21] A. Yu. Teplinskii and K. M. Khanin, Robust rigidity for circle diffeomorphisms with singularities, Inventiones mathematicae, 169 (2007), 193-218. doi: 10.1007/s00222-007-0047-0.  Google Scholar [22] J. C. Yoccoz, Il n'y a pas de contre-exemple de Denjoy analytique, C. R. Acad. Sci. Paris, 298 (1984), 141-144.  Google Scholar
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