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 - A, 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, 25 (1961), 21. 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. doi: 10.1017/etds.2012.159. Google Scholar [3] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory,, Springer-Verlag, (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. 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. doi: 10.1007/BF02463336. Google Scholar [6] A. A. Dzhalilov and I. Liousse, Circle homeomorphismswith two break points,, Nonlinearity, 19 (2006), 1951. 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. 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. 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. Google Scholar [10] E. de Faria and W. de Melo, Rigidity of critical circle mappings,, I. J. Eur. Math. Soc. (JEMS), 1 (1999), 339. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1088/0951-7715/1/4/004. Google Scholar [19] G. Swiatek, Rational rotation number for maps of the circle,, Commun. Math. Phys., (1988), 109. doi: 10.1007/BF01218263. Google Scholar [20] M. Stein, Groups of piecewise linear homeomorphisms,, Trans. A.M.S., 332 (1992), 477. 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. 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. Google Scholar

show all references

References:
 [1] V. I. Arnol'd, Small denominators: I. Mappings from the circle onto itself,, Izv. Akad. Nauk SSSR, 25 (1961), 21. 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. doi: 10.1017/etds.2012.159. Google Scholar [3] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory,, Springer-Verlag, (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. 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. doi: 10.1007/BF02463336. Google Scholar [6] A. A. Dzhalilov and I. Liousse, Circle homeomorphismswith two break points,, Nonlinearity, 19 (2006), 1951. 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. 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. 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. Google Scholar [10] E. de Faria and W. de Melo, Rigidity of critical circle mappings,, I. J. Eur. Math. Soc. (JEMS), 1 (1999), 339. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1088/0951-7715/1/4/004. Google Scholar [19] G. Swiatek, Rational rotation number for maps of the circle,, Commun. Math. Phys., (1988), 109. doi: 10.1007/BF01218263. Google Scholar [20] M. Stein, Groups of piecewise linear homeomorphisms,, Trans. A.M.S., 332 (1992), 477. 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. 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. Google Scholar
 [1] Malo Jézéquel. Parameter regularity of dynamical determinants of expanding maps of the circle and an application to linear response. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 927-958. doi: 10.3934/dcds.2019039 [2] Akhtam Dzhalilov, Isabelle Liousse, Dieter Mayer. Singular measures of piecewise smooth circle homeomorphisms with two break points. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 381-403. doi: 10.3934/dcds.2009.24.381 [3] Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 2018, 12: 175-191. doi: 10.3934/jmd.2018007 [4] Qiudong Wang. The diffusion time of the connecting orbit around rotation number zero for the monotone twist maps. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 255-274. doi: 10.3934/dcds.2000.6.255 [5] Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617 [6] Abdelhamid Adouani, Habib Marzougui. Computation of rotation numbers for a class of PL-circle homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3399-3419. doi: 10.3934/dcds.2012.32.3399 [7] Rafael De La Llave, Michael Shub, Carles Simó. Entropy estimates for a family of expanding maps of the circle. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 597-608. doi: 10.3934/dcdsb.2008.10.597 [8] Liviana Palmisano. Unbounded regime for circle maps with a flat interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2099-2122. doi: 10.3934/dcds.2015.35.2099 [9] Alena Erchenko. Flexibility of Lyapunov exponents for expanding circle maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2325-2342. doi: 10.3934/dcds.2019098 [10] Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343 [11] Wenxian Shen. Global attractor and rotation number of a class of nonlinear noisy oscillators. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 597-611. doi: 10.3934/dcds.2007.18.597 [12] Joachim Escher, Boris Kolev. Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle. Journal of Geometric Mechanics, 2014, 6 (3) : 335-372. doi: 10.3934/jgm.2014.6.335 [13] Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017 [14] Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979 [15] Jan J. Dijkstra and Jan van Mill. Homeomorphism groups of manifolds and Erdos space. Electronic Research Announcements, 2004, 10: 29-38. [16] M . Bartušek, John R. Graef. Some limit-point/limit-circle results for third order differential equations. Conference Publications, 2001, 2001 (Special) : 31-38. doi: 10.3934/proc.2001.2001.31 [17] Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621 [18] Grzegorz Graff, Piotr Nowak-Przygodzki. Fixed point indices of iterations of $C^1$ maps in $R^3$. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 843-856. doi: 10.3934/dcds.2006.16.843 [19] Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793 [20] Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039

2018 Impact Factor: 1.143