American Institute of Mathematical Sciences

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

Renormalizations of circle hoemomorphisms with a single break point

Abdumajid Begmatov 1, , Akhtam Dzhalilov 2, and  Dieter Mayer 3,

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.
Keywords: break point, renormalizations, Circle homeomorphism, rotation number, fractional linear maps..
Mathematics Subject Classification: Primary: 37E10, 37E20; Secondary: 37C15, 37C4.
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

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences