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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, 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

show all references

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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