Renormalizations of circle hoemomorphisms with a single break point

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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

Abstract
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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 and 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.
[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.
[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.
[4]	A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl., 11 (1932), 333-375.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	K. M. Khanin and E. B. Vul, Circle homeomorphisms with weak discontinuities, Advances in Soviet Mathematics, 3 (1991), 57-98.
[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.
[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.
[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.
[19]	G. Swiatek, Rational rotation number for maps of the circle, Commun. Math. Phys., textbf{119} (1988), 109-128. doi: 10.1007/BF01218263.
[20]	M. Stein, Groups of piecewise linear homeomorphisms, Trans. A.M.S., 332 (1992), 477-514. doi: 10.1090/S0002-9947-1992-1094555-4.
[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.
[22]	J. C. Yoccoz, Il n'y a pas de contre-exemple de Denjoy analytique, C. R. Acad. Sci. Paris, 298 (1984), 141-144.

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.
[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.
[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.
[4]	A. Denjoy, Sur les courbes définies par les équations différentielles à la surface du tore, J. Math. Pures Appl., 11 (1932), 333-375.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	K. M. Khanin and E. B. Vul, Circle homeomorphisms with weak discontinuities, Advances in Soviet Mathematics, 3 (1991), 57-98.
[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.
[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.
[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.
[19]	G. Swiatek, Rational rotation number for maps of the circle, Commun. Math. Phys., textbf{119} (1988), 109-128. doi: 10.1007/BF01218263.
[20]	M. Stein, Groups of piecewise linear homeomorphisms, Trans. A.M.S., 332 (1992), 477-514. doi: 10.1090/S0002-9947-1992-1094555-4.
[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.
[22]	J. C. Yoccoz, Il n'y a pas de contre-exemple de Denjoy analytique, C. R. Acad. Sci. Paris, 298 (1984), 141-144.

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