September  2017, 37(9): 4587-4609. doi: 10.3934/dcds.2017197

Notes on a theorem of Katznelson and Ornstein

1. 

School of Quantitative Sciences, University Utara Malaysia, CAS 06010, UUM Sintok, Kedah Darul Aman, Malaysia

2. 

Turin Polytechnic University, Kichik Halka yuli 17, Tashkent 100095, Uzbekistan

3. 

Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada

* Corresponding author: Habibulla Akhadkulov

Received  June 2016 Revised  April 2017 Published  June 2017

Let $\log f'$ be an absolutely continuous and $f"/f'∈ L_{p}(S^{1}, d\ell)$ for some $p>1, $ where $\ell$ is Lebesgue measure. We show that there exists a subset of irrational numbers of unbounded type, such that for any element $\widehat{ρ}$ of this subset, the linear rotation $R_{\widehat{ρ}}$ and the shift $f_{t}=f+t\mod 1, $ $t∈ [0, 1)$ with rotation number $\widehat{ρ}, $ are absolutely continuously conjugate. We also introduce a certain Zygmund-type condition depending on a parameter $γ$, and prove that in the case $γ>\frac{1}{2}$ there exists a subset of irrational numbers of unbounded type, such that every circle diffeomorphism satisfying the corresponding Zygmund condition is absolutely continuously conjugate to the linear rotation provided its rotation number belongs to the above set. Moreover, in the case of $γ> 1, $ we show that the conjugacy is $C^{1}$-smooth.

Citation: Habibulla Akhadkulov, Akhtam Dzhalilov, Konstantin Khanin. Notes on a theorem of Katznelson and Ornstein. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4587-4609. doi: 10.3934/dcds.2017197
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]

W. de Melo and S. van Strien, A structure theorem in one-dimensional dynamics, Ann. Math., 129 (1989), 519-546. doi: 10.2307/1971516. Google Scholar

[3]

R. Durrett, Probability Theory and Examples Second edition. Duxbury Press, Belmont, CA, 1996. doi: 10.1017/CBO9780511779398. Google Scholar

[4]

J. Hawkins and K. Schmidt, On $C^{2}$-diffeomorphisms of the circle which are of type $Ⅲ_{1}$, Invent. Math., 66 (1982), 511-518. doi: 10.1007/BF01389227. Google Scholar

[5]

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

[6]

J. Hu and D. Sullivan, Topological conjugacy of circle diffeomorphisms, Ergodic Theory Dynam. Systems, 17 (1997), 173-186. doi: 10.1017/S0143385797061002. Google Scholar

[7]

F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426. doi: 10.1002/cpa.3160140317. Google Scholar

[8]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187. Google Scholar

[9]

Y. Katznelson and D. Ornstein, The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergod. Theor. Dyn. Syst., 9 (1989), 643-680. doi: 10.1017/S0143385700005277. Google Scholar

[10]

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

[11]

K. M. Khanin and Ya. G. Sinai, A new proof of M. Herman's theorem, Commun. Math. Phys., 112 (1987), 89-101. doi: 10.1007/BF01217681. Google Scholar

[12]

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: 10.1007/s00222-009-0200-z. Google Scholar

[13]

K. M. Khanin and A. Yu. Teplinsky, Herman's theory revisited, Invent. Math., 178 (2009), 333-344. doi: 10.1007/s00222-009-0200-z. Google Scholar

[14]

J. Moser, A rapid convergent iteration method and non-linear differential equations. Ⅱ, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 499-535. Google Scholar

[15]

E. M. Stein, Singular Integrals and Differentaibility Properties of Functions, Princeton University Press, Princeton, N. J. , 1970. Google Scholar

[16]

D. Sullivan, Bounds, quadratic differentials and renormalization conjectures, American Mathematical Society Centennial Publications, (Providence, RI, 1988), Amer. Math. Soc., Providence, RI, 2 (1992), 417-466. Google Scholar

[17]

G. Świątek, Rational rotation number for maps of the circle, Commun. Math. Phys., 119 (1988), 109-128. doi: 10.1007/BF01218263. Google Scholar

[18]

A. Teplinsky, On cross-ratio distortion and Schwartz derivative, Nonlinearity, 21 (2008), 2777-2783. doi: 10.1088/0951-7715/21/12/003. Google Scholar

[19]

M. Weiss and A. Zygmund, A note on smooth functions, Indag. Math., 21 (1959), 52-58. Google Scholar

[20]

J. C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. École Norm. Sup.(4), 17 (1984), 333-359. doi: 10.24033/asens.1475. Google Scholar

[21]

A. Zygmund, Trigonometric Series Third edition, Cambridge University Press, 2002. 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]

W. de Melo and S. van Strien, A structure theorem in one-dimensional dynamics, Ann. Math., 129 (1989), 519-546. doi: 10.2307/1971516. Google Scholar

[3]

R. Durrett, Probability Theory and Examples Second edition. Duxbury Press, Belmont, CA, 1996. doi: 10.1017/CBO9780511779398. Google Scholar

[4]

J. Hawkins and K. Schmidt, On $C^{2}$-diffeomorphisms of the circle which are of type $Ⅲ_{1}$, Invent. Math., 66 (1982), 511-518. doi: 10.1007/BF01389227. Google Scholar

[5]

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

[6]

J. Hu and D. Sullivan, Topological conjugacy of circle diffeomorphisms, Ergodic Theory Dynam. Systems, 17 (1997), 173-186. doi: 10.1017/S0143385797061002. Google Scholar

[7]

F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math., 14 (1961), 415-426. doi: 10.1002/cpa.3160140317. Google Scholar

[8]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187. Google Scholar

[9]

Y. Katznelson and D. Ornstein, The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergod. Theor. Dyn. Syst., 9 (1989), 643-680. doi: 10.1017/S0143385700005277. Google Scholar

[10]

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

[11]

K. M. Khanin and Ya. G. Sinai, A new proof of M. Herman's theorem, Commun. Math. Phys., 112 (1987), 89-101. doi: 10.1007/BF01217681. Google Scholar

[12]

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: 10.1007/s00222-009-0200-z. Google Scholar

[13]

K. M. Khanin and A. Yu. Teplinsky, Herman's theory revisited, Invent. Math., 178 (2009), 333-344. doi: 10.1007/s00222-009-0200-z. Google Scholar

[14]

J. Moser, A rapid convergent iteration method and non-linear differential equations. Ⅱ, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 499-535. Google Scholar

[15]

E. M. Stein, Singular Integrals and Differentaibility Properties of Functions, Princeton University Press, Princeton, N. J. , 1970. Google Scholar

[16]

D. Sullivan, Bounds, quadratic differentials and renormalization conjectures, American Mathematical Society Centennial Publications, (Providence, RI, 1988), Amer. Math. Soc., Providence, RI, 2 (1992), 417-466. Google Scholar

[17]

G. Świątek, Rational rotation number for maps of the circle, Commun. Math. Phys., 119 (1988), 109-128. doi: 10.1007/BF01218263. Google Scholar

[18]

A. Teplinsky, On cross-ratio distortion and Schwartz derivative, Nonlinearity, 21 (2008), 2777-2783. doi: 10.1088/0951-7715/21/12/003. Google Scholar

[19]

M. Weiss and A. Zygmund, A note on smooth functions, Indag. Math., 21 (1959), 52-58. Google Scholar

[20]

J. C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition diophantienne, Ann. Sci. École Norm. Sup.(4), 17 (1984), 333-359. doi: 10.24033/asens.1475. Google Scholar

[21]

A. Zygmund, Trigonometric Series Third edition, Cambridge University Press, 2002. Google Scholar

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