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 and Continuous Dynamical Systems, 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. 

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[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. 

[17]

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

[18]

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

[19]

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

[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.

[21]

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

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]

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

[3]

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

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[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.

[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.

[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.

[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.

[14]

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

[15]

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

[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. 

[17]

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

[18]

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

[19]

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

[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.

[21]

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

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