Notes on a theorem of Katznelson and Ornstein

Habibulla Akhadkulov; Akhtam Dzhalilov; Konstantin Khanin

doi:10.3934/dcds.2017197

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2017, Volume 37, Issue 9: 4587-4609. Doi: 10.3934/dcds.2017197

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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
* Corresponding author: Habibulla Akhadkulov

Received: May 31, 2016

Revised: March 31, 2017

Published: October 2017

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Abstract

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.

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Mathematics Subject Classification: Primary: 37E10, 37C15; Secondary: 37C40.

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References

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