Notes on a theorem of Katznelson and Ornstein
Let logf′ be an absolutely continuous and f′′/f′∈Lp(S1,dℓ) for some p>1, where ℓ is Lebesgue measure. We show that there exists a subset of irrational numbers of unbounded type, such that for any element ρˆ of this subset, the linear rotation Rρˆ and the shift ft=f+tmod1, t∈[0,1) with rotation nu...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Published: |
American Institute of Mathematical Sciences
2017
|
| Subjects: |
| _version_ | 1825804854870671360 |
|---|---|
| author | Akhadkulov, Habibulla Dzhalilov, Akhtam Khanin, Konstantin |
| author_facet | Akhadkulov, Habibulla Dzhalilov, Akhtam Khanin, Konstantin |
| author_sort | Akhadkulov, Habibulla |
| collection | UUM |
| description | Let logf′ be an absolutely continuous and f′′/f′∈Lp(S1,dℓ) for some p>1, where ℓ is Lebesgue measure. We show that there exists a subset of irrational numbers of unbounded type, such that for any element ρˆ of this subset, the linear rotation Rρˆ and the shift ft=f+tmod1, t∈[0,1) with rotation number ρˆ, are absolutely continuously conjugate.We also introduce a certain Zygmund-type condition depending on a parameter γ, and prove that in the case γ>12 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 C1-smooth. |
| first_indexed | 2024-07-04T06:22:30Z |
| format | Article |
| id | uum-23043 |
| institution | Universiti Utara Malaysia |
| last_indexed | 2024-07-04T06:22:30Z |
| publishDate | 2017 |
| publisher | American Institute of Mathematical Sciences |
| record_format | eprints |
| spelling | uum-230432018-02-13T01:21:25Z https://repo.uum.edu.my/id/eprint/23043/ Notes on a theorem of Katznelson and Ornstein Akhadkulov, Habibulla Dzhalilov, Akhtam Khanin, Konstantin QA75 Electronic computers. Computer science Let logf′ be an absolutely continuous and f′′/f′∈Lp(S1,dℓ) for some p>1, where ℓ is Lebesgue measure. We show that there exists a subset of irrational numbers of unbounded type, such that for any element ρˆ of this subset, the linear rotation Rρˆ and the shift ft=f+tmod1, t∈[0,1) with rotation number ρˆ, are absolutely continuously conjugate.We also introduce a certain Zygmund-type condition depending on a parameter γ, and prove that in the case γ>12 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 C1-smooth. American Institute of Mathematical Sciences 2017 Article PeerReviewed Akhadkulov, Habibulla and Dzhalilov, Akhtam and Khanin, Konstantin (2017) Notes on a theorem of Katznelson and Ornstein. Discrete and Continuous Dynamical Systems, 37 (9). pp. 4587-4609. ISSN 1078-0947 http://doi.org/10.3934/dcds.2017197 doi:10.3934/dcds.2017197 doi:10.3934/dcds.2017197 |
| spellingShingle | QA75 Electronic computers. Computer science Akhadkulov, Habibulla Dzhalilov, Akhtam Khanin, Konstantin Notes on a theorem of Katznelson and Ornstein |
| title | Notes on a theorem of Katznelson and Ornstein |
| title_full | Notes on a theorem of Katznelson and Ornstein |
| title_fullStr | Notes on a theorem of Katznelson and Ornstein |
| title_full_unstemmed | Notes on a theorem of Katznelson and Ornstein |
| title_short | Notes on a theorem of Katznelson and Ornstein |
| title_sort | notes on a theorem of katznelson and ornstein |
| topic | QA75 Electronic computers. Computer science |
| work_keys_str_mv | AT akhadkulovhabibulla notesonatheoremofkatznelsonandornstein AT dzhalilovakhtam notesonatheoremofkatznelsonandornstein AT khaninkonstantin notesonatheoremofkatznelsonandornstein |