### Abstract

Let f be an orientation-preserving circle diffeomorphism with an irrational rotation number and with a break point that is, its derivative has a jump discontinuity at this point. Suppose that satisfies a certain Zygmund condition dependent on a parameter We prove that the renormalizations of f are approximated by Möbius transformations in C ^{1}-Topology if and in C ^{2}-Topology if Moreover, it is shown that, in case of the coefficients of Möbius transformations get asymptotically linearly dependent. Further, consider two circle diffeomorphisms with a break point, with the same size of the break and satisfying Zygmund condition with We prove that, under a certain technical condition on rotation numbers, the renormalizations of these diffeomorphisms approach each other in C ^{2}-Topology.

Original language | English |
---|---|

Pages (from-to) | 2687-2717 |

Number of pages | 31 |

Journal | Nonlinearity |

Volume | 30 |

Issue number | 7 |

DOIs | |

Publication status | Published - 23 May 2017 |

### Fingerprint

### Keywords

- 37C15
- 37F25
- break point
- circle diffeomorphism
- convergence
- Möbius transformations
- renormalization
- rotation number Mathematics Subject Classification numbers: 37E10

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics

### Cite this

*Nonlinearity*,

*30*(7), 2687-2717. https://doi.org/10.1088/1361-6544/aa6d94

**Renormalization of circle diffeomorphisms with a break-Type singularity.** / Akhadkulov, Habibulla; Md. Noorani, Mohd. Salmi; Akhatkulov, Sokhobiddin.

Research output: Contribution to journal › Article

*Nonlinearity*, vol. 30, no. 7, pp. 2687-2717. https://doi.org/10.1088/1361-6544/aa6d94

}

TY - JOUR

T1 - Renormalization of circle diffeomorphisms with a break-Type singularity

AU - Akhadkulov, Habibulla

AU - Md. Noorani, Mohd. Salmi

AU - Akhatkulov, Sokhobiddin

PY - 2017/5/23

Y1 - 2017/5/23

N2 - Let f be an orientation-preserving circle diffeomorphism with an irrational rotation number and with a break point that is, its derivative has a jump discontinuity at this point. Suppose that satisfies a certain Zygmund condition dependent on a parameter We prove that the renormalizations of f are approximated by Möbius transformations in C 1-Topology if and in C 2-Topology if Moreover, it is shown that, in case of the coefficients of Möbius transformations get asymptotically linearly dependent. Further, consider two circle diffeomorphisms with a break point, with the same size of the break and satisfying Zygmund condition with We prove that, under a certain technical condition on rotation numbers, the renormalizations of these diffeomorphisms approach each other in C 2-Topology.

AB - Let f be an orientation-preserving circle diffeomorphism with an irrational rotation number and with a break point that is, its derivative has a jump discontinuity at this point. Suppose that satisfies a certain Zygmund condition dependent on a parameter We prove that the renormalizations of f are approximated by Möbius transformations in C 1-Topology if and in C 2-Topology if Moreover, it is shown that, in case of the coefficients of Möbius transformations get asymptotically linearly dependent. Further, consider two circle diffeomorphisms with a break point, with the same size of the break and satisfying Zygmund condition with We prove that, under a certain technical condition on rotation numbers, the renormalizations of these diffeomorphisms approach each other in C 2-Topology.

KW - 37C15

KW - 37F25

KW - break point

KW - circle diffeomorphism

KW - convergence

KW - Möbius transformations

KW - renormalization

KW - rotation number Mathematics Subject Classification numbers: 37E10

UR - http://www.scopus.com/inward/record.url?scp=85021198020&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021198020&partnerID=8YFLogxK

U2 - 10.1088/1361-6544/aa6d94

DO - 10.1088/1361-6544/aa6d94

M3 - Article

AN - SCOPUS:85021198020

VL - 30

SP - 2687

EP - 2717

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 7

ER -