### Abstract

Let ^{fi}, i=1,2, be piecewise ^{C1} circle homeomorphisms with two break points, logD^{fi}, i=1,2, are absolutely continuous on each continuity interval of D^{fi} and DlogD ^{fi}â̂̂^{Lp} for some p>1. Suppose, the jump ratios of ^{f1} and ^{f2} at their break points do not coincide but ^{f1},^{f2} have the same total jumps (i.e. the product of jump ratios) and identical irrational rotation number of bounded type. Then the map h conjugating ^{f1} and ^{f2} is a singular function, that is, it is continuous on ^{S1}, but Dh(x)=0 almost everywhere with respect to Lebesgue measure.

Original language | English |
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Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 99 |

DOIs | |

Publication status | Published - Apr 2014 |

### Fingerprint

### Keywords

- Break point
- Circle homeomorphism
- Conjugating map
- Invariant measure
- Rotation number
- Singular function

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Nonlinear Analysis, Theory, Methods and Applications*,

*99*, 1-15. https://doi.org/10.1016/j.na.2013.12.013

**On conjugacies between piecewise-smooth circle maps.** / Akhadkulov, Habibulla; Dzhalilov, Akhtam; Md. Noorani, Mohd. Salmi.

Research output: Contribution to journal › Article

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 99, pp. 1-15. https://doi.org/10.1016/j.na.2013.12.013

}

TY - JOUR

T1 - On conjugacies between piecewise-smooth circle maps

AU - Akhadkulov, Habibulla

AU - Dzhalilov, Akhtam

AU - Md. Noorani, Mohd. Salmi

PY - 2014/4

Y1 - 2014/4

N2 - Let fi, i=1,2, be piecewise C1 circle homeomorphisms with two break points, logDfi, i=1,2, are absolutely continuous on each continuity interval of Dfi and DlogD fiâ̂̂Lp for some p>1. Suppose, the jump ratios of f1 and f2 at their break points do not coincide but f1,f2 have the same total jumps (i.e. the product of jump ratios) and identical irrational rotation number of bounded type. Then the map h conjugating f1 and f2 is a singular function, that is, it is continuous on S1, but Dh(x)=0 almost everywhere with respect to Lebesgue measure.

AB - Let fi, i=1,2, be piecewise C1 circle homeomorphisms with two break points, logDfi, i=1,2, are absolutely continuous on each continuity interval of Dfi and DlogD fiâ̂̂Lp for some p>1. Suppose, the jump ratios of f1 and f2 at their break points do not coincide but f1,f2 have the same total jumps (i.e. the product of jump ratios) and identical irrational rotation number of bounded type. Then the map h conjugating f1 and f2 is a singular function, that is, it is continuous on S1, but Dh(x)=0 almost everywhere with respect to Lebesgue measure.

KW - Break point

KW - Circle homeomorphism

KW - Conjugating map

KW - Invariant measure

KW - Rotation number

KW - Singular function

UR - http://www.scopus.com/inward/record.url?scp=84892618208&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892618208&partnerID=8YFLogxK

U2 - 10.1016/j.na.2013.12.013

DO - 10.1016/j.na.2013.12.013

M3 - Article

VL - 99

SP - 1

EP - 15

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

ER -