On conjugacies between piecewise-smooth circle maps

Habibulla Akhadkulov, Akhtam Dzhalilov, Mohd. Salmi Md. Noorani

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1-15
Number of pages15
JournalNonlinear Analysis, Theory, Methods and Applications
Volume99
DOIs
Publication statusPublished - Apr 2014

Fingerprint

Circle Map
Jump
Singular Functions
Irrational number
Rotation number
Lebesgue Measure
Absolutely Continuous
Circle
Interval

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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

In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 99, 04.2014, p. 1-15.

Research output: Contribution to journalArticle

@article{e53caed5ff024edb8e6f65ee875da4fb,
title = "On conjugacies between piecewise-smooth circle maps",
abstract = "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 fi{\^a}̂̂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.",
keywords = "Break point, Circle homeomorphism, Conjugating map, Invariant measure, Rotation number, Singular function",
author = "Habibulla Akhadkulov and Akhtam Dzhalilov and {Md. Noorani}, {Mohd. Salmi}",
year = "2014",
month = "4",
doi = "10.1016/j.na.2013.12.013",
language = "English",
volume = "99",
pages = "1--15",
journal = "Nonlinear Analysis, Theory, Methods and Applications",
issn = "0362-546X",
publisher = "Elsevier Limited",

}

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 -