# On the invariant measure of a piecewise-smooth circle homeomorphism of Zygmund class

Volume 10, Issue 1, pp 48--59
Publication Date: January 26, 2017 Submission Date: February 10, 2016
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### Authors

Sokhobiddin Akhatkulov - School of Mathematical Sciences, Faculty of Science and Technology, University Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia. Mohd. Salmi Md. Noorani - School of Mathematical Sciences, Faculty of Science and Technology, University Kebangsaan Malaysia, 43600 UKM Bangi, Selangor DE, Malaysia. Habibulla Akhadkulov - School of Quantitative Sciences, University Utara Malaysia, CAS 06010, UUM Sintok, Kedah DA, Malaysia.

### Abstract

We prove that the invariant probability measure of an orientation preserving circle homeomorphism f with several break points (at which the derivative $\acute{f}$ has jumps) is singular with respect to Lebesgue measure, if $\acute{f}$ satisfies certain condition and the product of jump ratios at break points is non-trivial.

### Share and Cite

##### ISRP Style

Sokhobiddin Akhatkulov, Mohd. Salmi Md. Noorani, Habibulla Akhadkulov, On the invariant measure of a piecewise-smooth circle homeomorphism of Zygmund class, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 1, 48--59

##### AMA Style

Akhatkulov Sokhobiddin, Noorani Mohd. Salmi Md., Akhadkulov Habibulla, On the invariant measure of a piecewise-smooth circle homeomorphism of Zygmund class. J. Nonlinear Sci. Appl. (2017); 10(1):48--59

##### Chicago/Turabian Style

Akhatkulov, Sokhobiddin, Noorani, Mohd. Salmi Md., Akhadkulov, Habibulla. "On the invariant measure of a piecewise-smooth circle homeomorphism of Zygmund class." Journal of Nonlinear Sciences and Applications, 10, no. 1 (2017): 48--59

### Keywords

• Break point
• circle homeomorphism
• invariant measure
• rotation number.

•  37E10
•  37C15
•  26D99

### References

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