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

Volume 10, Issue 1, pp 48--59 http://dx.doi.org/10.22436/jnsa.010.01.05

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.

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Keywords

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

MSC

•  37E10
•  37C15
•  26D99

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