A remark on Denjoy's inequality for PL circle homeomorphisms with two break points

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:458

Article

Dzhalilov, Akhtam¹  Jalilov, Alisher²  Mayer, Dieter³ 

[1] Turin Polytech Univ, Kichik Halka Yuli 17, Tashkent 100095, Uzbekistan

[2] Ajou Univ, Dept Math, Suwon, South Korea

[3] Tech Univ Clausthal, Inst Theoret Phys, D-38678 Clausthal Zellerfeld, Germany

关键词: Piecewise-linear circle;    homeomorphism;    Break point;    Rotation number;    Invariant measure;   

DOI  :  10.1016/j.jmaa.2017.09.003

来源: Elsevier

【 摘 要 】

It is well known that for a P-homeomorphism f of the circle S-1=R/Z with irrational rotation number (pf) the Denjoy's inequality |log Df(qn)| <= V holds, where V is the total variation of log Df and q(n), n >= 1, are the first return times of f. Let h be a piecewise-linear (PL) circle homeomorphism with two break points a(0), c(0), irrational rotation number p(h) and total jump ratio sigma h = 1. Denote by B-n(h) the partition determined by the break points of h(qn) and by mu(h) the unique h-invariant probability measure. It is shown that the derivative Dh(qn) is constant on every element of B-n(h) and takes either two or three values. Furthermore we prove, that log Dh(qn) can be expressed in terms of mu(h)-measures of some intervals of the partition B-n(h) multiplied by the logarithm of the jump ratio sigma(h)(a(0)) of h at the break point a(0). (C) 2017 Elsevier Inc. All rights reserved.

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