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Crossratio inequality with infinite type of singularities
 Sokhobiddin Akhatkulov^{1}Email author and
 Mohd Salmi Md Noorani^{1}
https://doi.org/10.1186/s1366001506316
© Akhatkulov and Md Noorani; licensee Springer. 2015
Received: 1 April 2014
Accepted: 5 January 2015
Published: 21 March 2015
Abstract
Let f be a preserving orientation circle homeomorphism with infinite number of break points, i.e., the points at which the derivative of f has jumps, and finite number of singular points, i.e., the points \(x_{i}\), \(i=1,2,\ldots,n\), such that \(f'(x_{i})=\infty\), \(i=1,2,\ldots,n\). Then the crossratio inequality with respect to f holds.
Keywords
MSC
1 Introduction
Early results about the existence of conjugacy to the linear rotation \(f_{\rho} : x\rightarrow x +\rho \operatorname{mod} 1\) are the following theorems.
Poincaré’s theorem
[2]
Let f be a circle homeomorphism with irrational rotation number ρ. Then f is semiconjugate to the rotation \(f_{\rho}\).
Denjoy’s theorem
[3]
A circle diffeomorphism f with irrational rotation number ρ and \(\log Df\) of bounded variation is topologically conjugate to the linear rotation \(f_{\rho}\).
Denjoy’s result has been extended for different classes of circle homeomorphisms. Indepth results have been found, see [4–8]. Two wellknown classes of these are the following.

there exist some constants \(0<\gamma<\zeta<\infty\) such that \(\gamma< f'(x)<\zeta\) for all \(x\in S^{1}\setminus BP(f)\), \(\gamma< f'_{+}(b)<\zeta\) and \(\gamma< f'_{}(b)<\zeta\) for all \(b\in BP(f)\), where \(BP(f)\) denotes the set of the break points of f;

\(\log f'\) has bounded variation. In this situation \(\log f'\), \(\log f'_{}\), \(\log f'_{+}\) and \(\log(f^{1})'\), \(\log(f^{1}_{})'\), \(\log(f^{1}_{+})'\) have the same total variation denoted by \(V = \operatorname{Var} \log f'\).
Critical circle homeomorphisms. These are orientation preserving \(C^{1}\) circle maps f such that for each \(x_{0}\in S^{1}\) there exist \(\alpha\geq1\), a neighborhood \(U(x_{0})\) of \(x_{0}\), and a homeomorphism \(\phi:U(x_{0})\rightarrow{R}\) such that \(\phi(x_{0})=0\), and if \(\alpha>1\) then \(f(x)=\pm\phi(x)^{\alpha}+f(x_{0})\), \(\forall x\in U(x_{0})\) if \(\alpha=1\) then \(f(x)=\phi(x)^{\alpha}+f(x_{0})\), \(\forall x\in U(x_{0})\).
For the class of Phomeomorphisms, the classical result of Denjoy can be easily extended, the existence of the conjugating map for this class was proved by Herman in [5]. The existence of the conjugating map for the class of critical real analytic circle maps was proved by Yoccoz [8] and extended by Świątek [6, 7]. In these works the existence of conjugation was shown by estimating crossratio distortions (i.e., proving crossratio inequality). Note that the crossratio distortions were used in dynamical systems for the first time by Yoccoz [8]. The crossratio distortions are the most powerful tools to study the existence and smoothness of a conjugating map for the critical circle homeomorphisms.
Our aim in this work is to prove the crossratio inequality for a new class of circle homeomorphisms, which will be defined below with the aid of the above two classes and to show the existence of a conjugating map for this new class. We shall talk about the crossratio distortions in the next section.
 (i)
f is a preserving orientation circle homeomorphism on \(S^{1}\).
 (ii)
There are points \(x_{i}\in S^{1}\) and \(\alpha_{i}\in(0, 1)\), \(i=1,2,\ldots,n\), such that \(f(x)=(xx_{i})^{\alpha_{i}}+f(x_{i})\) for some \(\epsilon _{i}\)neighborhoods of each \(x_{i}\).
 (iii)
f is a Phomeomorphism on \(S^{1}\setminus \bigcup^{n}_{i=1}U_{\epsilon_{i}}(x_{i})\).
2 Crossratio inequality
The main results of this paper are the following theorems.
Theorem 2.1
Inequality (3) is called crossratio inequality with respect to f. As an application of this theorem, we provide the following second result of this paper.
Theorem 2.2
In the proof of the second main theorem, we use the properties of a dynamical partition, and therefore we introduce this concept.
3 Dynamical partition of the circle
4 Proof of the main results
In this section, first we prove few lemmas, and then using these lemmas we give the proof of the main results. Note that the first main theorem is proved in a similar way as that of [10].
Lemma 4.1
Proof
Lemma 4.2
Proof
 (1)
\((a, b)\subset E\);
 (2)
\((a, b)\subset V\);
 (3)
neither (1) nor (2).

the interval \((a, b)\) contains the point \(x_{i^{*}}\),

the interval \((a, b)\) does not contain \(x_{i^{*}}\).
Now we consider case (3). It is clear that \(ba>\frac{\epsilon _{i^{*}}}{2} \), and since f is strictly increasing we have that \(\frac {f(b)f(a)}{ba}\leq\mbox{const}\). Besides that \(\frac{1}{f'(a)f'(b)}\) cannot be sufficiently large. Hence, from this and from (6), (8), (10) the proof of the lemma follows. □
Proof of Theorem 2.1
 (a)
\(A_{1}=\{i\in A: (a_{i},b_{i})\subset V^{*}\}\),
 (b)
\(A_{2}=\{i\in A: (a_{i},b_{i})\subset E^{*}\}\),
 (c)
\(A_{3}=A\setminus(A_{1}\cup A_{2})\).
Before we prove the next lemma, we introduce a concept of wandering interval.
Definition 4.3

the intervals \(I, f(I),\ldots,f^{n}(I),\ldots \) are pairwise disjoint;

the ωlimit set of I is not equal to a single periodic orbit.
Lemma 4.4
Suppose that a circle homeomorphism f with irrational rotation number satisfies conditions (i)(iii) without break points. Then f has no wandering interval.
Proof
Declarations
Acknowledgements
The authors would like to acknowledge the financial support received from the Government of Malaysia under the research Grants FRGS/1/2014/ST06/UKM/01/1, DIP2014034.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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