#### Abstract

We prove that a critical circle homeomorphism with infinite number of break points without periodic orbits is conjugated to the linear rotation by a quasisymmetric map if and only if its rotation number is of bounded type. And we also prove that any two adjacent atoms of dynamical partition of a unit circle are comparable.

#### 1. Introduction

Let with clearly defined orientation, metric, Lebesgue measure, and the operation of addition be the unit circle. Let denote the corresponding projection mapping that “winds” a straight line on the circle. An arbitrary homeomorphism that preserves the orientation of the unit circle can “be lifted” on the straight line in the form of the homeomorphism with property that is connected with by relation . This homeomorphism is called the* lift* of the homeomorphism and is defined up to an integer term. The most important arithmetic characteristic of the homeomorphism of the unit circle is the* rotation number*:
where is the lift of with to . Here and below, for a given map , denotes its th iterate. The rotation number is rational if and only if has periodic points. Denjoy proved that if is a circle diffeomorphism with irrational rotation number and is of bounded variation, then is* topologically conjugate* to the pure rotation ; that is, there exists an essentially unique homeomorphism of the circle with (see [1]). Since the conjugating map and the unique -invariant measure are related by (see [1]), regularity properties of the conjugating map imply corresponding properties of the density of the absolutely continuous invariant measure . The problem of relating the smoothness of to that of has been studied extensively. In-depth results have been found; see [2–5].

Other classes of circle homeomorphisms are known to satisfy the conclusion of Denjoy’s theorem (see [6], Chapters I and IV, and [2], Chapter VI) and the study of the regularity of their conjugation maps arises naturally. Two of these classes are commonly referred to as the following.

##### 1.1. Critical Circle Homeomorphisms

The orientation preserving circle homeomorphisms , such that , , have finite number of critical points , around which, in some coordinate system, has the form , where are the odd integers. Such critical points we say are of* polynomial type* of order .

##### 1.2. -Homeomorphisms

That is, orientation preserving circle homeomorphisms are differentiable except in many countable points called* break points* admitting left and right derivatives (denoted by and , resp.) such that(i)there exist some constants such that for all , and for all , where denotes the set of the break points of ;(ii) has bounded variation. In this situation, , , and , , have the same total variation denoted by .The ratio is called* jump ratio* of in .

The existence of the conjugating map for the class critical circle homeomorphisms was proved by Yoccoz in [7] and for the class -homeomorphisms the existence of conjugating map was proved by Herman in [2].

The singularity of the conjugating map for critical circle homeomorphisms was shown by Graczyk and Świątek in [8]. They proved that if is smooth circle homeomorphism with finitely many critical points of polynomial type and an irrational rotation number of bounded type, then the conjugating map is a singular function. For the -homeomorphisms, the situation is different; that is, in this case, the conjugating map can be singular or absolutely continuous. Indeed, in the works [9–11], it was shown that the conjugating map is singular. The deeper result in this area was obtained by Dzhalilov et al. [12]. They proved that if is piecewise-smooth -homeomorphism with finite number of break points and the product of jump ratios at these break points is nontrivial, then the conjugating map is a singular function. But in the works [9, 13], it was shown that if is piecewise-smooth -homeomorphism with finite number of break points having the -property (see for the definition [13]) and the product of the jump ratios on each orbit is equal to 1, then the conjugating map is an absolutely continuous function. Now, we discuss the symmetric property of a given function.

*Definition 1. *A homeomorphism is called -quasisymmetric if and only if, for every real and ,

The criteria of quasisymmetry of the conjugating map of the critical circle homeomorphisms were obtained by Świątek in [14]. Due to [14], if the circle homeomorphism with an irrational rotation number is analytic and has finitely many critical points, then the conjugating map is quasisymmetric if and only if the rotation number is of bounded type.

The quasisymmetric property of the conjugating map of -homeomorphisms is also different from the case of critical circle homeomorphisms. More precisely, if the rotation number of -homeomorphism is irrational of bounded type, then conjugating map is quasisymmetric, but there is a -homeomorphism with irrational rotation number of unbounded type such that the conjugating map is quasisymmetric. In this paper, we introduce a new class of circle homeomorphisms with the aid of the above two classes. Our aim in this work is to show the existence of conjugating map for this new class and study the quasisymmetric property of this conjugating map. Now, we introduce our class.

Let be a circle homeomorphism.(a) has one critical point polynomial type of order ; that is, there is a point such that and .(b) for some -neighborhood of the point .(c) is a -homeomorphism on the set .

Note that all the above results were obtained for the class -homeomorphisms with finite number of break points, but in our work it is not necessary for the number of break points to be finite. Now, we state our main results.

Theorem 2. *Suppose that a circle homeomorphism satisfies the conditions (a)–(c) and the rotation number is irrational. Then, there exists circle homeomorphism , such that the functional equation
**
is satisfied. Moreover, is quasisymmetric if and only if is of bounded type.*

The proof of Theorem 2 is based on the method of cross-ratio distortion estimates. Note that the cross-ratio estimates were used in dynamical systems for the first time by Yoccoz [7] and later by Świątek [14]. In fact, the proof of Theorem 2 follows closely that of Świątek [14]. Our second result below is also proved by using cross-ratio estimates.

Theorem 3. *Suppose that a circle homeomorphism satisfies the conditions (a)–(c) and the rotation number is irrational. Then, there exists universal constant such that any two adjacent atoms and of a dynamical partition (see, for the definition, below) are -comparable; that is,
*

#### 2. Dynamical Partition

We will assume that the rotation number is irrational throughout this paper. We use the continued fraction representation of the rotation number, which is understood as a limit of the sequence of convergents . The sequence of positive integers with , which are called* partial quotients*, are uniquely determined for each . The coprimes and satisfy the recurrence relations and for , where we set, for convenience, , and , . For we define the th* fundamental segment* as the circle arc if is even and if is odd. We denote two sets of closed intervals of order * ‘‘long’’* intervals: and * ‘‘short’’* intervals: . The long and short intervals are mutually disjoint except for the endpoints and cover the whole circle. The partition obtained by the above construction will be denoted by and called the th* dynamical partition* of . Obviously, the partition is a refinement of the partition . Indeed, the short intervals are members of and each long interval , , is partitioned into intervals belonging to such that

#### 3. Cross-Ratio Inequality

Now, we equip with the usual metric {, where , range over the lifts of , resp.}. Our main analytic tools are ratio and cross-ratio distortions. Let be four points of the circle which preserve orientation; that is, on the circle; we define a ratio of three points of , , by and we define a cross-ratio of four points of , , , by

The distortions and of the ratio and the cross-ratio by the function are defined by respectively. It is clear that Notice that the ratio and the cross-ratio distortions have the following properties.(i)The ratio and the cross-ratio are invariant by translation: (ii)The ratio and the cross-ratio distortions are multiplicative with respect to composition for two functions and on ; we have

Now, let us formulate the following theorem, which plays an important role during studying the properties of dynamical partitions.

Theorem 4. *Suppose that a homeomorphism satisfies the conditions (a)–(c). Consider a system of quadruples {, , } on the circle . Suppose that the system of intervals {, } covers each point of the circle at most times. Then, there exists a constant such that the following inequality holds:
*

Inequality (12) is called the* cross-ratio inequality*. This inequality was proved for the critical circle homeomorphisms by Świątek [15]. Now, we prove the following three important lemmas to be used in the proof of the main results.

Lemma 5. *Suppose that a homeomorphism satisfies the conditions (a)–(c). Consider a system of quadruples {, , } on the circle . Suppose that the system of intervals {, } covers each point of the set at most times. Then, there exists a constant such that the following inequality holds:
*

*Proof. *Since is a -homeomorphism on the set for any quadruples satisfying assumption of lemma, we have
Using this inequality, we get

By assumption, the system of intervals covers each point of the set at most times. Now, we describe this system of intervals as a union of subsystems of , in the following way: first, we take as an element of and then consider the intersection ; if this intersection is empty, then we count the interval an element of ; otherwise, we count an element of . Next, consider (here and below, it is considered the intersection with each element of ); if it is empty, we count an element of ; otherwise, we check the intersection . Again, if is empty, we count an element of ; otherwise, we count an element of . Continuing this process, we get all , . By construction of subsystems , of , the elements of each subsystem do not intersect with each other and , . Therefore
where .

We will use the following definition and fact to formulate the next lemma.

*Definition 6. *Let be a function such that . The Schwarzian derivative of is defined by

*Fact *(see [6]). If on interval , then
for any quadruples .

Lemma 7. *Suppose that a circle homeomorphism satisfies the conditions (a)–(c). Then, there is , such that satisfies the following conditions:*(i)* in ;*(ii)*there exist constants , such that for all .*

*Proof. *Consider the following function:
By using Taylor’s formula, we have
where , . Since is a nonzero, continuous function and if we take , , then the second assertion of this lemma follows. In order to prove the first assertion of this lemma we calculate the Schwarzian derivative of the function as follows:
Since is nonzero and continuous on , there is a sufficient small such that the ratios and are uniformly close to 1 for any . It ensures that , .

Lemma 8. *Suppose that a circle homeomorphism satisfies the conditions (a)–(c). Let and . Then, there is a constant such that
*

*Proof. *By the property of the cross-ratio distortion, we know that . Now, to prove this lemma, it is enough to show that is bounded above. The second ratio distortion is analogously estimated. Denote . Consider the following three cases:

(1) , (2) , and (3) neither (1) nor (2).

If , then

where and . It is clear that . Let . Using Lemma 7, we get
where , , , and . It is easy to check that for any . Without any loss of generality, one can assume that . Then,
Here, there can be four cases: , , , and . It is easy to check that for all cases the ratio distortion is bounded above. Now, consider case (3). It is clear that . From this it is implied that . Hence, from , (23), and (24), we get the proof of Lemma 8.

*Proof of Theorem 4. *Let . We divide the set into three disjoint subsets , , and in the following way: (i);(ii);(iii).

We estimate each product , separately. Let . Then, by Lemma 8, we have

Denote ; that is, . By Lemma 7, we have

Let . Then, using Lemma 5, we get

Let . Note that the number of intervals does not exceed . This and Lemma 8 imply that
Hence, from (26) to (29), the assertion of Theorem 4 follows.

#### 4. Proof of Main Theorems

*Proof of Theorem 2. *The proof of Theorem 2 follows from assertion of Theorem 4 together with the following proposition which was proved by Świątek in [14].

Proposition 9. *Let be a circle homeomorphism with irrational rotation number . Assume that satisfies the cross-ratio inequality with bound . Then, there is a circle homeomorphism , which conjugates to the linear rotation . Furthermore, is quasisymmetric if is of bounded type. If has at least one critical point of polynomial type, then is quasisymmetric if and only if is of bounded type.*

Before we prove Theorem 3, we formulate two lemmas and use them to prove this theorem. Note that these lemmas also were obtained by Świątek [14].

Lemma 10. *Let be a circle homeomorphism with irrational rotation number. Assume that satisfies the cross-ratio inequality with bound . Then, there is a constant such that for every *

Lemma 11. *Let be a circle homeomorphism with irrational rotation number. Assume that satisfies the cross-ratio inequality with bound . Then, there is a constant such that
*

*Proof of Theorem 3. *It is clear that any two adjacent atoms of can be of the following three types:
In (I), the adjacent atoms are and , and in this case by Lemma 10 these intervals are -comparable. Consider case (II). Using the property of dynamical partition, it is easy to see that (it suffices to prove it for the linear rotation , which follows from arithmetical properties of ).
for any . This together with Lemma 10 implies that
for any . If we take instead of , we get

Now, we show an opposite inequality to (35). Let us take , , and . Applying to these intervals, we have , , and , respectively. Due to Lemma 10, the intervals and are -comparable. Besides that, by Lemma 11, we have

From (34) and (36), it follows that the intervals and are comparable. Since the intervals and are -comparable and , the intervals and are -comparable. From this it is implied that the intervals and are comparable. Hence, , , and are comparable.

Since the intervals , , and are adjacent on and the system of intervals , , covers each point of the circle at most three times, by the cross-ratio inequality (12), we have

Since the intervals , , are -comparable and the intervals , are -comparable, and by inequality (37) for constant , we have

Hence, case follows from (35) and (38). Case . By using Lemma 10 twice, we get the comparability of the following intervals: , and , . By case , the intervals and are comparable and it gives the proof of case . Theorem 3 is proved.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

The authors would like to acknowledge the financial support received from the Government of Malaysia under the Research Grant FRGS/2/2013/ST06/UKM/02/2.