On Critical Circle Homeomorphisms with Infinite Number of Break Points

and Applied Analysis 3 intervals are mutually disjoint except for the endpoints and cover the whole circle. The partition obtained by the above construction will be denoted by P n = P n (ξ, f) and called the nth dynamical partition of S. Obviously, the partition P n+1 is a refinement of the partition P n . Indeed, the short intervals are members of P n+1 and each long interval I i ∈ P n , 0 ≤ i < q n , is partitioned into k n+1 + 1 intervals belonging to P n+1 such that


Introduction
Let  1 = R/Z with clearly defined orientation, metric, Lebesgue measure, and the operation of addition be the unit circle.Let  : R →  1 denote the corresponding projection mapping that "winds" a straight line on the circle.An arbitrary homeomorphism  that preserves the orientation of the unit circle  1 can "be lifted" on the straight line R in the form of the homeomorphism  : R → R with property ( + 1) = () + 1 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  1 is the rotation number: where  is the lift of  with  1 to R.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 log  is of bounded variation, then  is topologically conjugate to the pure rotation   :  →  +  mod 1; 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 () =   ([0, ]) (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.Indepth results have been found; see [2][3][4][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  ∈   ,  ≥ 3, have finite number of critical points   , around which, in some   coordinate system,  has the form  →    , where   ≥ 3 are the odd integers.Such critical points we say are of polynomial type of order   .
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  3 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][10][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 (D)-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.
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.
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  :  1 →  1 , such that the functional equation The proof of Theorem 2 is based on the method of crossratio 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  = () > 1 such that any two adjacent atoms  1 and  2 of a dynamical partition P  (  , ) (see, for the definition, below) are -comparable; that is,

Dynamical Partition
We will assume that the rotation number  = () is irrational throughout this paper.We use the continued fraction For  ∈  1 we define the th fundamental segment   0 =   0 () as the circle arc [,    ()] if  is even and [   (), ] if  is odd.We denote two sets of closed intervals of order  :   "long" 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 P  = P  (, ) and called the th dynamical partition of  1 .Obviously, the partition P +1 is a refinement of the partition P  .Indeed, the short intervals are members of P +1 and each long interval  −1  ∈ P  , 0 ≤  <   , is partitioned into  +1 + 1 intervals belonging to P +1 such that

Cross-Ratio Inequality
Now, we equip  1 with the usual metric | − | = inf{| x − ỹ|, where x, ỹ range over the lifts of ,  ∈  1 , resp.}.Our main analytic tools are ratio and cross-ratio distortions.Let , , ,  ∈  1 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.
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 {(  ,   ,   ,   ),   ≺   ≺   ≺   ≺   , 1 ≤  ≤ } on the circle  1 .Suppose that the system of intervals {(  ,   ), 1 ≤  ≤ } covers each point of the circle at most  times.Then, there exists a constant  1 =  1 (, ) 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 {(  ,   ,   ,   ),   ≺   ≺   ≺   ≺   , 1 ≤  ≤ } on the circle  1 .Suppose that the system of intervals {(  ,   ), 1 ≤  ≤ } covers each point of the set  1 \  (  ) at most  times.Then, there exists a constant  2 =  2 (, ) such that the following inequality holds: Proof.Since  is a -homeomorphism on the set By assumption, the system of intervals F := {[  ,   ], 1 ≤  ≤ } covers each point of the set  1 \   (  ) at most  times.Now, we describe this system of intervals as a union of subsystems of F  ,  ≤  in the following way: first, we take [ 1 ,  1 ] as an element of F 1 and then consider the intersection [ 1 ,  1 ] ⋂[ 2 ,  2 ]; if this intersection is empty, then we count the interval [ 2 ,  2 ] an element of F 1 ; otherwise, we count an element of F 2 .Next, consider F 1 ⋂[ 3 ,  3 ] (here and below, it is considered the intersection with each element of F 1 ); if it is empty, we count [ 3 ,  3 ] an element of F 1 ; otherwise, we check the intersection is empty, we count [ 3 ,  3 ] an element of F 2 ; otherwise, we count [ 3 ,  3 ] an element of F 3 .Continuing this process, we get all F  ,  ≤ .By construction of subsystems F  ,  ≤  of F, the elements of each subsystem do not intersect with each other and where  = Var log .
We will use the following definition and fact to formulate the next lemma.Definition 6.Let  be a  3 function such that  ̸ = 0.The Schwarzian derivative of  is defined by Fact (see [6]).If S < 0 on interval , then for any quadruples , , ,  ∈ .

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].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  1 .Then, there is a constant  6 =  6 ( 1 , ) ≥ 1 such that for every  ∈ In (I), the adjacent atoms are  and   −1 (), and in this case by Lemma 10 these intervals are  6 -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 ).

Proposition 9 .
Let  be a circle homeomorphism with irrational rotation number .Assume that  satisfies the cross-ratio inequality with bound  1 .Then, there is a circle homeomorphism  :  1 →  1 , 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.