On Absolute Continuity of Conjugations between Circle Maps with Break Points

and Applied Analysis 3 i there exist constants 0 < c1 < c2 < ∞ with c1 < DT x < c2 for all x ∈ S1 \ BP T , c1 < DT− xb < c2 and c1 < DT xb < c2 for all xb ∈ BP T , with BP T the set of break points of T on S1; ii DT has bounded variation. The ratio σT c : DT− c / DT c is called the jump of T in c or the T -jump. General B-homeomorphisms with one break point was first studied by Khanin and Vul in 3 . Among other results it was proved by these authors that their renormalizations approximate fractional linear transformations. Let T be an orientation preserving C1-diffeomorphism of the circle. If the rotation number ρ is irrational and DT is of bounded variation then, by a well-known theorem of Denjoy, T is conjugate to the rigid rotation Rρ see 1 . The conjugationmeans that there exists an essentially unique homeomorphism h of the circle such that T h−1 ◦ Rρ ◦ h. In this context, a natural question to ask is under what condition the conjugacy is smooth? Several authors, for example 4–6 have shown that if T is C2 , α > 0 and ρ satisfies certain diophantine condition then the conjugacy will be at least C1. The classical result of Denjoy can be easily extended to the case of B-homeomorphisms. Next we consider the problem of the regularity of the conjugating map between two class B-homeomorphisms with one break point and coinciding irrational rotation numbers. The case of one break point with the same jump ratios, so called rigidity problem, was studied in detail by Teplinskii and Khanin in 7 . Let ρ k1, k2, . . . , kn, . . . be the continued fraction expansion of the irrational rotation number ρ and define Mo { ρ : ∃C > 0, ∀n ∈ N, k2n−1 ≤ C } , Me { ρ : ∃C > 0, ∀n ∈ N, k2n ≤ C } . 1.6 The main result of 7 is as follows. Theorem 1.1. Let Ti ∈ C2 α S1 \ {bi} , i 1, 2, α > 0 be B-homeomorphisms with one break point that have the same jump ratio σ and the same irrational rotation number ρ ∈ 0, 1 . In addition, let one of the following restrictions be true: either σ > 1 and ρ ∈Me or σ < 1 and ρ ∈Mo. Then the map h conjugating the homeomorphisms T1 and T2 is a C1-diffeomorphism. In the case of different jump ratios, the following theorem was proved in 8 by Dzhalilov et al. Theorem 1.2. Let Ti ∈ C2 α S1 \ {bi} , i 1, 2, α > 0 be B-homeomorphisms with one break point that have different jump ratio and the same irrational rotation number ρ ∈ 0, 1 . Then the map h conjugating the homeomorphisms T1 and T2 is a singular function, that is, is continuous on S1 and Dh x 0 a.e. with respect to Lebesgue measure. Let T1 and T2 beB-homeomorphismswith identical irrational rotation number ρ. Now, we consider dynamical partitions Pn ξ, T1 Pn T1 and Pn h ξ , T2 Pn T2 appropriate to the homeomorphisms T1 and T2. Denote by Δ̂ intervals of partition of Pn T2 . Since the function h is a conjugation function between T1 and T2, so we have h Δ Δ̂ for any Δ ∈ Pn T1 . Denote by |A| the Lebesgue measure of the corresponding set of A ⊂ S1. Our purpose in this paper is to give some criteria for the absolute continuity of the conjugation map h. Our first main result is the following. 4 Abstract and Applied Analysis Theorem 1.3. Assume the rotation number ρ is irrational of bounded type. Suppose that there exist a sequence τn such that ∑∞ n 1 τ 2 n <∞ with ∣∣∣∣∣∣ |Δ1| |Δ2| − ∣∣Δ̂1 ∣∣ ∣∣Δ̂2 ∣∣ ∣∣∣∣∣∣ ≤ τn 1.7 for each pair of adjacent intervals Δ1,Δ2 ∈ Pn T1 for all n > 1. Then the conjugation map h is absolutely continuous function. In the proof of Theorem 1.3, we will use the consideration of theory of martingales. The idea of using theory of martingales was established in 9 by Katznelson and Ornstein. Our second main result is the following. Theorem 1.4. Let T1 and T2 be B-homeomorphisms with identical irrational rotation number ρ. If the conjugation map h is a absolutely continuous function, then for all δ > 0, the sequence of Lebesgue measure of the set |{x : |logDT 2 h x − logDT qn 1 x | ≥ δ}| tends to 0 when n goes to ∞. 2. The Denjoy Theory and Ergodicity of B-Homeomorphisms The assertions listed below, which are valid for any orientation-preserving homeomorphism T ∈ B with irrational rotation number ρ, constitute classical Denjoy theory. Their elementary proofs can be found in 10, 11 . a Generalized Denjoy estimate; let ξ0 ∈ S1 be a continuity point of DTn , then the following inequality holds: e−v ≤ DTn ξ0 ≤ e, where v VarS1 logDT . b Exponential refinement; there exists a universal constant C1 C1 T such that |Δn m 0 | ≤ C1λ|Δ0 |, where λ 1 e−v −1/2. c Bounded geometry; let rotation number ρ is bounded type that is the coefficients in continued fraction expansion of ρ are bounded. Then there exist universal constants C2 C2 T , C3 C3 T such that 0 < C2 < 1 and C3 ≤ 1 with i each pair of adjacent intervals of Pn T are C2-comparable that is their ratio of lengths belongs to C2, C−1 2 ; ii an interval Δ 1 of Pn 1 T is C3-comparable to the interval Δ of Pn T that contains it: C3|Δ| ≤ |Δn 1|. d Generalized Finzi estimate; suppose ξ ∈ S1, η ∈ Δn−1 ξ and ξ, η are continuity points of DTn . Then for any 0 ≤ k < qn, the following inequality holds: | logDT ξ − logDT η | ≤ v. Let S1,G, μ be a measure space and F : S1 → S1 be a measurable map. Definition 2.1. The set A ∈ G is said to be invariant with respect to the measurable F, if A F−1A. Definition 2.2. A measurable map F : S1 → S1 is said to be ergodic with respect measure μ if the measure μ A of any invariant set A equals 0 or 1. Let ξ0 ∈ S1, denote by Vn Δ ξ0 ∪Δn−1 ξ0 . Abstract and Applied Analysis 5 Lemma 2.3. Let T be a B-homeomorphism with irrational rotation number ρ. Suppose ξ ∈ Vn and ξ be a continuity point of DTn . Then for any 0 ≤ k < qn, the following inequality holds: e−v ∣∣Tk Vn ∣∣ |Vn| ≤ DT k ξ ≤ e ∣∣Tk Vn ∣∣ |Vn| . 2.1and Applied Analysis 5 Lemma 2.3. Let T be a B-homeomorphism with irrational rotation number ρ. Suppose ξ ∈ Vn and ξ be a continuity point of DTn . Then for any 0 ≤ k < qn, the following inequality holds: e−v ∣∣Tk Vn ∣∣ |Vn| ≤ DT k ξ ≤ e ∣∣Tk Vn ∣∣ |Vn| . 2.1 Proof. Let the system of intervals I {I : I ⊂ Vn, and the map DTn is continuous on I} be continuity intervals of DTn . Let ξ ∈ Δn−1 ξ0 . Then, by the mean value theorem, for any 0 ≤ k < qn, we have ∣∣Tk(Δn−1 ξ0 )∣∣ DTk ξ |Δn−1 ξ0 | DT z1 |I1| DT z2 |I2| · · · DT zd |Id| DTk ξ |Δn−1 ξ0 | , 2.2 where zi ∈ Ii ⊂ Δn−1 ξ0 and Ii ∈ I, 1 ≤ i ≤ d. If ξ ∈ Δ ξ0 then we have ∣∣Tk Δ ξ0 ∣∣ DTk ξ |Δn ξ0 | DT ( y1 |J1| DT ( y2 |J2| · · · DT ( yt |Jt| DTk ξ |Δn ξ0 | , 2.3 where yi ∈ Ji ⊂ Δ ξ0 and Ji ∈ I, 1 ≤ i ≤ t. Apply generalized Finzi estimate to the righthand side of relations 2.2 and 2.3 , we get e−v ≤ ∣∣Tk(Δn−1 ξ0 )∣∣ DTk ξ |Δn−1 ξ0 | ≤ e, e−v ≤ ∣∣Tk Δ ξ0 ∣∣ DTk ξ |Δn ξ0 | ≤ e. 2.4


Introduction and Statement of Results
Let S 1 R/Z with clearly defined orientation, metric, Lebesgue measure, and the operation of addition be the unit circle. Let π : R → S 1 denote the corresponding projection mapping that "winds" a straight line on the circle. An arbitrary homeomorphism T that preserves the orientation of the unit circle S 1 can "be lifted" on the straight line R in the form of the homeomorphism L T : R → R with property L T x 1 L T x 1 that is connected with T by relation π • L T T • π. This homeomorphism L T is called the lift of the homeomorphism T and is defined up to an integer term. The most important arithmetic characteristic of the homeomorphism T of the unit circle S 1 is the rotation number where L T is the lift of T with S 1 to R. Here and below, for a given map F, F i denotes its ith iteration. Poincaré proved that the above limit exists, does not depend on the initial point 2 Abstract and Applied Analysis x ∈ R of the lifted trajectory, and, up to additional of an integer, does not depend on the lift L T see 1 . The rotation number ρ ρ T is irrational if and only if the homeomorphism T has no periodic point. Hereafter, we will always assume that ρ is irrational and use its decomposition in an infinite continued fraction see 2 ρ 1 k 1 1 k 2 1 · · · k n 1 · · · : k 1 , k 2 , . . . , k n , . . . .

1.2
The value of a "countable-floor" fraction is the limit of the sequence of rational convergents p n /q n k 1 , k 2 , . . . , k n . The positive integers k n , n ≥ 1, called incomplete multiples, are defined uniquely for irrational ρ. The mutually prime positive integers p n and q n satisfy the recurrent relations p n k n p n−1 p n−2 and q n k n q n−1 q n−2 for n ≥ 1, where it is convenient to define p −1 0, q −1 1 and p 0 1, q 0 k 1 . Given a circle homeomorphism T with irrational rotation number ρ, one may consider a marked trajectory i.e., the trajectory of a marked point ξ i T i ξ 0 ∈ S 1 , where i ≥ 0, and pick out of it the sequence of the dynamical convergents ξ q n , n ≥ 0, indexed by the denominators of consecutive rational convergents to ρ. We will also conventionally use ξ q −1 ξ 0 − 1. The well-understood arithmetical properties of rational convergents and the combinatorial equivalence between T and rigid rotation R ρ : ξ → ξ ρ mod 1 imply that the dynamical convergents approach the marked point, alternating their order in the following way: ξ q −1 < ξ q 1 < ξ q 3 < · · · < ξ q 2m 1 < · · · < ξ 0 < · · · < ξ q 2m < · · · < ξ q 2 < ξ q 0 . 1.3 We define the nth fundamental interval Δ n ξ 0 as the circle arc ξ 0 , T q n ξ 0 for even n and as T q n ξ 0 , ξ 0 for odd n. For the marked trajectory, we use the notation Δ n 0 Δ n ξ 0 , Δ n i Δ n ξ i T i Δ n 0 . It is well known that the set P n ξ 0 , T P n T of intervals with mutually disjoint interiors defined as determines a partition of the circle for any n. The partition P n T is called the nth dynamical partition of the point ξ 0 . Obviously the partition P n 1 T is a refinement of the partition P n T : indeed the intervals of order n are members of P n 1 T and each interval Δ n−1 i ∈ P n T 0 ≤ i < q n , is partitioned into k n 1 1 intervals belonging to P n 1 T such that Class B-homeomorphisms. These are orientation-preserving circle homeomorphisms T differentiable except in finite number break points at which left and right derivatives, denoted, respectively by DT − and DT , exist, and such that i there exist constants 0 < c 1 < c 2 < ∞ with c 1 < DT x < c 2 for all x ∈ S 1 \ BP T , c 1 < DT − x b < c 2 and c 1 < DT x b < c 2 for all x b ∈ BP T , with BP T the set of break points of T on S 1 ; ii DT has bounded variation.
The ratio σ T c : DT − c / DT c is called the jump of T in c or the T -jump. General B-homeomorphisms with one break point was first studied by Khanin and Vul in 3 . Among other results it was proved by these authors that their renormalizations approximate fractional linear transformations. Let T be an orientation preserving C 1 -diffeomorphism of the circle. If the rotation number ρ is irrational and DT is of bounded variation then, by a well-known theorem of Denjoy, T is conjugate to the rigid rotation R ρ see 1 . The conjugation means that there exists an essentially unique homeomorphism h of the circle such that T h −1 • R ρ • h. In this context, a natural question to ask is under what condition the conjugacy is smooth? Several authors, for example 4-6 have shown that if T is C 2 α , α > 0 and ρ satisfies certain diophantine condition then the conjugacy will be at least C 1 .
The classical result of Denjoy can be easily extended to the case of B-homeomorphisms. Next we consider the problem of the regularity of the conjugating map between two class B-homeomorphisms with one break point and coinciding irrational rotation numbers. The case of one break point with the same jump ratios, so called rigidity problem, was studied in detail by Teplinskii and Khanin in 7 . Let ρ k 1 , k 2 , . . . , k n , . . . be the continued fraction expansion of the irrational rotation number ρ and define The main result of 7 is as follows.
, α > 0 be B-homeomorphisms with one break point that have the same jump ratio σ and the same irrational rotation number ρ ∈ 0, 1 . In addition, let one of the following restrictions be true: either σ > 1 and ρ ∈ M e or σ < 1 and ρ ∈ M o . Then the map h conjugating the homeomorphisms T 1 and T 2 is a C 1 -diffeomorphism.
In the case of different jump ratios, the following theorem was proved in 8 by Dzhalilov et al.
homeomorphisms with one break point that have different jump ratio and the same irrational rotation number ρ ∈ 0, 1 . Then the map h conjugating the homeomorphisms T 1 and T 2 is a singular function, that is, is continuous on S 1 and Dh x 0 a.e. with respect to Lebesgue measure.
Let T 1 and T 2 be B-homeomorphisms with identical irrational rotation number ρ. Now, we consider dynamical partitions P n ξ, T 1 P n T 1 and P n h ξ , T 2 P n T 2 appropriate to the homeomorphisms T 1 and T 2 . Denote by Δ n intervals of partition of P n T 2 . Since the function h is a conjugation function between T 1 and T 2 , so we have h Δ n Δ n for any Δ n ∈ P n T 1 . Denote by |A| the Lebesgue measure of the corresponding set of A ⊂ S 1 . Our purpose in this paper is to give some criteria for the absolute continuity of the conjugation map h. Our first main result is the following.

4
Abstract and Applied Analysis Theorem 1.3. Assume the rotation number ρ is irrational of bounded type. Suppose that there exist a sequence τ n such that ∞ n 1 τ 2 n < ∞ with for each pair of adjacent intervals Δ 1 , Δ 2 ∈ P n T 1 for all n > 1. Then the conjugation map h is absolutely continuous function.
In the proof of Theorem 1.3, we will use the consideration of theory of martingales. The idea of using theory of martingales was established in 9 by Katznelson and Ornstein. Our second main result is the following.

The Denjoy Theory and Ergodicity of B-Homeomorphisms
The assertions listed below, which are valid for any orientation-preserving homeomorphism T ∈ B with irrational rotation number ρ, constitute classical Denjoy theory. Their elementary proofs can be found in 10, 11 . a Generalized Denjoy estimate; let ξ 0 ∈ S 1 be a continuity point of DT q n , then the following inequality holds: c Bounded geometry; let rotation number ρ is bounded type that is the coefficients in continued fraction expansion of ρ are bounded. Then there exist universal constants C 2 C 2 T , C 3 C 3 T such that 0 < C 2 < 1 and C 3 ≤ 1 with i each pair of adjacent intervals of P n T are C 2 -comparable that is their ratio of lengths belongs to C 2 , C −1 2 ; ii an interval Δ n 1 of P n 1 T is C 3 -comparable to the interval Δ n of P n T that contains it: d Generalized Finzi estimate; suppose ξ ∈ S 1 , η ∈ Δ n−1 ξ and ξ, η are continuity points of DT q n . Then for any 0 ≤ k < q n , the following inequality holds: Let S 1 , G, μ be a measure space and F : S 1 → S 1 be a measurable map. Let ξ 0 ∈ S 1 , denote by V n Δ n ξ 0 ∪ Δ n−1 ξ 0 .

Lemma 2.3.
Let T be a B-homeomorphism with irrational rotation number ρ. Suppose ξ ∈ V n and ξ be a continuity point of DT q n . Then for any 0 ≤ k < q n , the following inequality holds: Proof. Let the system of intervals I {I : I ⊂ V n , and the map DT q n is continuous on I} be continuity intervals of DT q n . Let ξ ∈ Δ n−1 ξ 0 . Then, by the mean value theorem, for any 0 ≤ k < q n , we have where y i ∈ J i ⊂ Δ n ξ 0 and J i ∈ I, 1 ≤ i ≤ t. Apply generalized Finzi estimate to the righthand side of relations 2.2 and 2.3 , we get Finally, we get Abstract and Applied Analysis point of the circle belongs to at most two intervals of this cover. Hence, the set A c is invariant with respect to T , using the above lemma, we get Since was arbitrary, |A c | 0. The theorem is proved. Proof. Consider two B-homeomorphisms T 1 and T 2 of the circle S 1 with identical irrational rotation number ρ. Let ϕ 1 and ϕ 2 be maps conjugating T 1 and T 2 with the rigid rotation T ρ , that is, It is easy to check that the map h ϕ −1 2 • ϕ 1 conjugates T 1 and T 2 , that is We know that conjugation function h is strictly increasing function on S 1 . Then Dh exists almost everywhere on S 1 . Denote by A {x : x ∈ S 1 , Dh x > 0}. It is clear that the set A is mod 0 invariant with respect to T 1 . Since the class B-homeomorphism is ergodic with respect to the Lebesgue measure. Hence, the Lebesgue measure of set A is either null or full. If Lebesque measure of A is null then h is a singular function, if it is full then h is an absolutely continuous function.
Remark 2.6. Let T 1 and T 2 be B-homeomorphisms with identical irrational rotation number. Then conjugation map h −1 between T 2 and T 1 is either absolutely continuous or singular.

Martingales and Martingale Convergence Theorem
Our objective in this section is to develop the fundamentals of the theory of martingales and prepare for the main results and applications that will be presented in the subsequent sections.
The sequence of algebras generated by dynamical partitions, which is also denoted by P m by abuse of notation is a filtration in B, where B is a Borel σ-algebra on S 1 .

Definition 3.3.
Let R m be a sequence of random variables on a measurable space X, F and F m a filtration in F. We say that R m is adapted to F m if, for each positive integer m, R m is F m -measurable.
Denote by E R | F conditional expectation of random variables R with respect to partition F. Definition 3.4. Let R m be a sequence of random variables on a probability space X, F, P and F m a filtration in F. The sequence R m is said to be a martingale with respect to F m if, for every positive integer m,

Lemma 3.5 see 12 .
Let R m be a sequence of random variables on a probability space X, F, P . If sup m E |R m | p < ∞ for some p > 1 and R m is a martingale, then there exists an integrable Suppose f is a homeomorphism not necessary to be B-homeomorphism of the circle S 1 . Using the homeomorphism f and sequence of dynamical partitions P m , we define the sequence of random variables on the circle which is generating a martingales. For any m ≥ 1, we set Lemma 3.6. The sequence R m of random variables is a martingale with respect to P m .
Proof. To prove the martingale, it suffices to check E R m 1 | P m R m , for any m ≥ 1, because the sequence of random variables R m is sequence of step functions, so the sequence of step functions is integrable and adapted to P m . Denote by χ I indicator function of interval I. Using definition of conditional expectation of random variables R m with respect to partition P m , we get

3.4
Now, we calculate each sum of 3.4 separately. Note, that each interval of P m order m is member of P m 1 and each interval Δ m−1 i ∈ P m , 0 ≤ i < q m , is partitioned into k m 1 1 intervals belonging to P m 1 such that Abstract and Applied Analysis Using this, we get

3.7
Finally, summing 3.4 , 3.6 , and 3.7 , we get The following inequality sometimes called "parallelogram inequality" is useful for estimating fractions, and we will use it in the proof of the next statement.
Proof. Consider points A a, b , B c, d , and C a c, b d on the plan xOy. The slope of the ray OC lies between slops of rays OA and OB.

Proof of Main Theorems
Let h be the conjugation homeomorphism between T 1 and T 2 , that is, h • T 1 T 2 • h. Without loss of generality, we assume h 0 0. Consider dynamical partition P m T 1 . Define sequence of random variables R m on the S 1 by this formula Proof. It is clear that where Δ m s ⊂ Δ m−1 . Using Lemma 3.7, we get It is clear that for any 0 ≤ s ≤ k m holds Since, each pair of adjacent intervals of P m T 1 are C 2 -comparable. By the assumption of Theorem 1.3, we get Hence, the rotation number ρ k 1 , k 2 , . . . , k m , . . . is of bounded type, and an easy trick gives us where K sup k m and C 4 KC 2 . A similar lower bound holds true for min R m Δ m s : max R m Δ m s . Therefore, we have for all 0 ≤ s ≤ k m .
Proof of Theorem 1.3. For the proof of Theorem 1.3, we use the above reasonings. By Lemma 3.6, the sequence R m of random variables is a martingale with respect to P m . We want to show that R m converges to Dh in the norm L 1 S 1 , d when m → ∞. By direct calculation, it is easy to see that Θ m x and R m−1 x is orthogonal, that is Using the assertion of Statement 4.1, we get Iterating the last relation, we have R m 2 L 2 ≤ m j 1 1 C 4 τ 2 j . So far as the series ∞ j 1 τ 2 j converges. From this implies that the sequence of random variables R m is bounded in L 2 norm. By Lemma 3.5, the sequence of random variables R m converges to some function R in L 1 norm. We prove that sequence of random variables R m converges to the Dh. Indeed, denote by α m and β m end points of interval Δ m of dynamical partition P m T 1 . By definition of R m , we have Moreover, using last inequality, we obtain

4.13
From this taking the limit when m → ∞, we get h x x 0 R x dx. Since, R ∈ L 1 S 1 , d , then h is absolutely continuous function and Dh x R x almost everywhere on S 1 . Thus, Theorem 1.3 is completely proved. 4.14 Proof. It is a well-known fact that the class C a, b of continuous functions on a, b is dense in · L 1 in L 1 a, b , d see 13 . From this fact it implies that if ψ ∈ L 1 S 1 , d , then for any > 0 there exists a continuous function ψ ∈ C S 1 and φ ∈ L 1 S 1 , d such that ψ ψ φ and φ L 1 ≤ . Using this and Denjoy estimate, we obtain

4.15
Abstract and Applied Analysis

11
As ψ is uniformly continuous on S 1 and by exponential refinement T q m 1 x uniformly tends to x, there exists a positive integer m 0 m 0 such that for all m ≥ m 0 , the ψ • T q m 1 − ψ L 1 ≤ . Therefore, ψ • T q m 1 − ψ L 1 ≤ 2 e v . Since > 0 was arbitrary and sufficiently small. x | ≥ δ} does not converge to 0 when n goes to infinity. Hence, for all positive integer n:

4.18
But |S n δ | does not tend to 0 when n goes to ∞. Hence S 1 |ψ T q n 1 x − ψ x |dx does not tend to 0 when n goes to ∞, this contradicts Statement 4.2 and ends the proof of Theorem 1.4.