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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 134701, 12 pages
On Absolute Continuity of Conjugations between Circle Maps with Break Points
1School of Mathematical Sciences, Faculty of Science and Technology, University Kebangsaan Malaysia, Bangi, 43600 Selangor, Malaysia
2Faculty of Mathematics and Mechanics, Samarkand State University, Boulevard Street 15, 703004 Samarkand, Uzbekistan
Received 31 August 2012; Accepted 23 October 2012
Academic Editor: Ivanka Stamova
Copyright © 2012 Habibulla Akhadkulov and Mohd Salmi Md Noorani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let and be piecewise smooth circle homeomorphisms with break points and identical irrational rotation numbers. We provide one sufficient and necessary condition for the absolute continuity of conjugation map between and .
1. Introduction and Statement of Results
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 iteration. Poincaré proved that the above limit exists, does not depend on the initial point of the lifted trajectory, and, up to additional of an integer, does not depend on the lift (see ). The rotation number is irrational if and only if the homeomorphism has no periodic point. Hereafter, we will always assume that is irrational and use its decomposition in an infinite continued fraction (see ) The value of a “countable-floor” fraction is the limit of the sequence of rational convergents . The positive integers , called incomplete multiples, are defined uniquely for irrational . The mutually prime positive integers and satisfy the recurrent relations and for , where it is convenient to define and . Given a circle homeomorphism with irrational rotation number , one may consider a marked trajectory (i.e., the trajectory of a marked point) , where , and pick out of it the sequence of the dynamical convergents , indexed by the denominators of consecutive rational convergents to . We will also conventionally use . The well-understood arithmetical properties of rational convergents and the combinatorial equivalence between and rigid rotation mod 1 imply that the dynamical convergents approach the marked point, alternating their order in the following way: We define the th fundamental interval as the circle arc for even and as for odd . For the marked trajectory, we use the notation . It is well known that the set of intervals with mutually disjoint interiors defined as determines a partition of the circle for any . The partition is called the th dynamical partition of the point . Obviously the partition is a refinement of the partition : indeed the intervals of order are members of and each interval , is partitioned into intervals belonging to such that Class -homeomorphisms. These are orientation-preserving circle homeomorphisms differentiable except in finite number break points at which left and right derivatives, denoted, respectively by and , exist, and such that(i)there exist constants with for all , and for all , with the set of break points of on ; (ii) has bounded variation. The ratio is called the jump of in or the -jump. General -homeomorphisms with one break point was first studied by Khanin and Vul in . Among other results it was proved by these authors that their renormalizations approximate fractional linear transformations. Let be an orientation preserving -diffeomorphism of the circle. If the rotation number is irrational and is of bounded variation then, by a well-known theorem of Denjoy, is conjugate to the rigid rotation (see ). The conjugation means that there exists an essentially unique homeomorphism of the circle such that . 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 is and satisfies certain diophantine condition then the conjugacy will be at least .
The classical result of Denjoy can be easily extended to the case of -homeomorphisms. Next we consider the problem of the regularity of the conjugating map between two class -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 . Let be the continued fraction expansion of the irrational rotation number and define The main result of  is as follows.
Theorem 1.1. Let , be -homeomorphisms with one break point that have the same jump ratio and the same irrational rotation number . In addition, let one of the following restrictions be true: either and or and . Then the map conjugating the homeomorphisms and is a -diffeomorphism.
In the case of different jump ratios, the following theorem was proved in  by Dzhalilov et al.
Theorem 1.2. Let , be -homeomorphisms with one break point that have different jump ratio and the same irrational rotation number . Then the map conjugating the homeomorphisms and is a singular function, that is, is continuous on and a.e. with respect to Lebesgue measure.
Let and be -homeomorphisms with identical irrational rotation number . Now, we consider dynamical partitions and appropriate to the homeomorphisms and . Denote by intervals of partition of . Since the function is a conjugation function between and , so we have for any . Denote by the Lebesgue measure of the corresponding set of . Our purpose in this paper is to give some criteria for the absolute continuity of the conjugation map . Our first main result is the following.
Theorem 1.3. Assume the rotation number is irrational of bounded type. Suppose that there exist a sequence such that with for each pair of adjacent intervals for all . Then the conjugation map 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  by Katznelson and Ornstein. Our second main result is the following.
Theorem 1.4. Let and be -homeomorphisms with identical irrational rotation number . If the conjugation map is a absolutely continuous function, then for all , the sequence of Lebesgue measure of the set tends to 0 when goes to .
2. The Denjoy Theory and Ergodicity of -Homeomorphisms
The assertions listed below, which are valid for any orientation-preserving homeomorphism with irrational rotation number , constitute classical Denjoy theory. Their elementary proofs can be found in [10, 11].(a)Generalized Denjoy estimate; let be a continuity point of , then the following inequality holds: , where .(b)Exponential refinement; there exists a universal constant such that , where .(c)Bounded geometry; let rotation number is bounded type that is the coefficients in continued fraction expansion of are bounded. Then there exist universal constants , such that and with(i) each pair of adjacent intervals of are -comparable that is their ratio of lengths belongs to ; (ii)an interval of is -comparable to the interval of that contains it: .(d)Generalized Finzi estimate; suppose and are continuity points of . Then for any , the following inequality holds: .
Let be a measure space and be a measurable map.
Definition 2.1. The set is said to be invariant with respect to the measurable , if .
Definition 2.2. A measurable map is said to be ergodic with respect measure if the measure of any invariant set equals or .
Let , denote by .
Lemma 2.3. Let be a -homeomorphism with irrational rotation number . Suppose and be a continuity point of . Then for any , the following inequality holds:
Proof. Let the system of intervals be continuity intervals of . Let . Then, by the mean value theorem, for any , we have where and . If then we have where and . Apply generalized Finzi estimate to the right-hand side of relations (2.2) and (2.3), we get Finally, we get
Lemma 2.4. Let be a -homeomorphism of the circle with irrational rotation number , then is ergodic with respect to Lebesgue measure.
Proof. Suppose that there exist an invariant set of positive but not full Lebesgue measure . Then by the Lebesgue Density Theorem, has a density point . We fix an arbitrary . By definition of density points, we can find a such that for any interval satisfying the conditions , we have , or, in other words, , where denotes the complement of . Now, we choose such that . We can check that and each point of the circle belongs to at most two intervals of this cover. Hence, the set is invariant with respect to , using the above lemma, we get Since was arbitrary, . The theorem is proved.
Lemma 2.5. Let and are -homeomorphisms with identical irrational rotation number. Then the conjugation map between and is either absolutely continuous or singular function.
Proof. Consider two -homeomorphisms and of the circle with identical irrational rotation number . Let and be maps conjugating and with the rigid rotation , that is, and . It is easy to check that the map conjugates and , that is We know that conjugation function is strictly increasing function on . Then exists almost everywhere on . Denote by . It is clear that the set is mod invariant with respect to . Since the class -homeomorphism is ergodic with respect to the Lebesgue measure. Hence, the Lebesgue measure of set is either null or full. If Lebesque measure of is null then is a singular function, if it is full then is an absolutely continuous function.
Remark 2.6. Let and be -homeomorphisms with identical irrational rotation number. Then conjugation map between and is either absolutely continuous or singular.
3. 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.
Definition 3.1. Let be a measurable space. A sequence of -algebras on is said to be a filtration in , if
Statement 3.2. The sequence of algebras generated by dynamical partitions, which is also denoted by (by abuse of notation) is a filtration in , where is a Borel -algebra on .
Definition 3.3. Let be a sequence of random variables on a measurable space and a filtration in . We say that is adapted to if, for each positive integer is -measurable.
Denote by conditional expectation of random variables with respect to partition .
Definition 3.4. Let be a sequence of random variables on a probability space and a filtration in . The sequence is said to be a martingale with respect to if, for every positive integer ,(i) is integrable; (ii) is adapted to ; (iii).
Lemma 3.5 (see ). Let be a sequence of random variables on a probability space . If for some and is a martingale, then there exists an integrable such that
Suppose is a homeomorphism (not necessary to be -homeomorphism) of the circle . Using the homeomorphism and sequence of dynamical partitions , we define the sequence of random variables on the circle which is generating a martingales. For any , we set
Lemma 3.6. The sequence of random variables is a martingale with respect to .
Proof. To prove the martingale, it suffices to check , for any , because the sequence of random variables is sequence of step functions, so the sequence of step functions is integrable and adapted to . Denote by indicator function of interval . Using definition of conditional expectation of random variables with respect to partition , we get
Now, we calculate each sum of (3.4) separately. Note, that each interval of order is member of and each interval , is partitioned into intervals belonging to such that
Using this, we get
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.
Lemma 3.7. Given , the following inequalities hold
Proof. Consider points , and on the plan . The slope of the ray lies between slops of rays and .
4. Proof of Main Theorems
Let be the conjugation homeomorphism between and , that is, . Without loss of generality, we assume . Consider dynamical partition . Define sequence of random variables on the by this formula Denote by and .
Statement 4.1. Let the sequence be defined in Theorem 1.3. Then there exists a universal constant such that for all , the following inequality holds
Proof. It is clear that Now, we estimate . Denote by and . Thus, we have where . Using Lemma 3.7, we get It is clear that for any holds Since, each pair of adjacent intervals of are -comparable. By the assumption of Theorem 1.3, we get Hence, the rotation number is of bounded type, and an easy trick gives us where and . A similar lower bound holds true for . Therefore, we have for all .
Proof of Theorem 1.3. For the proof of Theorem 1.3, we use the above reasonings. By Lemma 3.6, the sequence of random variables is a martingale with respect to . We want to show that converges to in the norm when . By direct calculation, it is easy to see that and is orthogonal, that is Using the assertion of Statement 4.1, we get Iterating the last relation, we have . So far as the series converges. From this implies that the sequence of random variables is bounded in norm. By Lemma 3.5, the sequence of random variables converges to some function in norm. We prove that sequence of random variables converges to the . Indeed, denote by and end points of interval of dynamical partition . By definition of , we have Moreover, using last inequality, we obtain From this taking the limit when , we get . Since, , then is absolutely continuous function and almost everywhere on . Thus, Theorem 1.3 is completely proved.
Statement 4.2. For all hold this equality
Proof. It is a well-known fact that the class of continuous functions on is dense (in ) in (see ). From this fact it implies that if , then for any there exists a continuous function and such that and . Using this and Denjoy estimate, we obtain As is uniformly continuous on and by exponential refinement uniformly tends to , there exists a positive integer such that for all , the . Therefore, . Since was arbitrary and sufficiently small.
Proof of Theorem 1.4. Assume that conjugation map is absolutely continuous, then and almost everywhere. For all positive integers , the function satisfies Taking the logarithm, we obtain Denote by , it is clear that . Suppose, by contradiction, that there exists , such that the Lebesgue measure of the set does not converge to 0 when goes to infinity. Hence, for all positive integer : But does not tend to 0 when goes to . Hence does not tend to 0 when goes to , this contradicts Statement 4.2 and ends the proof of Theorem 1.4.
The authors are grateful to A. A. Dzhalilov for useful discussions and grateful to Universiti Kebangsaan Malaysia for providing financial support via the Grants UKM-MI-OUP-2011 and UKM-DIP-2012-31. They also thank the referee for comments that have improved the presentation of this paper.
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