Notes on a theorem of Katznelson and Ornstein

. Let log f (cid:48) be an absolutely continuous and f (cid:48)(cid:48) /f (cid:48) ∈ L p ( S 1 ,d(cid:96) ) for some p > 1 , where (cid:96) is Lebesgue measure. We show that there exists a subset of irrational numbers of unbounded type, such that for any element (cid:98) ρ of this subset, the linear rotation R (cid:98) ρ and the shift f t = f + t mod 1 , t ∈ [0 , 1) with rotation number (cid:98) ρ, are absolutely continuously conjugate. We also introduce a certain Zygmund-type condition depending on a parameter γ , and prove that in the case γ > 12 there exists a subset of irrational numbers of unbounded type, such that every circle diﬀeomorphism satisfying the corresponding Zygmund condition is absolutely continuously conjugate to the linear rotation provided its rotation number belongs to the above set. Moreover, in the case of γ > 1 , we show that the conjugacy is C 1 -smooth.

1. Introduction. We study orientation-preserving aperiodic diffeomorphisms f of the circle S 1 = R/Z. Poincaré (1885) noticed that the orbit structure of such f is determined by some irrational mod 1, called the rotation number of f and denoted ρ = ρ(f ), in the following sense: for any ξ ∈ S 1 , the mapping f j (ξ) → jρ mod 1, j ∈ Z, is orientation-preserving. Some fifty years later, Denjoy proved that if f is an orientation-preserving C 1 -diffeomorphism of the circle with irrational rotation number ρ and log f has bounded variation then the orbit {f j (ξ)} j∈Z is dense and the mapping f j (ξ) → jρ mod 1 can therefore be extended by continuity to a homeomorphism h of S 1 , which conjugates f to the linear rotation R ρ : ξ → ξ + ρ mod 1, that is, such that In this context, a natural question to ask is under what condition one can obtain higher smoothness for the conjugation h? The first local result asserting regularity of the conjugation of the circle diffeomorphism to the linear rotation was obtained by Arnold [1]. He proved that, for typical irrational rotation number ρ and for analytic diffeomorphism f sufficiently close to the linear rotation R ρ , the conjugation h is analytic. Later on, Moser extended this result for sufficiently smooth but not analytic diffeomorphisms [14].
In the end of the 70's the first global result, that is, a result which is not requiring the closeness of diffeomorphism to the linear rotation, was proved by Herman [5]. It was shown that if f ∈ C k (k ≥ 3), its rotation number is irrational and satisfies a certain Diophantine condition i.e., ρ ∈ D = {α : |e 2πinα − 1| ≥ C δ |n| −1−δ , for any δ > 0 and n ∈ Z}, then h is in fact k − 1 − times differentiable for any > 0, and is analytic if f is analytic. We notice that (D) = 1, that is Herman's theorem holds for a set of rotation numbers of full measure. The proof of Herman's theorem is based on an application of the Schwarzian derivative and therefore the condition k ≥ 3 is crucial. Later, Yoccoz [20] extended Herman's theorem for all Diophantine numbers. At the same time Hawkins and Schmidt [4] showed that, for every irrational number α ∈ [0, 1) of unbounded type, there exists a C 2 circle diffeomorphism f with ρ(f ) = α, for which the conjugating map h between f and R ρ is singular.
In the end of the 80's two different approaches to the Herman's theory were developed by Katznelson and Ornstein [9,10] and Khanin and Sinai [11,12]. These approaches gave sharp results on the smoothness of the conjugacy in the case of diffeomorphisms with low smoothness. Recently, Khanin and Teplinsky [13] developed a conceptually new approach which is entirely based on the idea of cross-ratio distortion estimates. Quite surprisingly this simple and elementary approach allows to prove stronger results. Let us briefly recall some details of above results. It is well known that the smoothness of a conjugacy h is strongly related to sharp estimates of K n = max ξ | log(f qn (ξ)) |, where q n is a first return time of f and it is defined as: q n = min{k ∈ N : kρ < q n−1 ρ }, q 0 = 1. More precisely, in the works [5], [9], [11,12], [13] and [20] it was shown that if f ∈ C k , k ≥ 2 + and the rotation number ρ is irrational, then the sequence (K n ) tends to zero exponentially fast. This fact together with Diophantine-type conditions on rotation numbers ensure that the conjugating map h is at least C 1 -smooth. Following Katznelson and Ornstein [10] we define a class of low smoothness circle diffeomorphisms as follows.
Definition 1.1. We say that a circle diffeomorphism f belongs to KO class if log f is absolutely continuous, f /f ∈ L p (S 1 , d ) for some p > 1, and the rotation number ρ of f is irrational.
Katznelson and Ornstein [10] proved that within KO class the sequence (K n ) belongs to 2 . Moreover, if the rotation number is of bounded type, then h is absolutely continuous. In view of the above results the following question arises naturally.
Is it possible to extend the results on absolute continuous linearization for a larger class of rotation numbers which will include some rotation number of unbounded type? Seemingly the results of Hawkins and Schmidt forbid such an extension. However, here we are interested not in all but in typical diffeomorphisms satisfying the KO conditions. A natural way to introduce typical diffeomorphisms is to consider one-parameter families such that the rotation number is changed monotonically with the parameter. The simplest example of such family is given by a shift corresponding to an additive constant, f t = f + t, t ∈ [0, 1]. In other words, we are interested whether the result of Katznelson and Ornstein [10] can be extended to a larger class of rotation numbers within a family f t .
In this paper, we give an affirmative answer to the above question. We show that for any KO diffeomorphism f, that is, the diffeomorphism satisfying KO conditions, there exists a subset I of irrational numbers of unbounded type, such that for any ρ ∈ I, the linear rotation R ρ and the shift f t = f + t with rotation number ρ, are absolutely continuously conjugate. To formulate this statement we need the following notions. Let α ∈ (0, 1) be an irrational number. We use the continued fraction representation α = 1/(a 1 + 1/(a 2 + ...)) := [a 1 , a 2 , ..., a s , ...) of a given number α. The sequence of positive integers (a s ) with s ≥ 1, called partial quotients, and is uniquely determined for each α. Now we define a subset of irrational numbers by using two given sequences of natural numbers. Let (i n ) be a strictly increasing sequence of natural numbers, (v n ) be an unbounded sequence of natural numbers and M be a natural number. Denoting the set of all irrational numbers α = [a 1 , a 2 , ..., a s , ...) such that a in ≤ v n and a s ≤ M for any s ∈ N \ {i n , n = 1, 2, ...} by I(i n , v n , M ), we set Our first main result is given by the following theorem.
Theorem 1.2. Let f be a KO diffeomorphism of the circle. Then for any unbounded sequence of natural numbers (v n ), there exists a strictly increasing sequence i n = i n (f, v n ) of natural numbers, such that for any ρ ∈ I(i n , v n ), the conjugating map h between f t0 and R ρ and its inverse h −1 are absolutely continuous and h , This result extends the result of Katznelson and Ornstein [10]. It is clear that the union of the sets I(i n , v n ) contains the set of all irrational numbers of bounded type. Since the set I(i n , v n ) depends on v n , generally we cannot say much about its Lebesgue measure. We may say that the main feature of this theorem is the arbitrariness of KO diffeomorphism f , and the fact that (v n ) can tend to infinity arbitrarily fast. Further, we show that the set of rotation numbers can be extended to an unbounded set for a certain subclass of KO. For this we impose a certain Zygmund condition on f which defines a subclass within the KO class. Let us consider the following one-parameter family of functions: Φ γ : [0, 1) → [0, +∞), Φ γ (0) = 0 and where ξ ∈ S 1 and τ ∈ [0, 1 2 ]. Suppose that there exists a constant C > 0 such that the following inequality holds: Denote by Z Φγ the class of circle diffeomorphisms f, whose derivatives f satisfy (1). Below we work with this class. We note that the class of continuous functions satisfying (1) is a subclass of Zygmund class Λ * (see [21] for the definition). The Zygmund class Λ * plays a key role in analysis of the trigonometric series. The class Λ * was applied to the circle homeomorphisms for the first time by Hu and Sullivan (see [6], [16]) who extended the classical Denjoy's theorem to this class. The main motivation comes from the idea that for many rigidity type problems, the Zygmund condition is more natural than the usual smoothness spaces when one tries to obtain sharp results. For example, as we have seen above in the C 2 -class in general one can guarantee absolute continuity of a linearizing conjugacy only for rotation numbers of bounded type. Below we study related questions in the Zygmund classes which allow us to extend the class of rotation numbers beyond the bounded type.
Note that for the case of γ ∈ ( 1 2 , 1] the class of continuous functions satisfying the condition (1) was studied by Weiss and Zygmund in [19]. They proved that in this case the function satisfies KO conditions (see Theorem 6.1 below). Our next main result is the following theorem. Theorem 1.3. Let f ∈ Z Φγ be a circle diffeomorphism with irrational rotation number ρ and γ ∈ ( 1 2 , 1]. Suppose that for some α ∈ (0, γ − 1 2 ) the partial quotients of ρ satisfies a n ≤ Cn α , C > 0. Then the conjugating map h between f and R ρ and its inverse h −1 are absolutely continuous and h , This theorem extends the result of Katznelson and Ornstein [10]. The theorem is applicable to a set of rotation numbers which includes some irrational numbers of unbounded type. However, the Lebesgue measure of this set is equal to zero.
We next consider the case of C 1 -smooth linearization. We again consider the Zygmund class Z Φγ but now assume that γ > 1. Note that in this case, the class Z Φγ is a subclass of C 2 (see Theorem 6.2) and it is wider than C 2+ . Our next main result is as follows.
Theorem 1.4. Let f ∈ Z Φγ be a circle diffeomorphism with irrational rotation number ρ and γ > 1. Suppose that for some α ∈ (0, γ − 1) the partial quotients of ρ satisfies a n ≤ Cn α , C > 0. Then the conjugating map h between f and R ρ and its inverse h −1 are C 1 diffeomorphisms.
In this theorem, if 1 < γ ≤ 2 then the Lebesgue measure of the set of rotation numbers is equal to zero, but if γ > 2 and 1 < α < γ − 1 then the set of rotation numbers has full Lebesgue measure.
The paper is organized as follows. In Section 2, we present the basic notions and classical inequalities. We also estimate the ratio of lengths of intervals of dynamical partition. In Section 3, we derive an estimate for K n (t) corresponding to f t , (Theorem 3.2) which plays an important role in the proof of the first main theorem. In Section 4, we obtain a uniform (in parameter) estimate for K n (t). In Section 5, we prove Theorem 1.2. The next sections are devoted to the proofs of the second and third main theorems. In Section 6, we briefly discuss the properties of the class Z Φγ . Using these properties, in Section 7 we get a sharp estimate for K n (Theorem 7.1). In Section 8 we prove Theorem 1.3 and Theorem 1.4. In the last section we discuss on some extensions of our main theorems.

Dynamical partition and universal estimates. Dynamical partition.
Let f be a circle homeomorphism with irrational rotation number ρ. Taking a point ξ 0 ∈ S 1 we define the n-th fundamental segment I n 0 := I n 0 (ξ 0 ) as the circle arc [ξ 0 , f qn (ξ 0 )] if n is even and [f qn (ξ 0 ), ξ 0 ] if n is odd. We denote two sets of closed intervals of order n: q n "long" intervals I n−1 i := f i (I n−1 0 ), 0 ≤ i < q n and q n−1 "short" intervals I n j := f j (I n 0 ), 0 ≤ j < q n−1 . 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 n := P n (ξ 0 , f ) and it is called the n-th dynamical partition of S 1 . Obviously, partition P n+1 is a refinement of partition P n . Indeed, the short intervals are members of P n+1 and each long interval I n−1 i ∈ P n , 0 ≤ i < q n , is partitioned into a n+1 + 1 intervals belonging to P n+1 such that Denjoy's theory. We first state the following definition which was introduced in [10].
Next we introduce two quantities which were also defined in [10]. Then we provide estimates for these quantities which are valid for any circle diffeomorphisms f ∈ C 1+BV (f has bounded variation) with irrational rotation number ρ. These estimates have very important applications in the theory of circle homeomorphisms. Their elementary proofs can be found in [9], [10] and [12].
) | the supremum being taken for all k, 0 ≤ k < q n and intervals (ξ, η) which are q n -small.
The following inequalities hold for any circle diffeomorphisms f ∈ C 1+BV (a) Denjoy's inequality: where v = V ar S 1 log f . Family of circle diffeomorphisms and universal estimates. Let f be a C 1+BV diffeomorphism of the circle. Consider a family of circle Similarly as above we can define K n (t) := K n (f t ) and K n (t) := K n (f t ) for every f t , t ∈ I. An important note is that both Denjoy's and Finzi's inequalities hold uniformly for every f t with the same constant v, that is (a ) Uniform Denjoy's inequality: Our next discussion is related to the study of some properties of dynamical Note that we equip S 1 with the usual metric |x − y| = inf{| x − y|, where x, y range over the lifts of x, y ∈ S 1 respectively}. We will need the following elementary but important theorem.
Theorem 2.2. Let f be a C 1+BV diffeomorphism of the circle. Assume that its rotation number is irrational. Then the following statements hold: (a) for any intervals I n+m,t = I n+m,t 0 (η) and I n,t = I n,t 0 (ξ) such that I n+m,t ⊂ I n,t , we have (b) for any n ≥ 1 and m ≥ 0 we have (I n,t 0 (ξ)). By applying uniform Denjoy's inequality to the last two relations, we get To prove (b), first notice that for even m a stronger statement follows immediately from (3) that is If m is odd the proof is similar but requires a little modification. In this case the interval I n+m,t 0 (ξ) is not inside I n,t 0 (ξ). Therefore, Next we prove (c). By the property of dynamical partition, it is easy to see that for any ζ ∈ S 1 and t ∈ I, we have |I n−1,t 0 (ζ)| ≥ |I n+1,t 0 (ζ)| + |I n,t 0 (ζ qn+1(t)−qn(t) )| and I n+1,t 0 (ζ) ⊂ I n,t 0 (ζ qn+1(t) ). The last two relations and uniform Denjoy's inequality imply By induction, we get |I n+m,t 0 (ζ)| ≤ (1 + e −v ) − m 2 |I n,t 0 (ζ)| for m even. If we pick out the point ζ = ξ * ∈ S 1 such that d n+m (t) = |I n+m,t 0 (ξ * )| then we get In case of odd m we have since d n+1 (t) ≤ d n (t).
Denote d n = sup t∈I d n (t).
Remark. The inequality (5) implies d n ≤ λ 2 d n−2 , n ≥ 2. Furthermore, inequalities (6) and (7) imply d n+m ≤ λ m d n for m even and d n+m ≤ λ m−1 d n for m odd. The useful convention q −1 = 0, q 0 = 1 imply that . It follows that d n ≤ λ n for n even, and d n ≤ λ n+1 for n odd.
The ratio distortion. Various types of ratio and cross-ratio distortion estimates are used in dynamical systems. The cross-ratio distortions were used for the first time by Yoccoz in [20] and later by de Melo and van Strien in [2] and byŚwiatek in [17]. The asymptotic estimates for a cross-ratio distortion with respect to smooth monotone function were studied in [18]. The ratio of three points a, b, c is Their ratio distortion with respect to the function f is Let f ∈ C 1 where the derivative of f does not have zeros on S 1 . Taking the limit in (8) when b → c we obtain This ratio distortion is one of main tools of the proofs in this paper. Notice that this distortion is multiplicative with respect to composition that is, for any two functions f and g we have R(c, I; f • g) = R(c, I; g) · R(g(c), g(I); f ).
Martingale convergence in L p . Suppose f satisfies KO conditions. Using dynamical partitions P t n we define a sequence of step functions on the circle as follows: M t 0 (x) ≡ 0, x ∈ S 1 and for any n ≥ 1 we set ds, if x ∈ I n,t , I n,t ∈ P t n .
Denoted by P t n the sequence of algebras generated by dynamical partitions. A simple calculation shows that the sequence of M t n is a martingale with respect to P t n for any t ∈ I. Moreover, using Hölder's inequality we obtain Hence, M t n is a L p -bounded martingale for any t ∈ I. According to Doob's theorem [3] we have the following. Following Katznelson and Ornstein, one can define the difference of martingales, that is The martingale property implies the following.

Statement 1.
Let a diffeomorphism f satisfy the KO conditions. Then for any t ∈ I the following equality holds: I Θ t n (x)dx = 0 f or any I ∈ P t n−1 .
The following proposition was proven by Katznelson and Ornstein in [10].
Since M t n is a L p -bounded martingale for any t ∈ I, Proposition 1 has the following immediate consequence. 3. 2 -convergence of K n (t). In this section we prove the 2 convergence of K n (t). We use the following sequences: Let a diffeomorphism f satisfy the KO conditions. Then the sequences (ε n (t)), (η n (t)) and (τ n (t)) belong to 2 for any t ∈ I.

Proof. By Theorem 2.2 we get
This implies

NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN 4595
On the other hand applying Cauchy-Schwarz inequality we obtain Similarly one can show that the sequences (η n (t)) and (τ n (t)) belong to 2 .

Theorem 3.2.
Let a diffeomorphism f satisfy the KO conditions. Then for any t ∈ I there exists a constant C 3 = C 3 (f ) > 0 such that It is easy to see that, by Lemma 3.1, K n (t) belongs to 2 . The proof of Theorem 3.2 will be provided in the next section after we prove the following two lemmas.
log R(ξ 0 , I n,t 0 ; f Proof. We prove the first inequality. Take any t ∈ I and fix it. To simplify the notations, below we write the formulae without t. By multiplicativity of R(ξ 0 , I n−1 0 ; f qn ) with respect to composition, we have log R(ξ 0 , I n−1 where η s = f s (ξ qn−1 ) and ξ s = f s (ξ 0 ) are end-points of the interval I n−1 s . Since the diffeomorphism f satisfies the KO conditions we have for any 0 ≤ s < q n . Using (15) and (16)

H. AKHADKULOV, A. DZHALILOV AND K. KHANIN
We note that the interval [ξ s , η s ] is a (n − 1)-th fundamental segment. It follows that the integral ηs ξs changes sign depending on the parity of n. More precisely, Using this we rewrite the right hand side of (17) in the following form Next we estimate the second sum of (18). It is obvious that for any natural number N one has First we estimate |B n |. We set

It is clear that
According to Theorem 2.3 we have So, one can choose a sufficiently large number N such that Thus, |B n | ≤ d 1 q n−1 . Now we estimate |A n |. For this, we divide the first sum of A n into three terms corresponding to summation over: 1 ≤ k ≤ n, k = n + 1 and n + 2 ≤ k ≤ N and we estimate each term separately.
Let 1 ≤ k ≤ n. In this case, since I n−1 s ⊆ I k ∈ P k the piecewise constant function Θ k takes a constant value on I n−1 s . Therefore Next, let k = n + 1. It is obvious that Finally, let n + 2 ≤ k ≤ N. We define a new piecewise constant function L k,s on I n−1 s as follows where ∂ r (I) is the right end-point of interval I. By construction, the function L k,s takes a constant value on every interval of P k−1 . Therefore, from Statement 1 it follows that Moreover, by Theorem 2.2 we get

Using this inequality and (22) we have
Using (20), (21) and (23) we obtain Then, this inequality together with (18) imply the inequality (13). The inequality (14) can be proved in the same manner as above, but there will be some changes in the estimate of A n . Since each short interval of P n is preserved when it is being passed from partition P n to P n+1 , the first sum of (19) is divided into three parts: 1 ≤ k ≤ n + 1, k = n + 2 and n + 3 ≤ k ≤ N. These three sums will be estimated similarly to the above and the estimate of |A n |+|B n | will be ε n+1 .
We also need the following lemma for the proof of Theorem 3.2.
Lemma 3.4. Suppose a diffeomorphism f satisfies the KO conditions. Then for any t ∈ I there exists Proof. We prove only the first inequality, the second inequality can be handled similarly. For simplicity we again omit t from the notations. The following three exact relations are crucial for our proof: dy. f as required.

Proof of Theorem 3.2.
Proof. In fact the proof of Theorem 3.2 follows closely to the proof in Khanin and Teplinsky [13]. We need the following two relations: ) . (28)
4. Uniform estimates for ε n (t), η n (t) and τ n (t). In the following theorem we will show that the sequences ε n (t), η n (t) and τ n (t) tend to zero uniformly in t ∈ I as n tends to infinity. Taking the limit as n → ∞ we get lim n→∞ supΘ n ≤ 4 .
Next, we estimateε n . According to Theorem 2.2 there exists a constant C 1 > 0 such thatε LetΘ n = sup{Θ m : m ≥ n}. It is easy to see thatΘ n ≥Θ n+1 for all n ≥ 1 and lim n→∞Θ n = 0. By monotonicity ofΘ n and the above inequality, we havẽ Hence lim n→∞ε n = 0. Next, we estimateη n . Due to Theorem 2.2, the monotonicity of Θ n and inequality (37) we get where [·] is the integer part of a given number. Therefore lim n→∞η n = 0. Now we are going to estimateτ n . From Theorem 2.2 it follows that Similarly to the proof of inequality (38) it can be shown that n k=1 λ n−kΘ Hence, the last three inequalities implỹ Thus lim n→∞τ n = 0, which concludes the proof of Theorem 4.1.

5.
Proof of Theorem 1.2. To prove our first main theorem we use a theory which was developed by Katznelson and Ornstein in [10]. The following sufficient condition for absolute continuity of the conjugacy was proved there.
Theorem 5.1. Let the diffeomorphism f satisfies the Denjoy's conditions that is, log f has bounded variation and the rotation number ρ is irrational. Assume ∞ n=1 (a n K n ) 2 < ∞.
Then the conjugating map h between f and R ρ and its inverse h −1 are absolutely continuous and h , (h −1 ) ∈ L 2 .
Proof of Theorem 1.2. Let the diffeomorphism f satisfy the KO conditions and (v n ) be an unbounded sequence of natural numbers. According to Theorem 4.1 we can find a subsequence (τ in ) of (τ n ) such that nv nτin ≤ 1 for all n = 1, 2, ... . Without loss of generality we may assume that the sequence (i n ) is a strictly increasing sequence. Let I(i n , v n ) be the set of irrational numbers which was defined in Section 1. It is clear that for any ρ ∈ I(i n , v n ) there exists a natural number M, such that ρ ∈ I(i n , v n , M ). Now we consider the family of diffeomorphisms f t = f + t, t ∈ I. Note that ρ(f t ) is a continuous and nondecreasing function of t. Moreover, it is strictly increasing at irrational values (see [8]). Due to this note for any ρ ∈ I(i n , v n , M ) there exits a unique t 0 ∈ I such that ρ(f t0 ) = ρ. By Theorem 3.2 and Lemma 3.1 we get On the other hand we have τ n (t 0 ) ≤τ n .
6. Theorem of Weiss-Zygmund. In this section we provide brief facts about continuous functions K : R 1 → R 1 satisfying inequality (1) for different values of parameter γ > 0. These facts will be used in the proofs of Theorems 1.3 and 1.4. First we consider the case γ ∈ ( 1 2 , 1]. The following theorem was proved by Weiss and Zygmund in [19]. Theorem 6.1. Let K : R 1 → R 1 be 1-periodic and continuous on R 1 . Assume that for some γ ∈ ( 1 2 , 1] the function K satisfies the inequality Then K is absolutely continuous and K ∈ L p ([0, 1]) for every p > 1.
The proof of this theorem in [19] is rather short but relies on a theorem of Littlewood and Paley. A more direct and general proof of this theorem can also be found in [7]. The statement of this theorem does not hold in the case γ ∈ (0, 1 2 ]. Indeed, in this case using the Weieratrass function one can construct a function satisfying (41) but almost nowhere differentiable. Similar examples can be found in [21]. Next we formulate a theorem on differentiability of K in the case of γ > 1.
Although this result is probably not new, we were not able to find a proper reference for it. Therefore, we provide a complete proof here.
By iterating from n = 1 to N we obtain Since the point η is the Lebesgue point for K and γ > 1 Taking the limit as N → ∞ in (44) we get where P γ (τ ) = ∞ n=1 Q γ (τ 2 −n ).
This proves uniform continuity of K on the set of Lebesgue points, thus K coincides almost everywhere with a continuous function. It is obvious that this continuous function is a derivative of K.