DISCRETE AND CONTINUOUS
DYNAMICAL SYSTEMS
Volume 37, Number 9, September 2017
doi:10.3934/dcds.2017197
pp. 4587–4609
NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN
Habibulla Akhadkulov∗
School of Quantitative Sciences, University Utara Malaysia
CAS 06010, UUM Sintok, Kedah Darul Aman, Malaysia
Akhtam Dzhalilov
Turin Polytechnic University
Kichik Halka yuli 17, Tashkent 100095, Uzbekistan
Konstantin Khanin
Department of Mathematics, University of Toronto
40 St. George Street, Toronto, Ontario M5S 2E4, Canada
(Communicated by Rafael de la Llave)
Abstract. Let log f 0 be an absolutely continuous and f 00 /f 0 ∈ Lp (S 1 , d`) for
some p > 1, where ` is Lebesgue measure. We show that there exists a subset
of irrational numbers of unbounded type, such that for any element ρb of this
subset, the linear rotation Rρb and the shift ft = f + t mod 1, t ∈ [0, 1) with
rotation number ρb, are absolutely continuously conjugate. We also introduce a
certain Zygmund-type condition depending on a parameter γ, and prove that
in the case γ > 21 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 0 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 Rρ : ξ → ξ + ρ
mod 1, that is, such that f = h−1 ◦ Rρ ◦ h.
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 Rρ , the conjugation h
2010 Mathematics Subject Classification. Primary: 37E10, 37C15; Secondary: 37C40.
Key words and phrases. Circle diffeomorphisms, rotation number, Denjoy’s inequality, conjugating map.
∗ Corresponding author: Habibulla Akhadkulov.
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H. AKHADKULOV, A. DZHALILOV AND K. KHANIN
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 : |e2πinα − 1| ≥ Cδ |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 Rρ 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 Kn = maxξ | log(f qn (ξ))0 |, where qn is a first return time of f and it
is defined as: qn = min{k ∈ N : kkρk < kqn−1 ρk}, q0 = 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 (Kn ) 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 0 is absolutely continuous, f 00 /f 0 ∈ Lp (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 (Kn ) 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, ft = 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 ft .
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
NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN
4589
ρb ∈ I, the linear rotation Rρb and the shift ft = f + t with rotation number ρb,
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/(a1 + 1/(a2 + ...)) := [a1 , a2 , ..., as , ...)
of a given number α. The sequence of positive integers (as ) 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 (in ) be a strictly
increasing sequence of natural numbers, (vn ) be an unbounded sequence of natural
numbers and M be a natural number. Denoting the set of all irrational numbers
α = [a1 , a2 , ..., as , ...) such that ain ≤ vn and as ≤ M for any s ∈ N \ {in , n =
1, 2, ...} by I(in , vn , M ), we set
I(in , vn ) =
∞
[
I(in , vn , M ).
M =1
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 (vn ), there exists a strictly increasing sequence in =
in (f, vn ) of natural numbers, such that for any ρb ∈ I(in , vn ), the conjugating map h
between ft0 and Rρb and its inverse h−1 are absolutely continuous and h0 , (h−1 )0 ∈
L2 . Here t0 = t0 (f, ρb) is the unique value of a parameter t such that ρ(ft0 ) = ρb.
This result extends the result of Katznelson and Ornstein [10]. It is clear that
the union of the sets I(in , vn ) contains the set of all irrational numbers of bounded
type. Since the set I(in , vn ) depends on vn , 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 (vn ) 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 0 which defines a subclass within the KO class. Let us
consider the following one-parameter family of functions: Φγ : [0, 1) → [0, +∞),
Φγ (0) = 0 and
x
, where 0 < x < 1 and γ > 0.
Φγ (x) =
(log x1 )γ
Denote by ∆2 f 0 (ξ, τ ) the second symmetric difference of f 0 i.e.,
∆2 f 0 (ξ, τ ) = f 0 (ξ + τ ) + f 0 (ξ − τ ) − 2f 0 (ξ)
where ξ ∈ S 1 and τ ∈ [0, 21 ]. Suppose that there exists a constant C > 0 such that
the following inequality holds:
k∆2 f 0 (·, τ )kL∞ (S 1 ) ≤ CΦγ (τ ).
(1)
0
Denote by ZΦγ 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
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H. AKHADKULOV, A. DZHALILOV AND K. KHANIN
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 ∈ ( 21 , 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 ∈ ZΦγ be a circle diffeomorphism with irrational rotation
number ρ and γ ∈ ( 12 , 1]. Suppose that for some α ∈ (0, γ − 12 ) the partial quotients
of ρ satisfies an ≤ Cnα , C > 0. Then the conjugating map h between f and Rρ and
its inverse h−1 are absolutely continuous and h0 , (h−1 )0 ∈ L2 .
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 ZΦγ but now assume that γ > 1. Note that in this case, the class
ZΦγ 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 ∈ ZΦγ 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 an ≤ Cnα , C > 0. Then the conjugating map h between f and Rρ 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 Kn (t) corresponding
to ft , (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 Kn (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 ZΦγ . Using these properties, in Section 7 we get a sharp estimate for
Kn (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.
2. Dynamical partition and universal estimates. Dynamical partition.
Let f be a circle homeomorphism with irrational rotation number ρ. Taking a
point ξ0 ∈ S 1 we define the n-th fundamental segment I0n := I0n (ξ0 ) as the circle arc
[ξ0 , f qn (ξ0 )] if n is even and [f qn (ξ0 ), ξ0 ] if n is odd. We denote two sets
of closed
intervals of order n: qn “long” intervals Iin−1:= f i (I0n−1 ), 0 ≤ i < qn and qn−1
“short” intervals Ijn := f j (I0n ), 0 ≤ j < qn−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 Pn := Pn (ξ0 , f ) and it is
NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN
4591
called the n-th dynamical partition of S 1 . Obviously, partition Pn+1 is a refinement
of partition Pn . Indeed, the short intervals are members of Pn+1 and each long
interval Iin−1 ∈ Pn , 0 ≤ i < qn , is partitioned into an+1 + 1 intervals belonging to
Pn+1 such that
an+1 −1
Iin−1
=
Iin+1
∪
[
n
Ii+q
.
n−1 +sqn
(2)
s=0
Denjoy’s theory. We first state the following definition which was introduced in
[10].
Definition 2.1. An interval I = (a, b) is qn -small and its endpoints a, b are qn -close
if {f j (I)}0qn −1 are disjoint.
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 0 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].
- Kn := Kn (f ) = maxξ | log(f qn (ξ))0 | = k log(f qn )0 k0 .
b n := K
b n (f ) = sup | log(f k (ξ))0 − log(f k (η))0 | the supremum being taken for
- K
all k, 0 ≤ k < qn and intervals (ξ, η) which are qn -small.
The following inequalities hold for any circle diffeomorphisms f ∈ C 1+BV
(a) Denjoy’s inequality: Kn ≤ v;
b n ≤ v;
(b) Finzi’s inequality: K
where v = V ar
log f 0 .
1
S
Family of circle diffeomorphisms and universal estimates. Let f be a C 1+BV
diffeomorphism of the circle. Consider a family of circle diffeomorphisms ft = f + t,
where t ∈ I = {t ∈ [0, 1) : ρt = ρ(ft ) − is irrational}. Similarly as above we can
b n (t) := K
b n (ft ) for every ft , t ∈ I. An important note
define Kn (t) := Kn (ft ) and K
is that both Denjoy’s and Finzi’s inequalities hold uniformly for every ft with the
same constant v, that is
(a0 ) Uniform Denjoy’s inequality: supt∈I Kn (t) ≤ v;
b n (t) ≤ v;
(b0 ) Uniform Finzi’s inequality: supt∈I K
where v = V ar
log f 0 .
1
S
The proofs of these inequalities follow from a simple observation: v = V ar
log ft0
1
S
does not depend on t.
Our next discussion is related to the study of some properties of dynamical
q (t)
partition of S 1 generated by ft . Denote dn (t) := dn (f, t) = kft n − Idk0 =
n,t
maxξ∈S 1 |I0 (ξ)|. It is easy to see that the sequence dn (t) is monotone decreasing: dn+1 (t) ≤ dn (t).
Note that we equip S 1 with the usual metric |x − y| = inf{|e
x − ye|, where x
e, ye
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:
4592
H. AKHADKULOV, A. DZHALILOV AND K. KHANIN
(a) for any intervals I n+m,t = I0n+m,t (η) and I n,t = I0n,t (ξ) such that I n+m,t ⊂
I n,t , we have
|I n+m,t |
dn+m (t)
≤ ev (1 + ev )
;
n,t
|I |
dn (t)
(b) for any n ≥ 1 and m ≥ 0 we have
dn+m (t)
|I0n+m,t (ξ)|
≤ e2v (1 + ev )
;
dn (t)
|I0n,t (ξ)|
(c) Let λ = (1 + e−v )−1/2 . Then
dn+m (t)
dn+m (t)
≤ λm , for m even, and
≤ λm−1 , for m odd.
dn (t)
dn (t)
Proof. Select the point ξ ∗ ∈ S 1 such that dn (t) = |I0n,t (ξ ∗ )|. Due to the combinatorics of trajectories, there exists 0 ≤ i < qn+1 (t) such that either I0n,t (ξ ∗ ) ⊂
i−q (t)
i+q (t)
fti (I0n,t (ξ)) ∪ ft n (I0n,t (ξ)) or I0n,t (ξ ∗ ) ⊂ fti (I0n,t (ξ)) ∪ ft n (I0n,t (ξ)). By applying uniform Denjoy’s inequality to the last two relations, we get |I0n (ξ ∗ , t)| ≤
(1 + ev )|fti (I0n,t (ξ))|. Then uniform Finzi’s inequality implies
i n+m,t
(η))|
dn+m (t)
|I n+m,t |
v |ft (I0
≤
e
≤ ev (1 + ev )
.
n,t
n,t
i
|I |
dn (t)
|ft (I0 (ξ))|
(3)
To prove (b), first notice that for even m a stronger statement follows immediately
from (3) that is
|I0n+m,t (ξ)|
dn+m (t)
.
≤ ev (1 + ev )
n,t
dn (t)
|I0 (ξ)|
If m is odd the proof is similar but requires a little modification. In this case the
interval I0n+m,t (ξ) is not inside I0n,t (ξ). Therefore,
|I0n+m,t (ξ−qn+m (t) )|
dn+m (t)
|I0n+m,t (ξ)|
|I0n+m,t (ξ)|
≤ e2v (1 + ev )
=
· n+m,t
. (4)
n,t
n,t
dn (t)
|I0 (ξ)|
|I0 (ξ)|
|I0
(ξ−qn+m (t) )|
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 |I0n−1,t (ζ)| ≥ |I0n+1,t (ζ)| + |I0n,t (ζqn+1 (t)−qn (t) )| and
I0n+1,t (ζ) ⊂ I0n,t (ζqn+1 (t) ). The last two relations and uniform Denjoy’s inequality
imply
|I0n,t (ζqn+1 (t)−qn (t) )|
|I0n−1,t (ζ)|
≥
1
+
≥ 1 + e−v .
(5)
|I0n,t (ζqn+1 (t) )|
|I0n+1,t (ζ)|
m
By induction, we get |I0n+m,t (ζ)| ≤ (1 + e−v )− 2 |I0n,t (ζ)| for m even. If we pick out
the point ζ = ξ ∗ ∈ S 1 such that dn+m (t) = |I0n+m,t (ξ ∗ )| then we get
dn+m (t)
|I n+m,t (ξ ∗ )|
≤ 0 n,t
≤ λm .
dn (t)
|I0 (ξ ∗ )|
(6)
dn+m (t)
dn+m (t) dn+1 (t)
=
·
≤ λm−1
dn (t)
dn+1 (t)
dn (t)
(7)
In case of odd m we have
since dn+1 (t) ≤ dn (t).
NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN
4593
Denote den = supt∈I dn (t).
Remark. The inequality (5) implies den ≤ λ2 den−2 , n ≥ 2. Furthermore, inequalities
(6) and (7) imply den+m ≤ λm den for m even and den+m ≤ λm−1 den for m odd. The
useful convention q−1 = 0, q0 = 1 imply that I0−1 (ξ) = [ξ − 1, ξ], I00 (ξ) = [ξ, f (ξ)].
It follows that den ≤ λn for n even, and den ≤ λn+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 by Świa̧tek 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
Q(a, b, c) =
|[a, b]|
.
|[b, c]|
Their ratio distortion with respect to the function f is
R(a, b, c; f ) =
Q(f (a), f (b), f (c))
.
Q(a, b, c)
(8)
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
R(c, I; f ) =
|f (I)| 1
|I| f 0 (c)
where
I = [a, c].
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 ).
(9)
Martingale convergence in Lp . Suppose f satisfies KO conditions. Using dynamical partitions Ptn we define a sequence of step functions on the circle as follows:
Mt0 (x) ≡ 0, x ∈ S 1 and for any n ≥ 1 we set
Z
1
f 00 (s)
Mtn (x) = n,t
ds, if x ∈ I n,t , I n,t ∈ Ptn .
(10)
|I | I n,t f 0 (s)
Denoted by Ptn the sequence of algebras generated
by dynamical partitions. A
simple calculation shows that the sequence of Mtn is a martingale with respect to
Ptn for any t ∈ I. Moreover, using Hölder’s inequality we obtain
Z
X Z
kMtn kpp =
|Mtn (x)|p dx =
|Mtn (x)|p dx
S1
=
X
I n,t ∈Ptn
1
|I n,t |p−1
Z
I n,t
I n,t
I n,t ∈Ptn
f 00 (x) p
dx ≤
f 0 (x)
f 00 (x) p
f 00 p
0
dx = 0 .
f
(x)
f p
n,t
I
X Z
I n,t ∈Ptn
Hence, Mtn is a Lp -bounded martingale for any t ∈ I. According to Doob’s
theorem [3] we have the following.
Theorem 2.3. Suppose f satisfies the KO conditions. Then for any t ∈ I as
n→∞
f 00
Mtn → 0 almost surely and in Lp .
f
4594
H. AKHADKULOV, A. DZHALILOV AND K. KHANIN
Following Katznelson and Ornstein, one can define the difference of martingales,
that is
Θtn (x) = Mtn (x) − Mtn−1 (x), n ≥ 1.
(11)
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:
Z
Θtn (x)dx = 0 f or any I ∈ Ptn−1 .
I
The following proposition was proven by Katznelson and Ornstein in [10].
Proposition 1. Let Gn be a Lp -bounded martingale, 1 < p ≤ 2. Define gn =
Gn − Gn−1 . Then
∞
X
kgn k2p < ∞.
n=1
Since Mtn is a Lp -bounded martingale for any t ∈ I, Proposition 1 has the
following immediate consequence.
Corollary 1. Let the diffeomorphism f satisfies the KO conditions and 1 < p ≤ 2.
Then for any t ∈ I we have
∞
X
kΘtn k2p < ∞.
n=1
3. `2 -convergence of Kn (t). In this section we prove the `2 convergence of Kn (t).
We use the following sequences:
1
q
(t) +
εn (t) = dn−1
∞
X
dk−2 (t) t
kΘ kp ,
dn−1 (t) k
n ≥ 0,†
k=n+1
ηn (t) =
n
X
k=1
τn (t) =
dn−1 (t) dn (t)
·
εk−1 (t),
dk−1 (t) dk (t)
n
X
dn (t) k=1
dk (t)
ηk (t) + εk (t) ,
n ≥ 1,
n ≥ 1.
Lemma 3.1. Let a diffeomorphism f satisfy the KO conditions. Then the sequences
(εn (t)), (ηn (t)) and (τn (t)) belong to `2 for any t ∈ I.
Proof. By Theorem 2.2 we get
εn (t) ≤
∞
X
1 nq
λ +
λk−(n+1) kΘtk kp .
λ
k=n+1
This implies
ε2n (t) ≤
∞
X
2 2 2n
k−(n+1)
t
q +
λ
λ
kΘ
k
.
k p
λ2
k=n+1
†
Note that d−1 (t) = maxξ0 ∈S 1 |I0−1,t (ξ0 )| = 1.
NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN
4595
On the other hand applying Cauchy-Schwarz inequality we obtain
∞
∞
X
2
X
1
λk−(n+1) kΘtk kp ≤
λk−(n+1) kΘtk k2p .
1−λ
k=n+1
Therefore
∞
X
n=0
ε2n (t) ≤
k=n+1
2
2
λ2 (1 − λ q )
+
∞
∞
X
X
2
λk−(n+1) kΘtk k2p =
λ2 (1 − λ) n=0
k=n+1
∞
X
∞
X
2
k
λ
kΘtn k2p < ∞.
+ 2
2
λ2 (1 − λ q ) λ (1 − λ) k=0 n=k+1
2
Similarly one can show that the sequences (ηn (t)) and (τn (t)) belong to `2 .
Theorem 3.2. Let a diffeomorphism f satisfy the KO conditions. Then for any
t ∈ I there exists a constant C3 = C3 (f ) > 0 such that
Kn (t) ≤ C3 · τn (t).
(12)
It is easy to see that, by Lemma 3.1, Kn (t) belongs to `2 . The proof of Theorem
3.2 will be provided in the next section after we prove the following two lemmas.
Lemma 3.3. Suppose a diffeomorphism f satisfies the KO conditions. Then for
any t ∈ I there exists C4 = C4 (f ) > 0 such that
n−1
log R(ξ0 , I n−1,t ; ftqn (t) ) − (−1)
0
2
n
log R(ξ0 , I n,t ; ftqn−1 (t) ) − (−1)
0
2
qn (t)−1
s=0
qn−1 (t)−1
X
s=0
f 00 (x) dx ≤ C4 εn (t),
f 0 (x)
(13)
f 00 (x) dx ≤ C4 εn+1 (t).
f 0 (x)
(14)
Z
X
Isn−1,t
Z
Isn,t
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(ξ0 , I0n−1 ; f qn )
with respect to composition, we have
log R(ξ0 , I0n−1 ; f qn )
=
qX
n −1
s=0
log
h f (η ) − f (ξ ) 1 i
s
s
,
ηs − ξs
f 0 (ξs )
(15)
where ηs = f s (ξqn−1 ) and ξs = f s (ξ0 ) are end-points of the interval Isn−1 . Since the
diffeomorphism f satisfies the KO conditions we have
Z ηs
f (ηs ) = f (ξs ) + f 0 (ξs )(ηs − ξs ) +
f 00 (x)(ηs − x)dx
(16)
ξs
for any 0 ≤ s < qn . Using (15) and (16) together with Hölder’s inequality one can
show that
Z ηs 00
qX
n −1
h
f (x) ηs − x i
log R(ξ0 , I0n−1 ; f qn ) =
log 1 +
·
dx =
(17)
0
ξs f (ξs ) ηs − ξs
s=0
qX
n −1 h Z ηs
s=0
ξs
1 f 00 (x) ηs − x i
q
·
dx
+
O
dn−1
.
f 0 (x) ηs − ξs
4596
H. AKHADKULOV, A. DZHALILOV AND K. KHANIN
We note that theR interval [ξs , ηs ] is a (n − 1)-th fundamental segment. It follows
η
that the integral ξss changes sign depending on the parity of n. More precisely,
Z ηs 00
Z
f (x) ηs − x
f 00 (x) ηs − x
n−1
·
dx
=
(−1)
·
dx.
0
η s − ξs
f 0 (x) ηs − ξs
ξs f (x)
Isn−1
Using this we rewrite the right hand side of (17) in the following form
log R(ξ0 , I0n−1 ; f qn )
= (−1)
n−1
qX
n −1
s=0
+
qX
n −1
(−1)n−1
s=0
Z
Z
f 00 (x)
dx+
2f 0 (x)
(18)
Isn−1
1 f 00 (x) ηs − x
1
q
·
−
dx + O dn−1
.
0
f (x)
ηs − ξs
2
Isn−1
Next we estimate the second sum of (18). It is obvious that for any natural number
N one has
qX
N qX
n −1 Z
n −1 Z
η −x
X
f 00 (x) ηs − x
1
1
s
·
dx
=
Θk (x)dx+ (19)
−
−
f 0 (x)
η s − ξs
2
η s − ξs
2
s=0
s=0
k=1
Isn−1
+
qX
n −1
s=0
Isn−1
Z ηs − x
1 f 00 (x)
−
M
(x)
dx := An + Bn .
−
·
N
ηs − ξs
2
f 0 (x)
Isn−1
First we estimate |Bn |. We set
Un =
qn
−1
[
Isn−1 .
s=0
It is clear that
1/q f 00 |Bn | ≤ `(Un )
MN − 0 .
f p
According to Theorem 2.3 we have
f 00 lim Mn − 0 = 0.
n→∞
f p
So, one can choose a sufficiently large number N such that
1
f 00 q
.
MN − 0 ≤ dn−1
f p
1
q
Thus, |Bn | ≤ dn−1
. Now we estimate |An |. For this, we divide the first sum of An
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 Isn−1 ⊆ I k ∈ Pk the piecewise constant function
Θk takes a constant value on Isn−1 . Therefore
Z 1
ηs − x
−
Θk (x)dx = 0.
ηs − ξs
2
Isn−1
NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN
4597
n qX
n −1 Z
X
η −x
1
s
Θk (x)dx = 0.
−
ηs − ξs
2
s=0
(20)
Hence
k=1
Isn−1
Next, let k = n + 1. It is obvious that
n −1 Z
qX
η −x
1/q
1
s
kΘn+1 kp .
Θn+1 (x)dx ≤ `(Un )
−
ηs − ξs
2
s=0
(21)
Isn−1
Finally, let n + 2 ≤ k ≤ N. We define a new piecewise constant function Lk,s on
Isn−1 as follows
Lk,s (x) =
ηs − ∂r (I) 1
− ,
ηs − ξs
2
x ∈ I ⊂ [ξs , ηs ], 0 ≤ s < qn and I ∈ Pk−1 ,
where ∂r (I) is the right end-point of interval I. By construction, the function Lk,s
takes a constant value on every interval of Pk−1 . Therefore, from Statement 1 it
follows that
Z
Θk (x)Lk,s (x)dx = 0.
(22)
Isn−1
Moreover, by Theorem 2.2 we get
η −x
1
dk−2
s
− − Lk,s (x) ≤ ev (1 + ev )
.
ηs − ξs
2
dn−1
Using this inequality and (22) we have
Z
qX
N
n −1 h Z
X
η −x
i
1
s
Θk (x)Lk,s (x)dx (23)
− − Lk,s (x) Θk (x)dx +
η
−
ξ
2
s
s
s=0
k=n+2
Isn−1
Isn−1
≤ ev (1 + ev )
N
X
dk−2
kΘk kp .
dn−1
k=n+2
Using (20), (21) and (23) we obtain
|An | ≤ ev (1 + ev )
N
X
dk−2
kΘk kp .
dn−1
k=n+1
Hence
|An | + |Bn | ≤ ev (1 + ev )εn .
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 An . Since each short interval of Pn is preserved
when it is being passed from partition Pn to Pn+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 |An |+|Bn | will be εn+1 .
We also need the following lemma for the proof of Theorem 3.2.
4598
H. AKHADKULOV, A. DZHALILOV AND K. KHANIN
Lemma 3.4. Suppose a diffeomorphism f satisfies the KO conditions. Then for
any t ∈ I there exists C5 = C5 (f ) > 0 such that
#
" n−1,t
qn (t)−1 Z
|Iqn (t) | |I0n−1,t | − |I0n+1,t |
(−1)n X
f 00 (x) −
·
dx ≤ C5 εn (t),
log
2
f 0 (x)
|
| − |Iqn+1,t
|I0n−1,t | |Iqn−1,t
s=0
n (t)
n (t)
Isn+1,t
#
" n+1,t
|Iqn (t) | |I0n−1,t | − |I0n+1,t |
(−1)n
−
log
n+1,t
n+1,t ·
n−1,t
2
|I0
| |Iqn (t) | − |Iqn (t) |
qn (t)−1
X
s=0
Z
f 00 (x) dx ≤ C5 εn (t).
f 0 (x)
Isn−1,t
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:
# q −1
"
n
X
| |I0n−1 | − |I0n+1 |
|Iqn−1
|f (Isn−1 )|
|Isn−1 | − |Isn+1 |
n
·
=
log
·
,
log
|I0n−1 | |Iqn−1
|Isn−1 |
|f (Isn−1 )| − |f (Isn+1 )|
| − |Iqn+1
|
n
n
s=0
(24)
Z ξs+q
00
n−1
n−1
f (y) ξs+qn−1 − y
|f (Is )|
=1+
·
dy,
(25)
0 (ξ ) ξ
f
|Isn−1 | · f 0 (ξs )
s
s+qn−1 − ξs
ξs
|f (Isn−1 )| − |f (Isn+1 )|
i
h
=1+
|Isn−1 | − |Isn+1 | · f 0 (ξs+qn+1 )
Z
ξs+qn−1
ξs+qn+1
f 00 (y)
f 0 (ξ
ξs+qn−1 − y
·
dy.
ξ
)
s+qn+1
s+qn−1 − ξs+qn+1
(26)
Equality (24) comes from the multiplicativity of ratio distortion with respect to
composition. The equalities (25) and (26) follows from (16). Since |Isn−1 | ∼ |Isn−1 |−
|Isn+1 |, using (24)-(26) and similar arguments as in the proof of Lemma 3.3 we get
# q −1 Z
"
qX
n
n −1 Z ξs+q
ξs
X
n−1 f 00 (x)
|Iqn−1
| |I0n−1 | − |I0n+1 |
f 00 (x)
n
·
=
dx
+
dx−
log
n−1
n−1
n+1
0
f (x)
2f 0 (x)
|I0 | |Iqn | − |Iqn |
s=0 ξs+qn+1
s=0 ξs
−
qX
n −1 Z ξs+q
n−1
s=0
ξs+qn+1
qn −1 Z
f 00 (x)
(−1)n X
f 00 (x)
dx
+
O(ε
)
=
dx + O(εn )
n
2f 0 (x)
2
f 0 (x)
s=0
Isn+1
as required.
3.1. 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:
" n+1,t
#
n−1,t
n+1,t
n+1,t
n−1,t
n+1,t
|Iqn−1,t
|
|I
|
|I
|
−
|I
|
|I
|
|I
|
−
|I
|
(t)
q
(t)
n
n
0
0
0
0
· 0
n+1,t −1 =
n−1,t ·
n+1,t ·
n−1,t
n+1,t − 1 , (27)
|I0n−1,t | |Iqn−1,t
|
−
|I
|
|I
|
|I
|
|I
|
−
|I
0
0
qn (t)
qn (t)
qn (t) |
n (t)
q (t)
q (t)
(ξ0 ))0 R(ξ0 , I0n−1,t ; ft n ) − 1
i
|I0n,t | h
q
(t)
q
(t)
= n−1,t
· 1 − (ft n−1 (ξ0 ))0 R(ξ0 , I0n,t ; ft n−1 ) .
|I0
|
(ft n
(28)
NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN
4599
The equality (27) is easily verified. The equality (28) comes from the definitions of
q
(t)
q (t)
R(ξ0 , I0n−1,t ; ft n ) and R(ξ0 , I0n,t ; ft n−1 ). Define
mn (t) = exp
(−1)n
qn (t)−1
2
s=0
f 00 (x) dx .
f 0 (x)
Z
X
Isn−1,t
It is clear that
qn (t)−1
X
s=0
f 00 (x)
dx +
f 0 (x)
Z
Isn−1,t
qn−1 (t)−1
X
s=0
f 00 (x)
dx =
f 0 (x)
Z
Isn,t
Z
f 00 (x)
dx = 0.
f 0 (x)
S1
Therefore, we have
exp
qn−1 (t)−1
(−1)n+1
X
2
s=0
f 00 (x) dx = mn (t).
f 0 (x)
Z
Isn,t
Due to (27) and Lemma 3.4 we have
mn+1 (t) − 1 =
|I0n+1,t |
(mn (t) − 1) + O(εn (t)),
|I0n−1,t |
(29)
which is iterated into
mn+1 (t) − 1 = O(κn+1 (t)),
(30)
where
κn+1 (t) =
|I0n,t ||I0n+1,t |
n+1
X
εk−1 (t)
.
k−1,t
||I0k,t |
k=1 |I0
It is easy to see that κn+1 (t) ∈ `2 and by Theorem 2.2 κn = O(ηn ). Relations (28),
(30) and Lemma 3.3 imply
q (t)
(ft n
(ξ0 ))0 − 1 =
|I0n,t |
q
(t)
(1 − (ft n−1 (ξ0 ))0 ) + O(ζn (t)),
|I0n−1,t |
(31)
which is iterated into
q (t)
(ft n
n
X
ζk (t) ,
(ξ0 ))0 − 1 = O |I0n,t |
k,t
k=1 |I0 |
where ζn (t) = ηn (t) + εn (t)‡ . The proof of Theorem 3.2 now follows from Theorem
2.2.
4. Uniform estimates for εn (t), ηn (t) and τn (t). In the following theorem we
will show that the sequences εn (t), ηn (t) and τn (t) tend to zero uniformly in t ∈ I
as n tends to infinity.
Theorem 4.1. Let ε̃n = supt∈I εn (t), η̃n = supt∈I ηn (t) and τ̃n = supt∈I τn (t).
Then
lim ε̃n = 0, lim η̃n = 0 and lim τ̃n = 0.
n→∞
‡ In
n→∞
n→∞
the inequality (14) the estimate is εn+1 (t), however here we use the inequality
≤ εn (t).
dn (t)
ε
(t)
dn−1 (t) n+1
4600
H. AKHADKULOV, A. DZHALILOV AND K. KHANIN
Proof. To prove this theorem, we first show that Θ̃n = supt∈I kΘtn kp → 0 as
n → ∞. It is clear that Mtn (x) = Mtn−1 (x) on the short intervals of Ptn−1 . Therefore
Z p
t
kΘtn kpp =
(32)
Mn (x) − Mtn−1 (x) dx =
S1
qn−1 (t)−1 Z
t
Mn (x) −
X
Iin−2,t
i=0
1
|Iin−2,t |
Z
Iin−2,t
f 00 (y) p
dy dx.
f 0 (y)
Utilizing Hölder’s inequality and Fubini’s theorem we obtain
Z
Z
1
f 00 (y) p
t
dy dx ≤
Mn (x) − n−2,t
| Iin−2,t f 0 (y)
|Ii
Iin−2,t
Z
Z
1
f 00 (y) p t
(x)
−
dx dy.
M
n
f 0 (y)
|Iin−2,t | Iin−2,t
Iin−2,t
From (2) and Hölder’s inequality it follows
Z
Z
f 00 (y) p 1
t
(x)
−
M
dx dy ≤
n
f 0 (y)
|Iin−2,t | Iin−2,t
Iin−2,t
Z
f 00 (s) f 00 (y) p X
Z
1
1
− 0
dy ≤
0
ds
n,t |p−1
|I
f
(s)
f
(y)
n,t
|Iin−2,t | Iin−2,t
I
n−2,t
(33)
(34)
I n,t ⊂Ii
1
Z
|Iin−2,t |
Iin−2,t
1
Z
X
I n,t ⊂Iin−2,t
f 00 (s) f 00 (y) p − 0
0
ds dy =
f (y)
I n,t f (s)
Z
f 00 (s) f 00 (y) p − 0
0
ds dy.
n−2,t
f
(s)
f
(y)
Ii
Z
|Iin−2,t | Iin−2,t
It is well known that continuous functions with compact support are dense in Lp .
Therefore for any > 0 there exists a continuous function W which has compact
support and an Lp integrable function V such that
f 00
= W + V f0
and kV kp ≤ .
Taking sufficiently small > 0 and using the above expansion of f 00 /f 0 we get
Z
f 00 (s) f 00 (y) p Z
1
− 0
(35)
0
ds dy ≤
n−2,t
f
(s)
f
(y)
|Iin−2,t | Iin−2,t
Ii
Z
Z
p Z
2p−1
p−1
|V (y)|p dy.
W (s) − W (y) ds dy + 2 · 4
|Iin−2,t | Iin−2,t
Iin−2,t
Iin−2,t
It is easy to see that
Z
p Z
2p−1
n−2,t
(s)
−
W
(y)
| · ω p (den−2 , W ) (36)
W
ds dy ≤ 2p−1 |Ii
|Iin−2,t | Iin−2,t
Iin−2,t
where ω(·, W ) is the modulus of continuity of W . Finally, summing relations (32)(36) we obtain
kΘtn kpp ≤ 4p (ω p (den−2 , W ) + p ).
This implies
Θ̃pn ≤ 4p (ω p (den−2 , W ) + p ).
NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN
4601
Taking the limit as n → ∞ we get
lim sup Θ̃n ≤ 4.
n→∞
Since > 0 is arbitrarily small, we have
lim Θ̃n = 0.
n→∞
Next, we estimate ε̃n . According to Theorem 2.2 there exists a constant C1 > 0
such that
∞
n
X
q
λk−(n+1) Θ̃k .
ε̃n ≤ C1 λ +
k=n+1
Let Θ̌n = sup{Θ̃m : m ≥ n}. It is easy to see that Θ̌n ≥ Θ̌n+1 for all n ≥ 1 and
lim Θ̌n = 0. By monotonicity of Θ̌n and the above inequality, we have
n→∞
∞
n
X
n
C1
λk−(n+1) Θ̃k ≤
(λ q + Θ̌n+1 ).
ε̃n ≤ C1 λ q +
1−λ
(37)
k=n+1
Hence lim ε̃n = 0. Next, we estimate η̃n . Due to Theorem 2.2, the monotonicity of
n→∞
Θ̌n and inequality (37) we get
n
η̃n ≤
X
k−1
C13
λ2(n−k) (λ q + Θ̌k ) ≤
2
λ (1 − λ)
(38)
k=1
[n]
2
n−1 X
C13
λ2(n−k) Θ̌k +
nλ q +
2
λ (1 − λ)
k=1
n
X
λ2(n−k) Θ̌k
k=[ n
2 ]+1
n−1
C13
n
q
n
nλ
+
λ
Θ̌
+
Θ̌
1
[
]
2
λ2 (1 − λ)2
where [·] is the integer part of a given number. Therefore lim η̃n = 0. Now we are
n→∞
going to estimate τ̃n . From Theorem 2.2 it follows that
n
n
X
X
τ̃n ≤ C1
λn−k ε̃k +
λn−k η̃k .
≤
k=1
k=1
Inequalities (37), (38) and Θ̌k ≤ Θ̌[ k ] imply
2
n
X
λn−k ε̃k +
k=1
n
X
n
λn−k η̃k ≤
k=1
X
C13
2 n−1
n−k
q (2 + Θ̌ ) +
n
λ
λ
Θ̌
.
k
1
[
]
2
λ2 (1 − λ)2
k=1
Similarly to the proof of inequality (38) it can be shown that
n
X
2 [n]
λ 2 Θ̌0 + Θ̌[ n4 ] .
λn−k Θ̌[ k ] ≤
2
1−λ
k=1
Hence, the last three inequalities imply
n
2C14
2 n−1
q (2 + Θ̌ ) + λ[ 2 ] Θ̌ + Θ̌ n
n
λ
τ̃n ≤ 2
.
1
0
[
]
4
λ (1 − λ)3
Thus lim τ̃n = 0, which concludes the proof of Theorem 4.1.
n→∞
4602
H. AKHADKULOV, A. DZHALILOV AND K. KHANIN
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 0 has bounded variation and the rotation number ρ is irrational. Assume
∞
X
(an Kn )2 < ∞.
n=1
Then the conjugating map h between f and Rρ and its inverse h−1 are absolutely
continuous and h0 , (h−1 )0 ∈ L2 .
Proof of Theorem 1.2. Let the diffeomorphism f satisfy the KO conditions and (vn )
be an unbounded sequence of natural numbers. According to Theorem 4.1 we can
find a subsequence (τ̃in ) of (τ̃n ) such that nvn τ̃in ≤ 1 for all n = 1, 2, ... . Without loss
of generality we may assume that the sequence (in ) is a strictly increasing sequence.
Let I(in , vn ) be the set of irrational numbers which was defined in Section 1. It
is clear that for any ρb ∈ I(in , vn ) there exists a natural number M, such that ρb ∈
I(in , vn , M ). Now we consider the family of diffeomorphisms ft = f + t, t ∈ I. Note
that ρ(ft ) is a continuous and nondecreasing function of t. Moreover, it is strictly
increasing at irrational values (see [8]). Due to this note for any ρb ∈ I(in , vn , M )
there exits a unique t0 ∈ I such that ρ(ft0 ) = ρb. By Theorem 3.2 and Lemma 3.1
we get
∞
X
(39)
Kn (t0 ) ≤ C3 · τn (t0 ) and
K2n (t0 ) < ∞.
n=1
On the other hand we have
τn (t0 ) ≤ τ̃n .
(40)
Inequalities (39), (40) and nvn τ̃in ≤ 1 imply
∞
∞
∞
X
X
X
2
2
(as Ks (t0 )) =
(as Ks (t0 )) +
(ain Kin (t0 ))2
s=1
s=1
n=1
s6=in
≤ M2
∞
X
K2s (t0 ) + C32
s=1
∞
X
(vn τ̃in )2 < ∞.
n=1
s6=in
Thus, the claim of the theorem follows from Theorem 5.1.
6. Theorem of Weiss-Zygmund. In this section we provide brief facts about
continuous functions K : R1 → R1 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 ∈ ( 21 , 1]. The following theorem was proved by Weiss
and Zygmund in [19].
Theorem 6.1. Let K : R1 → R1 be 1-periodic and continuous on R1 . Assume that
for some γ ∈ ( 21 , 1] the function K satisfies the inequality
k∆2 K(·, ν)kL∞ ([0,1]) ≤ CΦγ (ν).
0
Then K is absolutely continuous and K ∈ Lp ([0, 1]) for every p > 1.
(41)
NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN
4603
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, 12 ].
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.
Theorem 6.2. Let K : R1 → R1 be 1-periodic and continuous on R1 . Assume that
the function K satisfies (41) for some γ > 1. Then K ∈ C 1 (R1 ).
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.
Proof. According to Weiss-Zygmund theorem K is absolutely continuous, hence K0
exists almost everywhere and K is the antiderivative of K0 . To prove the theorem we
take any two points ξ and η that are Lebesgue points for K0 such that ξ − η ∈ (0, 1)
and obtain a uniform estimate for |K0 (ξ)−K0 (η)|. Hence we show that K0 is uniformly
continuous on its set of Lebesgue points. Thus, it can be continuously extended to
the whole of R1 . Let us consider the function Dτ K(x) = K(x + τ ) − K(x) where
x ∈ R1 and τ ∈ (0, 1). By inequality (41) we have
Dτ K(x) = Dτ K(x − τ ) + O(τ Qγ (τ ))
for all x ∈ R1 and τ ∈ (0, 1),
(42)
where Qγ (τ ) = Φγ (τ )/τ . We set τ := ξ − η. Replacing x by xn = η + τ 2−n and τ
by τ 2−n , n = 1, 2, ... in (42), we obtain
Dτ 2−n K(xn ) = Dτ 2−n K(η) + O(τ 2−n Qγ (τ 2−n )).
(43)
It is easy to see
Dτ 2−n K(xn ) − Dτ 2−n K(η) = Dτ 2−n+1 K(η) − 2Dτ 2−n K(η).
Thus
Dτ 2−n+1 K(η) = 2Dτ 2−n K(η) + O(τ 2−n Qγ (τ 2−n )).
By iterating from n = 1 to N we obtain
N
X
2N
Dτ K(η)
Qγ (τ 2−n ) .
=
Dτ 2−N K(η) + O
τ
τ
n=1
(44)
Since the point η is the Lebesgue point for K0 and γ > 1
2N
Dτ 2−N K(η) = K0 (η).
N →∞ τ
Taking the limit as N → ∞ in (44) we get
lim
Dτ K(η)
= K0 (η) + O(Pγ (τ )),
τ
where
Pγ (τ ) =
∞
X
(45)
Qγ (τ 2−n ).
n=1
Similarly, replacing x by xn = ξ − τ 2−n and τ by τ 2−n , n = 1, 2, ... in (42) we
obtain
Dτ K(ξ − τ )
= K0 (ξ) + O(Pγ (τ )).
(46)
τ
4604
H. AKHADKULOV, A. DZHALILOV AND K. KHANIN
Taking τ = ξ − η we get
|K0 (ξ) − K0 (η)| = O Pγ (|ξ − η|) .
This proves uniform continuity of K0 on the set of Lebesgue points, thus K0 coincides
almost everywhere with a continuous function. It is obvious that this continuous
function is a derivative of K.
It turns out that the functions satisfying relation (41) have “a considerable degree
of continuity”.
Theorem 6.3. Let K : R1 → R1 be 1-periodic and continuous on R1 , and satisfies
the inequality (41) for some γ ∈ (0, 1]. If γ ∈ (0, 1) then
1
ω(δ, K) = O δ(log )1−γ .
δ
(47)
1 ω(δ, K) = O δ(log log ) ,
δ
(48)
If γ = 1 then
where ω(·, K) is the modulus of continuity of K.
Proof. The proof follows closely to that of Theorem 6.2. Let us take any η ∈ R1
and fix it. Taking any ξ ∈ (η, η + 1) we set τ := ξ − η. In the same way as in the
proof of Theorem 6.2 we obtain
N
X
Qγ (τ 2−n )
Dτ K(η) = 2N Dτ 2−N K(η) + O τ
(49)
n=1
for any N ∈ N and τ ∈ (0, 1). Suppose δ > 0 is small. Choose a number n0 ∈ N
such that 21 < 2n0 δ < 1. Taking N := n0 , τ := 2n0 δ i.e., ξ = 2n0 δ + η one has
Dδ K(η) =
N
X
D2N δ K(η)δ
N −n
Q
(2
δ)
.
+
O
δ
γ
2N δ
n=1
(50)
It is clear that
D N K(η) 2 δ
≤ 4 max |K(x)|
2N δ
x∈[0,1]
and
N
X
n=1
Qγ (2N −n δ) = 2
N
X
Qγ (2N −n δ)
≤2
2N −n+1 2−(N −n)
n=1
1
Z
1
2−N
Qγ ( xδ )
dx.
x
Hence
|Dδ K(η)| = O
Z
1
2−N
δQγ ( xδ ) dx .
x
It is easy to see that the last integral is estimated by δ(log 1δ )1−γ if γ ∈ (0, 1) and
δ(log log 1δ ) if γ = 1. That concludes the proof of Theorem 6.3.
NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN
4605
7. Sharp estimate for Denjoy’s inequality. In this section we obtain a sharp
estimate for Kn when f ∈ ZΦγ and γ > 21 . For this we use the same strategy as
above. Define a function Ω : (0, 1) × (0, +∞) → R,

δ(log 1δ )1−γ if (δ, γ) ∈ (0, 1) × (0, 1);





δ(log log 1δ ) if (δ, γ) ∈ (0, 1) × {1};
Ω(δ, γ) =





δ
if (δ, γ) ∈ (0, 1) × (1, +∞).
In fact, Ω(δ, γ) is the modulus of continuity of functions satisfying inequality (41).
Using this function we define the following sequences:
Λn = Qγ (dn−1 ) + Ω(dn−1 , γ),
Υn =
n
X
dn−1
dk−1
k=1
Ψn =
n
X
dn k=1
dk
·
γ>
1
, n ≥ 0,
2
dn
Λk−1 , n ≥ 1,
dk
Υk + Λk , n ≥ 1,
where dn = kf qn − Idk0 . Now we provide a sharp estimate for Denjoy’s inequality.
Theorem 7.1. Let f ∈ ZΦγ , γ > 12 be a diffeomorphism of the circle with irrational
rotation number. There exists a constant C6 = C6 (f ) > 0 such that
Kn ≤ C6 · Ψn .
To prove this theorem we use the same strategy as in Theorem 3.2, that is using
Zygmund condition we strengthen the estimates of Lemmas 3.3 and 3.4, so that
they imply the proof of Theorem 7.1.
Lemma 7.2. Let f satisfy the conditions of Theorem 7.1. Then for any ξ0 ∈ S 1
there exists a constant C7 = C7 (f ) > 0 such that
n −1 Z
n−1 qX
f 00 (x) log R(ξ0 , I n−1 ; f qn ) − (−1)
dx ≤ C7 · Λn ,
(51)
0
2
f 0 (x)
s=0
Isn−1
n qn−1
X−1 Z f 00 (x) log R(ξ0 , I0n ; f qn−1 ) − (−1)
dx ≤ C7 · Λn+1 .
2
f 0 (x)
s=0
(52)
Isn
Proof. We prove only the first inequality. The second one can be proved similarly.
Since R(ξ0 , I0n−1 ; f qn ) is multiplicative with respect to composition, we have
log R(ξ0 , I0n−1 ; f qn ) =
qX
n −1
s=0
log
h f (η ) − f (ξ ) 1 i
s
s
,
ηs − ξs
f 0 (ξs )
(53)
where ηs = f s (ξqn−1 ) and ξs = f s (ξ0 ) are the end-points of interval Isn−1 . Setting
νs = (ηs + ξs )/2, ϑs = (ηs − ξs )/2 for every 0 ≤ s < qn and using inequality (1) one
has
Z ϑs 1
f (ηs ) − f (ξs ) 1
0
0
=
f
(ν
+
x)
+
f
(ν
−
x)
dx =
(54)
s
s
η s − ξs
f 0 (ξs )
2ϑs f 0 (ξs ) 0
4606
H. AKHADKULOV, A. DZHALILOV AND K. KHANIN
Z ϑs 1
f 0 (ηs ) + f 0 (ξs )
0
2f
=
(ν
)
+
O(Φ
(x))
dx
=
+ O(Φγ (|Isn−1 |)).
s
γ
2ϑs f 0 (ξs ) 0
2f 0 (ξs )
Taking the logarithm, we obtain
h f (η ) − f (ξ ) 1 i
s
s
log
η s − ξs
f 0 (ξs )
(55)
0
0
f (ηs ) − f (ξs )
n−1
0
0
2
=
+ O |f (ηs ) − f (ξs )| + O Φγ (|Is |) .
2f 0 (ξs )
According to Weiss-Zygmund theorem and Theorem 6.2 the function f 0 is absolute
continuous and f 00 ∈ Lp (S 1 ) for every p > 1 in the case γ ∈ ( 12 , 1] and it is
differentiable in the case γ > 1. Therefore
Z ηs 00
Z ηs 00
Z
f 0 (ηs ) − f 0 (ξs )
f (x)
f (x) x f 00 (z) (56)
=
dx +
dz dx.
0
0
0
2f 0 (ξs )
ξs 2f (x)
ξs 2f (x)
ξs f (ξs )
By Theorems 6.2 and 6.3 we have
qX
n −1 h
2 Z ηs f 00 (x) Z x f 00 (z) i
0
dz
dx
=
O
Ω(d
,
γ)
. (57)
f (ηs ) − f 0 (ξs ) + n−1
0
0
ξs 2f (x)
ξs f (ξs )
s=0
It is clear
Φγ (|Isn−1 |) ≤ |Isn−1 | · Qγ (dn−1 )
for all
0 ≤ s < qn .
Hence
qX
n −1
Φγ (|Isn−1 |) ≤ Qγ (dn−1 ).
(58)
s=0
Summing (53)-(58) we obtain
log R(ξ0 , I0n−1 ; f qn ) =
qX
n −1
(−1)n−1
s=0
Z
f 00 (x)
dx
+
O
Ω(d
,
γ)
+
Q
(d
)
n−1
γ n−1 .
2f 0 (x)
Isn−1
(59)
This proves the inequality (51).
Lemma 7.3. Let f satisfy the conditions of Theorem 7.1. Then for any ξ0 ∈ S 1
there exists a constant C8 = C8 (f ) > 0 such that
"
#
qn −1 Z
|Iqn−1
| |I0n−1 | − |I0n+1 |
(−1)n X
f 00 (x) n
·
−
log
dx ≤ C8 · Λn ,
n−1
n−1
n+1
2
f 0 (x)
|I0 | |Iqn | − |Iqn |
s=0
Isn+1
"
#
qn −1 Z
|Iqn+1
| |I0n−1 | − |I0n+1 |
(−1)n X
f 00 (x) n
·
−
dx ≤ C8 · Λn .
log
2
f 0 (x)
|I0n+1 | |Iqn−1
| − |Iqn+1
|
n
n
s=0
Isn−1
Proof. The proof of this lemma is similar to the proof of Lemma 3.4.
8. Proof of Theorems 1.3 and 1.4. To prove Theorem 1.3 we use the sufficient
condition provided in Theorem 5.1 and to prove Theorem 1.4 we use the following
sufficient condition for C 1 -smoothness of the conjugacy which was developed by
Khanin and Sinai in [12].
NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN
4607
Theorem 8.1. Let the diffeomorphism f satisfies the Denjoy’s conditions. Assume
that
∞
X
an Kn < ∞.
(60)
n=1
Then the conjugating map h between f and Rρ and its inverse h−1 are C 1 diffeomorphisms.
We use the following elementary lemma in the proof of Theorems 1.3 and 1.4.
Lemma 8.2. The following estimate holds
1 Ψn = O γ .
n
Proof. The proof of this lemma follows from Theorem 2.2 and the definitions of
Qγ (·) and Ω(·, γ).
Proof of Theorems 1.3 and 1.4. Let f ∈ ZΦγ be a circle diffeomorphism with
irrational rotation number ρ and γ ∈ (1/2, 1]. Suppose that for some α ∈ (0, γ − 12 )
the partial quotients of ρ satisfies an ≤ Cnα . According to Theorem 7.1 and Lemma
8.2 we have
∞ ∞
2
X
X
1
.
an Kn ≤ C
2(γ−α)
n
n=1
n=1
Since γ − α > 1/2 the last sum converges. Thus the claim of Theorem 1.3 implies
from Theorem 5.1. Similarly, if γ > 1 and for some α ∈ (0, γ − 1) the partial
quotients of ρ satisfies an ≤ Cnα then
∞
∞
X
X
1
an Kn ≤ C
.
γ−α
n
n=1
n=1
Since γ − α > 1 the last sum converges. By Theorem 8.1, h and h−1 are C 1
diffeomorphisms. Theorems 1.3 and 1.4 are proved.
9. Extensions of main theorems. One of the main results of this paper is an
extension the main result of Katznelson and Ornstein [10] to the larger class of
rotation numbers within the specific family ft = f + t, t ∈ [0, 1]. In fact, this result
remains valid for any family of circle diffeomorphisms f : S 1 × T → S 1 , where T is
a parameter space, which we assume to be a compact. Denote by ft (x) = f (x, t).
Thus, the following holds.
Theorem 9.1. Let f : S 1 × T → S 1 be a family of circle diffeomorphisms. Assume
(i)
(ii)
ft00 (x)
ft0 (x) ∈ Lp , p > 1 for any
f 00 (x)
t → ft0 (x) is a continuous
t
t ∈ T;
mapping from T into Lp , p > 1.
Then there exists a subset U of irrational numbers of unbounded type such that if
ρ(ft ) ∈ U for some t ∈ T then the conjugation map h between ft and Rρ(ft ) and its
inverse h−1 are absolute continuous and h0 , (h−1 )0 ∈ L2 .
Since S 1 × T is compact, the condition (ii) implies ṽ = supt∈T V arS 1 log ft0 < ∞.
Hence λ̃ = supt∈T (1+e−v(t) )−1/2 < 1. These uniform estimates ensure uniformity of
constants in Theorem 2.2. Further, again using compactness of T and the condition
f 00 (x)
(ii) one can show that ft0 (x) can be approximated by continuous functions such
t
that the modulus of continuities of those functions tend to zero uniformly in t. This
4608
H. AKHADKULOV, A. DZHALILOV AND K. KHANIN
implies uniform estimates for εn (t), ηn (t) and τn (t). The rest of the proof of this
theorem follows exactly the proof of Theorem 2.2.
Theorem 1.3 extends Katznelson and Ornstein’s theorem [10] to a larger class
of rotation numbers for the circle diffeomorphisms satisfying inequality (1). The
function Φγ in the inequality (1) has been chosen in specific form in order to describe
the set of rotation numbers in precise form. Theorem 1.3 can be extended in the
following way. Let Φ : [0, ς] → [0, +∞) be a non-decreasing function satisfying
Z ς 2
Φ (s)
ds < ∞.
(61)
s
0
Consider a set of circle diffeomorphisms f, such that those derivatives f 0 satisfy
k∆2 (f 0 (·, τ ))kL∞ (S 1 ) ≤ Cτ Φ(τ ),
(62)
for some constant C > 0. It is obvious that this class is wider than the class
considered in Theorem 1.3 and therefore the following theorem extends it.
Theorem 9.2. Let f be a circle diffeomorphism such that its derivative f 0 satisfies
the inequality (62) with some Φ satisfying (61). Then there exists a subset U of
irrational numbers of unbounded type such that if ρ(f ) ∈ U then the conjugating
map h between f and Rρ(f ) and its inverse h−1 are absolute continuous and h0 ,
(h−1 )0 ∈ L2 .
Note that Weiss - Zygmund’s theorem holds for functions satisfying inequality
(62) (see [15]). Using this fact and inequalities (61) and (62) one can show that
there exists a sequence (βn ) ∈ `2 such that
Kn ≤ Cβn .
(63)
Hence the claim of this theorem follows from Theorems 1.2 and 5.1. If we assume
that the function Φ in (62) satisfies
Z ς
Φ(s)
ds < ∞,
(64)
s
0
then it can be shown that the sequence (βn ) in (63) belongs to `1 . Therefore using
Theorems 1.2 and 8.1 one can prove the following theorem.
Theorem 9.3. Let f be a circle diffeomorphism such that its derivative f 0 satisfies
the inequality (62) with some Φ satisfying (64). Then there exists a subset U of
irrational numbers of unbounded type such that if ρ(f ) ∈ U then the conjugating
map h between f and Rρ(f ) and its inverse h−1 are C 1 diffeomorphisms.
This theorem extends Theorem 1.4.
Acknowledgments. The authors would like to acknowledge the grants: Fundamental Research Grant Scheme (FRGS) S/O 13558 and TWAS grant 14-121RG/MATHS/ AS− G UNESCOFR: 324028604, for the financial support. We also
wish to express our thanks to the referee for providing us with helpful comments.
REFERENCES
[1] V. I. Arnol’d, Small denominators: I. Mappings from the circle onto itself, Izv. Akad. Nauk
SSSR, Ser. Mat., 25 (1961), 21–86.
[2] W. de Melo and S. van Strien, A structure theorem in one-dimensional dynamics, Ann. Math.,
129 (1989), 519–546.
[3] R. Durrett, Probability Theory and Examples, Second edition. Duxbury Press, Belmont, CA,
1996.
NOTES ON A THEOREM OF KATZNELSON AND ORNSTEIN
4609
[4] J. Hawkins and K. Schmidt, On C 2 -diffeomorphisms of the circle which are of type III1 ,
Invent. Math., 66 (1982), 511–518.
[5] M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations,
Inst. Hautes Etudes Sci. Publ. Math., 49 (1979), 5–233.
[6] J. Hu and D. Sullivan, Topological conjugacy of circle diffeomorphisms, Ergodic Theory Dynam. Systems, 17 (1997), 173–186.
[7] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl.
Math., 14 (1961), 415–426.
[8] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,
Cambridge University Press, Cambridge, 1995.
[9] Y. Katznelson and D. Ornstein, The differentiability of the conjugation of certain diffeomorphisms of the circle, Ergod. Theor. Dyn. Syst., 9 (1989), 643–680.
[10] Y. Katznelson and D. Ornstein, The absolute continuity of the conjugation of certain diffeomorphisms of the circle, Ergod. Theor. Dyn. Syst., 9 (1989), 681–690.
[11] K. M. Khanin and Ya. G. Sinai, A new proof of M. Herman’s theorem, Commun. Math.
Phys., 112 (1987), 89–101.
[12] K. M. Khanin and Ya. G. Sinai, Smoothness of conjugacies of diffeomorphisms of the circle
with rotations, Russ. Math. Surv., 44 (1989), 69–99, translation of Usp. Mat. Nauk., 44
(1989), 57–82.
[13] K. M. Khanin and A. Yu. Teplinsky, Herman’s theory revisited, Invent. Math., 178 (2009),
333–344.
[14] J. Moser, A rapid convergent iteration method and non-linear differential equations. II, Ann.
Scuola Norm. Sup. Pisa, 20 (1966), 499–535.
[15] E. M. Stein, Singular Integrals and Differentaibility Properties of Functions, Princeton University Press, Princeton, N.J., 1970.
[16] D. Sullivan, Bounds, quadratic differentials and renormalization conjectures, American Mathematical Society Centennial Publications, (Providence, RI, 1988), Amer. Math. Soc., Providence, RI, 2 (1992), 417–466.
[17] G. Świa̧tek, Rational rotation number for maps of the circle, Commun. Math. Phys., 119
(1988), 109–128.
[18] A. Teplinsky, On cross-ratio distortion and Schwartz derivative, Nonlinearity, 21 (2008),
2777–2783.
[19] M. Weiss and A. Zygmund, A note on smooth functions, Indag. Math., 21 (1959), 52–58.
[20] J. C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de
rotation vérifie une condition diophantienne, Ann. Sci. École Norm. Sup., (4) 17 (1984),
333–359.
[21] A. Zygmund, Trigonometric Series, Third edition, Cambridge University Press, 2002.
Received June 2016; revised April 2017.
E-mail address: [email protected]
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E-mail address: [email protected]