4 Proof of Main Result

On the invariant measure of a piecewise-smooth circle homeomorphism of Zygmund class

Sokhobiddin Akhatkulov^∗2, Mohd Salmi Md Noorani², Habibulla Akhadkulov² Abstract

We prove that the invariant probability measure of an orientation preserving circle homeomorphism with several break points (i.e. derivatives have jumps at these points) such that the product of jumps at these break points is non-trivial and satisfies certain Zygmund condition, is singular with respect to Lebesgue measure.

1 Introduction

LetS¹ =R/Zbe the unit circle with clearly defined orientation, metric, Lebesgue measure and the operation of addition. Let π : R → S¹ denote the corresponding projection mapping that ”winds” a straight line on the circle. An arbitrary homeomorphismf that preserves the orientation of the unit circle S¹ can be ”lifted” on the straight line R in the form of the homeomorphism F :R → R with property F(x+ 1) =F(x) + 1 that is connected withf by relationπ◦F =f◦π.This homeomorphismF is called thelift of the homeomorphism f and is defined up to an integer term. The most important arithmetic characteristic of the homeomorphismf of the unit circleS¹ is the rotation number

ρ(f) = lim

n→∞

Fⁿ(x)

n mod1

whereF is the lift off withS¹ toR. Here and below, for a given mapF, Fⁿ denotes its n-th iterate. The rotation number is rational if and only iff has periodic points. Poincare proved that, if f does not have any periodic orbit then it is semi-conjugate to the linear rotation fρ : x → x+ρ mod 1. Denjoy [2] proved that, if f is a circle diffeomorphism with irrational rotation number ρ = ρ(f) and logf⁰ is of bounded variation, then f is topologically conjugate to the linear rotationf_ρ that is, there exists an essentially unique homeomorphism ϕ of the circle with ϕ◦f = fρ◦ϕ. This classical result can be easily extended in the case of a homeomorphism with singularities of break-point type.

It is well known that every circle homeomorphism with irrational rotation number ρ has a unique f−invariant probability measure µf. Furthermore, the conjugation ϕ and the invariant probability measure µ_f are connected by the relation ϕ(x) = µ_f([0, x]), x ∈ S¹ (see, for example, [1]). Because of this relation, the invariant measure µ_f is absolutely continuous with respect to the Lebesgue measurel if and only if ϕis given by an absolutely continuous function. Dzhalilov and Khanin in [3] proved that the invariant probability measures of C²⁺, > 0 circle homeomorphisms with one break point and irrational rotation number is singular with respect to Lebesgue measure. Later Dzhalilov et. al. in [4] extended this result for the case of circle homeomorphisms with several break points which the first derivative of the circle homeomorphism is absolutely continuous function on every connected components of the circle and the product of jumps at break

1MSC: 37E10, 37C15, 26D99 Keywords and phrases: break point, circle homeomorphism, invariant measure, rotation number.

2School of Mathematical Sciences, Faculty of Science and Technology, University Kebangsaan Malaysia.

43600 UKM Bangi, Selangor Darul Ehsan, Malaysia. e-mail: [email protected], [email protected]

∗Corresponding author, e-mail: [email protected]

points is non-trivial, that is not equal to one. Now we introduce a new class of circle homeomorphisms and we study, in this paper, the invariant probability measure of the circle homeomorphisms with beak points belonging to this new class. In particular, we extend the result of [4] to this class.

Let ψ :S¹ →S¹ be a continuous, non decreasing function with ψ(0) = 0. Using this function we define a class of orientation preserving circle homeomorphismsf such that (1) |f(x+t) +f(x−t)−2f(x)| ≤Ctψ(t)

for all x, t ∈S¹ such that x−t, x+t∈ S¹, here C > 0 is a constant. The class of real functions satisfying (1) with ψ ≡ 1 on real line is called Zygmund class and denoted by Λ∗ (see [9]). This class plays a key role to investigate the trigonometric series. The class Λ∗ was applied to the circle homeomorphisms for the first time by Jun Hu and Sullivan [6]. They extended the classical Denjoy’s theorem to this class. The functions satisfying (1) are not of bounded variation at all, the reverse also is not true. For example let us consider Weieratrass function:

W_β(x) =

∞

X

n=1

θnb^−nβcos(bⁿx) where b >1 and lim

n→∞θ_n = 0. The following fact can be found in [9]. Weieratrass proved that for a small enough β > 0 the functionWβ is nowhere differentiable. The extension toβ ≤1 was fist proved by Hardy. For β >1 the function W_β⁰ exists and continuous. If the sum of squares of the sequence θ_n is divergence then W₁ is differentiable in a set of measure zero. Thus making b even number and instead of θn taking the sequence n^−1/2 easily we may check that the function W1 satisfies the condition (1) but almost nowhere differentiable. The main result of this paper is as follows.

Let f : S¹ → S¹ be an orientation preserving circle homeomorphism with irrational rotation number and satisfies the following conditions:

1) f is differentiable except at finitely many points b1, b2, ..., bn ∈S¹, the so called break points of f, at which the left and right derivatives (denoted by f₋⁰ (bi), f₊⁰ (bi), i = 1,2, ..., n, respectively) are defined andf₋⁰ (b_i)/f₊⁰ (b_i) :=σ(b_i)6= 1, i= 1, ..., n;

2) logf⁰ has bounded variation on S¹ and f⁰ satisfies the condition (1);

3)

n

Q

i=1

σ(b_i)6= 1.

Where the numberσ(bi) is called jump off at the break pointbi.

Theorem 1.1. If a circle homeomorphism f with irrational rotation number satisfies the above conditions 1)-3) then its invariant probability measure µ_f is singular with respect to Lebesgue measure l.

2 Necessary facts and definitions

We consider a circle homeomorphismf that preserves orientation and has irrational rota- tion numberρ. Let {a_k, k∈N}denote the sequence of elements in the expansion ofρinto a continued fraction, that is, ρ = [a₁, a₂, ..., a_n, ...]. We set p_n/q_n = [a₁, a₂, ..., a_n], n ≥1.

The numbers pn/qn are called the convergents of ρ, and qn is the first return time. The numbers q_n satisfy the recurrence relation q_n+1 = a_n+1q_n+qn−1, n ≥1, with the initial conditions q0 = 1 and q1 = a1.For an arbitrary point x0 ∈ S¹, let I₀⁽ⁿ⁾(x0) denotes the closed interval with endpoints x₀ and x_qn =f^qn(x₀). Note that for odd n the point x_qn lies to the left of x₀, and for evenn to the right. We set I_i⁽ⁿ⁾=fⁱ(I₀⁽ⁿ⁾), i≥1.

Lemma 2.1. [8] Consider an arbitrary pointx0∈S¹.The segment{x_i,0≤i < qn+qn−1} of the trajectory of this point divides the circle into the following disjoint (except for the endpoints ) intervals: I_i⁽ⁿ⁾, 0≤i≤qn−1−1, I_i⁽ⁿ⁻¹⁾, 0≤i≤q_n−1.

We denote the resulting partition by ξ_n(x₀) and call it a dynamical partition of order n. We now describe the process of transition from ξ_n(x₀) to ξ_n+1(x₀). All the intervals I_j⁽ⁿ⁾, 0 ≤j ≤qn−1−1 are preserved, and each of the intervalsI_i⁽ⁿ⁻¹⁾, 0 ≤i≤qn−1 is divided into an+1+ 1 parts:

I_i⁽ⁿ⁻¹⁾(x0) =I_i⁽ⁿ⁺¹⁾(x0)∪

an+1−1

[

s=0

I_i+q⁽ⁿ⁾_n−1+sqn(x0).

Lemma 2.2. Consider a circle homeomorphismf with irrational rotation number. Sup- pose that at points b_i ∈ S¹, i = 1,2, ..., k, b₁ ≺ b₂ ≺ ... ≺ b_k, there exist positive and finite one-sided derivatives f₋⁰ (bi), f₊⁰(bi), f ∈ C¹([bi, bi+1]), i = 1,2, ..., k bi+1 =b1 and v=

Pk

i=1

var

[bi,bi+1]logf⁰+|lnσ(b_i)|<∞. Then the inequalities e^−v ≤

qn−1

Y

s=0

f⁰(y_s)≤e^v

hold for any y₀ such thatf^s(y₀)6=b_i, i= 1,2, ..., k, 0≤s < q_n.

These inequalities are called Denjoy’s inequalities. Lemma 2.2 is proved in the same fashion as the analogous assertion for diffeomorphisms (see [7]). It follows from this lemma that the intervals comprising the dynamical partition ξn(x0) have exponentially small lengths.

Corollary 2.3. Let Iⁿ be an arbitrary element of the dynamical partitionξn(x0). Then l(Iⁿ)≤C1λⁿ,

where the constant C₁ is independent of n, x₀ and λ= (1 +e^−v)^−1/2.

Denjoy’s Theorem [7].Suppose that the hypotheses of Lemma 2.2 hold. Then the homeomorphism f is topologically conjugate to the linear rotation f_ρ.

Definition 2.4. Let K > 1 be a constant. Two intervals I₁ and I₂ are said to be K- comparable on S¹ if the inequalities K⁻¹l(I2)≤l(I1)≤Kl(I2) hold.

Definition 2.5. An interval I = [τ, t] ⊂ S¹ is said to be q_n-small, and its endpoints qn-close, if the intervals fⁱ(I), 0≤i≤qn−1, are pairwise disjoint.

It follows from the structure of dynamical partitions that an interval I = [τ, t] is q_n-small if and only if eitherτ < t≤f^qn−1(τ) or f^qn−1(t)≤τ < t.

Now we mention a notion which is called cross-ratio distortion. This notion is the most powerful tool to study the existence and smoothness of conjugating map for the circle homeomorphisms with break points. Note that the cross-ratio distortions were used in dynamical systems for the firs time by Yoccoz [10]. Yoccoz showed the existence of conjugation for critical circle homeomorphisms.

Definition 2.6. The cross-ratio of four numbers (z₁, z₂, z₃, z₄), z₁< z₂ < z₃ < z₄,is the number

Cr(z1, z2, z3, z4) =(z2−z1)(z4−z3) (z3−z1)(z4−z2).

Definition 2.7. The cross-ratio distortion of four numbers for a strictly increasing func- tion F :R→Ris defined by

Dst(z₁, z₂, z₃, z₄;F) = Cr(F(z1), F(z2), F(z3), F(z4)) Cr(z₁, z₂, z₃, z₄) .

Let z1, z2, z3, z4 ∈ S¹, have lifts bz1,bz2,bz3,bz4 and z1 ≺ z2 ≺ z3 ≺ z4 ≺ z1 in the anti-clockwise order onS¹. Define z₁ =zb₁ and

z_i =

bzi, if zb1<bzi <1 1 +zb₁, if 0<zb₁ <bz_i

where i= 2,3,4. It is obvious thatz1 < z2 < z3 < z4. The vector (z1, z2, z3, z4) ∈R⁴ is called the lifted vector of (z₁, z₂, z₃, z₄)∈(S¹)⁴. Letf be a circle homeomorphism with lift F. We define the cross-ratio distortion of a four-tuple (z₁, z₂, z₃, z₄), z_i ∈S¹, i= 1, ...,4 z1≺z2 ≺z4≺z1, with respect to f, by

Dst(z1, z2, z3, z4;f) :=Dst(z1, z2, z3, z4;F).

3 Distortion lemmas and covering intervals theorem

In this section, we estimate the distortion of cross-ratios of four points, for the cases, when the break points of the circle homeomorphism f, are contained in an interval which created from end points of those four points and the break points are not contained in this interval. We also provide covering intervals theorem.

Let ω(δ;f) denotes a modulus of continuity of f in the closed interval I, that is ω(δ;f) = {sup|f(x₁) −f(x₂)| for x₁, x₂ ∈ I, |x₁ −x₂| ≤ δ}. If f⁰ satisfies (1) then ω(δ;f⁰) =o(δlog¹_δ) (see [9]).

Lemma 3.1. Suppose that a circle homeomorphism f satisfies the hypotheses of Theorem 1.1. Suppose also that zi ∈S¹, i= 1, ...,4 z1 ≺z2 ≺z3 ≺z4 ≺z1 and the interval [z1, z4] does not contain any break point of f. Then

(2) |Dst(z₁, z2, z3, z4;f)−1| ≤C2|z₄−z1|ψ(|z₄−z1|) +|f⁰(z4)−f⁰(z1)|ω(|z₄−z1|;f⁰) where the constant C2 depends on f.

Proof. We note that iff⁰ satisfies (1) then for each x, y∈S¹ f(x)−f(y)

x−y = 1

x−y

y

Z

x

f⁰(t)dt= f⁰(x) +f⁰(y)

2 +O(|z₁−z4|ψ(|z₁−z4|)).

This equality is proven in the same way that of [5]. Using this equality, we get

(3) f(z2)−f(z1) z2−z1

z4−z2

f(z4)−f(z2) = f⁰(z2) +f⁰(z1) +O(|z₂−z1|ψ(|z₂−z1|)) f⁰(z4) +f⁰(z2) +O(|z₄−z2|ψ(|z₄−z2|)) =

1−f⁰(z4)−f⁰(z1) f⁰(z4) +f⁰(z2)

1+O(|z₄−z₁|ψ(|z₄−z₁|))

=

1−f⁰(z4)−f⁰(z1) 2f⁰(z4)

1 1−^f0(z_2f^4)−f0(z4⁰)^(z2)

×

1 +O(|z₄−z₁|ψ(|z₄−z₁|))

=

1−f⁰(z₄)−f⁰(z₁)

2f⁰(z₄) (1 +O(f⁰(z₄)−f⁰(z₂)))

×

1 +O(|z₄−z1|ψ(|z₄−z1|))

= 1−f⁰(z4)−f⁰(z1)

2f⁰(z4) +O(|z₄−z1|ψ(|z₄−z1|).

In the same way can get that (4) z3−z1

f(z₃)−f(z₁)

f(z4)−f(z3)

z₄−z₃ = 1 +f⁰(z4)−f⁰(z1)

2f⁰(z₄) +O(|z₄−z1|ψ(|z₄−z1|).

From (3) and (4) we obtain f(z₂)−f(z₁)

z₂−z₁

z₄−z₂ f(z₄)−f(z₂)

z₃−z₁ f(z₃)−f(z₁)

f(z₄)−f(z₃)

z₄−z₃ = 1−f⁰(z₄)−f⁰(z₁) 2f⁰(z₄)

2

+O(|z₄−z1|ψ(|z₄−z1|).

Hence, from this equality and the modulus of continuity of f⁰ follows that

|Dst(z₁, z₂, z₃, z₄;f)−1| ≤const|z₄−z₁|ψ(|z₄−z₁|) +|f⁰(z₄)−f⁰(z₁)|ω(|z₄−z₁|;f⁰).

The lemma is proved withconst=C2.

Now we consider the case when the interval [z1, z4] contains just one break point bi0. More precisely, suppose that b_i0 lies outside the middle interval, that is b_i0 ∈ [z₁, z₂]∪ [z₃, z₄]. Suppose for definiteness that b_i0 ∈ [z₁, z₂]. We define the numbers α, β, γ, τ, η and z as follows: α:=z2−z1,β :=z3−z2,γ :=z4−z3,τ :=z2−bi0,η:= _α^β,ξ := _α^τ. Lemma 3.2. Suppose that a circle homeomorphism f satisfies the hypotheses of Theorem 1.1. Let z_i ∈S¹, i = 1, ...,4 with z₁ ≺z₂ ≺z₃ ≺z₄ ≺z₁. Suppose also that b_i0 ∈[z₁, z₂] and the other break points of f are not contained in [z₁, z₄]. Then

(5)

Dst(z1, z2, z3, z4;f)−(σ(bi0) + (1−σ(bi0)ξ))(1 +η) σ(b_i0) + (1−σ(b_i0))ξ+η

≤C₃|z₄−z₁|ψ(|z₄−z₁|) +ω(|z₄−z₁|;f⁰) where the constant C₃ >0 depends on f.

Proof. By assumption bi0 ∈[z1, z2].RewritingDst(z1, z2, z3, z4;f) in the form Dst(z1, z2, z3, z4;f) = Cr(f(z1), f(z2), f(z3), f(z4))

Cr(z₁, z₂, z₃, z₄) = f(z₂)−f(z₁)

z2−z1

· z₃−z₁ f(z3)−f(z1)

f(z₄)−f(z₃) z4−z3

· z₄−z₂ f(z4)−f(z2)

it is easy to check, that each multiplication in brackets equals to the following

(6) f(z₂)−f(z₁)

z₂−z₁ · z₃−z₁

f(z₃)−f(z₁) = f₊⁰(b_i0)(z₂−x_b) +f₋⁰ (b_i0)(x_b−z₁)

z₂−z₁ ×

z₃−z₁

f₊⁰(b_i0)(z₃−x_b) +f₋⁰ (b_i0)(x_b−z₁) = (σ(b_i0) + (1−σ(b_i0))ξ)(1 +η) σ(b_i0) + (1−σ(b_i0))ξ+η , whereσ(b_i0) = ^f

−0(bi0)

f₊⁰(bi0) the jump ratio off at the point b_i0. (7) f(z₄)−f(z₃)

z4−z3

· z₄−z₂

f(z4)−f(z2) =f⁰(z₄) +f⁰(z₃)

2 +O(|z₄−z₃|ψ(|z₄−z₃|)) : f⁰(z4) +f⁰(z2)

2 +O(|z₄−z2|ψ(|z₄−z2|))

= f⁰(z4) +f⁰(z3) +O(|z₄−z3|ψ(|z₄−z3|)) f⁰(z₄) +f⁰(z₂) +O(|z₄−z₂|ψ(|z₄−z₂|)) = 1 +f⁰(z3)−f⁰(z2)

f⁰(z₄) +f⁰(z₂) +O(|z₄−z1|ψ(|z₄−z1|)) From (6) and (7) we have

f(z₂)−f(z₁) z₂−z₁

z₃−z₁ f(z₃)−f(z₁)

f(z₄)−f(z₃) z₄−z₃

z₄−z₂

f(z₄)−f(z₂) = (σ(b_i0) + (1−σ(b_i0))ξ)(1 +η) σ(b_i0) + (1−σ(b_i0))ξ+η + f⁰(z3)−f⁰(z2)

f⁰(z4) +f⁰(z2)

(σ(bi0) + (1−σ(bi0))ξ)(1 +η)

σ(bi0) + (1−σ(bi0))ξ+η +O(|z₄−z1|ψ(|z₄−z1|)).

Hence, from this equality and the modulus of continuity of f⁰ follows that

Dst(z1, z2, z3, z4;f)−(σ(b_i0) + (1−σ(b_i0)ξ))(1 +η) σ(bi0) + (1−σ(bi0))ξ+η

≤const|z₄−z1|ψ(|z₄−z1|) +ω(|z₄−z1|;f⁰).

The lemma is proved withconst=C3.

Now we provide a theorem on covering intervals for the circle homeomorphisms with break points. Consider f with n break points b1, b2, ..., bn ∈ S¹ and irrational rotation number ρ. Suppose that all these break points lie in different orbits. If this were not the case, then we could achieve it by considering sufficiently high renormalizations. We set B(f) = {b₁, b2, ..., bn} and say a subset Bb ⊂ B(f) = {b₁, b2, ..., bn} is non-trivial if

Q

bi∈Bb

σ(b_i) 6= 1. We introduce the notion of a ’regular’ cover of the break points in B(f).

Suppose that z_i ∈ S¹, i = 1, ...,4, z₁ ≺ z₂ ≺ z₃ ≺ z₄ ≺ z₁ and r_n takes value in the

set {q_n−1, qn, qn−1+qn}.Suppose that the interval [z1, z4] is rn-small and the system of intervals {f^j([z₁, z₄]),0≤j < r_n−1} cover the elements ofBb. We denote the number of elements ofBb bym. For every elementb_is ∈Bbthere exists a numberl_is, 0≤l_is < r_nsuch that ¯b⁽ⁿ⁾_i

s =f^−lis(b_is) ∈ [z₁, z₄]. the point ¯b⁽ⁿ⁾_i

s is called the r_n- pre image of the element b_is in [z₁, z₄]. The set of r_n pre-images of elements of Bb also consists of m elements:

¯b⁽ⁿ⁾_i

1 ,¯b⁽ⁿ⁾_i

2 , ...,¯b⁽ⁿ⁾_i

m; we denote the maximal element of this set bybb⁽ⁿ⁾_t .Clearly,bb⁽ⁿ⁾_t = ¯b⁽ⁿ⁾_i

t for some 0≤t≤m.We introduce the following notations:

(8) η(j) = l([f^j(z2), f^j(z3)])

l([f^j(z₁), f^j(z₂)]), ξ^(is)(j) = l([f^j(¯b⁽ⁿ⁾_i

s ), f^j(z₂)]) l([f^j(z₁), f^j(z₂)]) where 1 ≤ s≤ m,0 ≤ j < r_n−1.In cases, where ¯b⁽ⁿ⁾_i

s ∈ [z₁, z₂], the numbersξ^is(j) are called normalized coordinates of the elements f^j(¯b⁽ⁿ⁾_is ). When the point ¯b⁽ⁿ⁾_is moves from z₂ toz₁, the normalized coordinateξ^is(j) varies from 0 to 1. It is easy to see that

e^−vη(0)≤η(j)≤e^vη(0), e^−vξ^is(0)≤ξ^is(j)≤e^vξ^is(0), i= 1,2, ..., n for all 0≤j < rn−1 and where v is the total variation of logf⁰ overS¹.

Definition 3.3. Let K > M ≥ 1, ζ ∈ (0,1), δ > 0 be constant numbers, let n be a positive integer and let x0 ∈ S¹. We say a triple of intervals ([z1, z2],[z2, z3],[z3, z4]), zi ∈ S¹, i = 1, ...4 (K, M, δ, ζ, x0)−regularly cover the break points in a subset Bb if for some r_n∈ {q_n−1, q_n, qn−1+q_n} the following conditions hold:

1) [z1, z4]⊂(x0−δ, x₀+δ)and the system of intervals{f^j([z1, z4]),0≤j < rn−1}covers every point in Bb only once.

2) z2=bb⁽ⁿ⁾_t and¯b⁽ⁿ⁾_is ∈[z1, z2), 1≤s≤n, s6=t.

3) M l([z₂, z₃])≤l([z₁, z₂])≤Kl([z₂, z₃])and K⁻¹l([z₃, z₄])≤l([z₂, z₃])≤Kl([z₃, z₄]).

4) The lengths of the intervals f^rn([z1, z2]), f^rn([z2, z3]) and f^rn([z3, z4]) are pairwise K−comparable.

5) max{l([f^rn(zi), x0]), l([zi, x0]), i= 1, ...,4} ≤Kl([z1, z2]).

6) max

1≤s≤m{z^(is)(0)}< ζ.

We now state a theorem on covering intervals which plays key role in the proof of main result. The proof of this theorem does not depend on the considered class of circle homeomorphisms and similar to the proof of [4]. That is why here we provide this theorem without proof.

Theorem 3.4. Suppose that a homeomorphism f satisfies the hypotheses of Theorem 1.1.

Let x0 ∈S¹ and let M ≥1,δ, ζ ∈(0,1)be constant numbers. Then there exists a constant K =K(f, M, ζ)> M such that for any sufficiently large n there exists non-trivial subset Bb = B(n) =b {b_i1, b_i2, ..., b_im}, points z_i ∈ S¹, i = 1, ...,4, z₁ ≺ z₂ ≺ z₃ ≺ z₄ ≺ z₁ and r_n =r_n(z₁, z₂, z₃, z₄) ∈ {qn−1, q_n, qn−1+q_n} such that the intervals [z_s, z_s+1], s = 1,2,3 (K, M, δ, ζ, x0)−regularly cover the break points of B.b

Note that in the proof of this theorem, the points z₁, z₂, z₃, z₄ were chosen and there was shown that the intervals [z1, z4] and [x0, xqn−1] are comparable. From this follows that the intervals [fⁱ(z1), fⁱ(z4)], i = 0,1, ..., rn cover S¹ finite times. We use from this statement in the proof of Lemma 4.3.

4 Proof of Main Result

Now we mention some necessary lemmas from [4] to prove Theorem 1.1.

Lemma 4.1. Suppose that at a point x = x₀ the conjugation ϕ has positive derivative, ϕ⁰(x₀) =ω₀ and the following conditions hold for some constant R₁ >1:

(i) the intervals [z₁, z₂],[z₂, z₃],[z₃, z₄] are pairwiseR₁−comparable.

(ii) max{|z₁−x0|,|z₄−x0|} ≤R1|z₁−z2|.

In this case, for any >0 there exist δ =δ(x₀, )>0 such that if all z_i, 1≤i≤4 belong to (x0−δ, x0+δ), the inequality

|Dist(z₁, z₂, z₃, z₄;ϕ)−1|< C₄

holds,where the constant C4 =C4(R1, ω0) depends onR1, ω0 and does not depend on. Before give next lemma we define the functions Fi(x, y), i= 1,2, ..., n on the domain {(x, y) :x >0, 0≤y≤1}as:

F_i(x, y) = [σ(b_i) + (1−σ(b_i))y](1 +x) σ(b_i) + (1−σ(b_i))y+x , whereσ(b_i) is the jump off at the point b_i.

Lemma 4.2. Let {b_i1, b_i2, ..., b_im} be an arbitrary non-trivial subset of break points of f, so that A :=

m

Q

s=1

σ(b_is)6= 1. Then there exist constants Ω₀ = Ω₀(σ_i1, σ_i2, ..., σ_im) >1 and τ₀ =τ₀(σ_i1, σ_i2, ..., σ_im)∈(0,1)such that the inequality

m

Y

s=1

F_is(x_s, y_s)−A

≤ |A−1|

8 holds for allx_s≥Ω₀, y_s∈[0, τ₀], s= 1,2, ..., m.

We useτ₀and Ω₀ to define two new constantsτ₀ and Ω₀, which will play an important role in the proof of Theorem 1.1. We set τ₀ = minτ₀(σ_i1, σ_i2, ..., σ_im) ∈ (0,1), Ω₀ = max Ω0(σi1, σi2, ..., σim) where the minimum and maximum are taken over all non-trivial subsets {b_i1, b_i2, ..., b_im}of break points of f.

Proof of Theorem 1.1. Suppose that a homeomorphism f satisfies the hypotheses of Theorem 1.1. Since the rotation number ρ is irrational, the invariant measure µf has no atoms and the conjugation ϕ(x) is given by monotonic function µ_f([0, x]), x ∈ S¹. The finite derivative ϕ⁰(x) of the conjugation exists by the monotonicity of the function ϕ(x) for almost all x with respect to Lebesgue measure. We claim that ϕ⁰(x) = 0 at all points x where the finite derivative exists. Suppose that ϕ⁰(x₀) = ω₀ > 0 at some point x₀ ∈ S¹. We fix > 0. Let δ = δ(x₀, ) > 0 be defined by Lemma 4.1. We use the constants Ω0 and τ0 to define new constants: M0 = Ω0e^v, ζ0 = τ0e^v where v > 0 is the total variation of logf⁰ overS¹. Let K0 = K0(f, M0, ζ0) > M0 > 1 be the constant defined Theorem 3.4. By that theorem, for sufficiently large n there exist a non-trivial subset Bb = {b_i1, bi2, ..., bim} of break points of f, points zi ∈ S¹, i = 1, ...,4 z1 ≺ z2 ≺ z3 ≺ z4 ≺ z1 and a number rn ∈ {q_n−1, qn, qn−1 +qn} such that the triple of intervals ([z₁, z₂],[z₂, z₃],[z₃, z₄]) (K₀, M₀, δ, ζ₀, x₀)−regularly cover the points of B.b Since after r_n steps the images of the triple of intervals ([z1, z2],[z2, z3],[z3, z4]) cover all points of the non- trivial subsetB, the cross-ratiob Cr(z1, z2, z3, z4) andCr(f^rn(z1), f^rn(z2), f^rn(z3), f^rn(z4)) are substantially different. More precisely, the following lemma holds.

Lemma 4.3. The inequality

(9) |Dst(z₁, z₂, z₃, z₄;f^rn)−1| ≥R₂

holds for sufficiently large n, where the constantR2 >0 depends only on f.

We will give the proof of this lemma later. Since the intervals [z_s, z_s+1], s = 1,2,3 (K0, M0, δ, ζ0, x0)−regularly cover the points in Bb these intervals along with

[f^rn(z_s), f^rn(z_s+1)], s = 1,2,3 satisfy conditions (i), (ii) of Lemma 4.1 with constant R₁ =K₀.Using the assertion of Lemma 4.1 we obtain

(10) |Dst(z₁, z2, z3, z4;ϕ)−1| ≤C4

(11) |Dst(f^rn(z1), f^rn(z2), f^rn(z3), f^rn(z4);ϕ)−1| ≤C4 where the constant C₄ >0 depends on R₁ and ω.

Since ϕeffects a conjugation to a linear rotation, it is easy to see that

(12) Cr(ϕ(f^rn(z₁)), ϕ(f^rn(z₂)), ϕ(f^rn(z₃)), ϕ(f^rn(z₄))) =Cr(ϕ(z₁), ϕ(z₂), ϕ(z₃), ϕ(z₄)) Formulae (10)−(12) immediately imply that

(13) |Dst(z₁, z₂, z₃, z₄;f^rn)−1| ≤C₅

where the constant C5 >0 is independent of and n. The relations (9) and (13) cannot hold simultaneously for sufficiently small. This contradiction proves Theorem 1.1.

Proof of Lemma 4.3. Recall that the triple of intervals ([z1, z2],[z2, z3],[z3, z4])

(K0, M0, δ, ζ0, x0)− regularly cover a non-trivial subset Bb = {b_i1, bi2, ..., bim} of break points. By Definition 3.3 we have z₂ = ¯b⁽ⁿ⁾_i

t and ¯b⁽ⁿ⁾_i

s ∈ [z₁, z₂), s = 1,2, ...m, s 6= t.

We rewrite Dst(z₁, z₂, z₃, z₄;f^rn) in the form (14) Dst(z₁, z₂, z₃, z₄;f^rn) =

m

Y

s=1

Dst(f^lis(z₁), f^lis(z₂), f^lis(z₃), f^lis(z₄);f)

×

rn−1

Y

p=0 p6=l_is,s=1,..,m

Dst(f^p(z1), f^p(z2), f^p(z3), f^p(z4);f).

Now we estimate the first factor in (14). The assertion of Lemma 3.2 and the definition of Fi(x, y) imply that

(15) Dst(f^lit(z1), f^lit(z2), f^lit(z3), f^lit(z4);f) =σ(¯b⁽ⁿ⁾_i

s )(1 +η(l_it)) σ(¯b⁽ⁿ⁾_is ) +η(lit)

+θt(z1, z4) = Fit(η(lit),0) +θt(z1, z4)

where

|θ_t(z₁, z₄)| ≤C₃|f^lit(z₄)−f^lit(z₁)|ψ(|f^lit(z₄)−f^lit(z₁)|) +ω(|f^lit(z₄)−f^lit(z₁)|;f⁰)

and

(16) Dst(f^lis(z1), f^lis(z2), f^lis(z3), f^lis(z4);f) = (σ(¯b⁽ⁿ⁾_is ) + (1−σ(¯b⁽ⁿ⁾_is ))ξ^is(lis))(1 +η(lis))

σ(¯b⁽ⁿ⁾_i

s ) + (1−σ(¯b⁽ⁿ⁾_i

s ))ξ^is(l_is) +η(l_is) +θs(z1, z4) = F_it(η(l_is), ξ^is(l_is)) +θ_s(z₁, z₄), s= 1,2, ..., m, s6=t where also

|θ_s(z₁, z₄)| ≤C₃|f^lis(z₄)−f^lis(z₁)|ψ(|f^lis(z₄)−f^lis(z₁)|+ω(|f^lis(z₄)−f^lis(z₁)|;f⁰) s= 1,2, ..., m, s6=t.

By construction, the interval [z1, z4] isrn-small and therefore the intervals [f^j(z1), f^j(z4)],0≤ j < r_n−1 are pairwise disjoint (except for endpoints). Hence, using assertion of Corollary 2.3 we have

(17) |f^j(z4)−f^j(z1)| ≤C1λⁿ, 0≤j < rn−1

where C1 is a constant and λ = (1 +e^−v)^−1/2. Because of the properties of modulus of continuity of f, that is

- ω(δ;f) is non-decreasing function ofδ;

- ω(aδ;f) ≤ ([a] + 1)ω(δ;f) for any a > 0 real number, where [·] is an integer part of number,

and (17) we have

(18) ω(|f^lis(z4)−f^lis(z1)|;f⁰)≤([C1] + 1)ω(λⁿ;f⁰),

for all 0≤l_is < r_n−1. In particular, this inequality holds for anys= 1,2, ..., m. Similarly, by monotonicity of ψwe have also

(19) ψ(|f^lis(z4)−f^lis(z1)|)≤ψ(C1λⁿ),

for all 0≤lis < rn−1 and particularly for any s= 1,2, ..., m. So if we fix , then there existsN =N()≥1 such that the estimate

(20) |θ_s(z₁, z₄)| ≤C₁λⁿψ(C₁λⁿ) + ([C₁] + 1)ω(λⁿ;f⁰)≤, s= 1,2, ..., m holds for all n≥N.

Suppose that η(0) and ξ^is(0), s = 1,2, ..., m satisfy the following relations η(0) >

Ω₀e^v=M₀and ξ^is(0)< τ₀e^−v, s= 1,2, ..., m. Hence using the relation (8) we obtain that η(l_is)>Ω₀ andξ^is(l_is)< τ₀, s= 1,2, ..., m. It follows from Lemma 4.2 that

(21)

F_it(η(l_it),0)

m

Y

s=1,s6=t

F_it(η(l_is), ξ^is(l_is))−A

≤ |A−1|

8 . By combining (15)-(21), for sufficiently small >0 we obtain

(22)

(F_it(η(l_it),0) +θ_t)

m

Y

s=1,s6=t

(F_it(η(l_is), ξ^is(l_is)) +θ_s)−A

≤ |A−1|

4 .

Now we estimate second factor in (14). Since [z1, z4]rn-small, we have (23)

rn−1

X

j=0

|f^j(z4)−f^j(z1)| ≤1.

We can write the second factor in (14) in the following form (24)

rn−1

Y

p=0 p6=l_is,s=1,..,m

Dst(f^p(z₁), f^p(z₂), f^p(z₃), f^p(z₄);f)−1 =

exp{

rn−1

X

p=0 p6=l_is,s=1,..,m

log[1 + (Dst(f^p(z₁), f^p(z₂), f^p(z₃), f^p(z₄);f)−1)]} −1 ≤

exp{

rn−1

X

p=0 p6=l_is,s=1,..,m

Dst(f^p(z1), f^p(z2), f^p(z3), f^p(z4);f)−1} −1 .

Using Lemma 3.1 and inequalities (18), (19) we obtain (25)

rn−1

X

p=0 p6=l_is,s=1,..,m

|Dst(f^p(z1), f^p(z2), f^p(z3), f^p(z4);f)−1| ≤

rn−1

X

p=0 p6=l_is,s=1,..,m

C2|f^p(z4)−f^p(z1)|ψ(|f^p(z4)−f^p(z1)|)+

rn−1

X

p=0 p6=l_is,s=1,..,m

|f⁰(f^p(z₄))−f⁰(f^p(z₁))|ω(|f^p(z₄)−f^p(z₁)|;f⁰)≤

ψ(C₁λⁿ)

rn−1

X

p=0 p6=l_is,s=1,..,m

|f^p(z₄)−f^p(z₁)|+

([C1] + 1)ω(λⁿ;f⁰)

rn−1

X

p=0 p6=l_is,s=1,..,m

|f⁰(f^p(z4))−f⁰(f^p(z1))|.

Because the intervals [f^p(z₁), f^p(z₄)], p = 0,1, ..., r_n−1, p6= l_is, s = 1,2, ..., m cover S¹ finite times, rewriting them as a non overlapping intervals and using (23), we obtain

(26)

rn−1

X

p=0 p6=l_is,s=1,..,m

|Dst(f^p(z1), f^p(z2), f^p(z3), f^p(z4);f)−1| ≤

ψ(C1λⁿ) +K([C1] + 1)var

S¹ f⁰·ω(λⁿ;f⁰)

whereK is the number of covers. Hence, for sufficiently large nthe right site of (26) less thanand from this it follows that the right site of (24) less than , that is

rn−1

Y

p=0 p6=l_is,s=1,..,m

Dst(f^p(z1), f^p(z2), f^p(z3), f^p(z4);f)−1 ≤

This inequality with sufficiently smalland (22) implies that

rn−1

Y

p=0

Dst(f^p(z1), f^p(z2), f^p(z3), f^p(z4);f)−1 =

^rYⁿ⁻¹

p=0

Dst(f^p(z1), f^p(z2), f^p(z3), f^p(z4);f)−A +

(A−1) ≥

rn−1

Y

p=0

Dst(f^p(z1), f^p(z2), f^p(z3), f^p(z4);f)−A

− |A−1|

≥ 3|A−1|

4 .

Hence, from this inequality follows the assertion of the lemma with constantR2= ^3|A−1|₄ . Acknowledgment

The authors would like to acknowledge the financial support received from Government of Malaysia under the research Grants FRGS/1/2014/ST06/UKM/01/1,

FRGS/2/2013/ST06/UKM/02/2.

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