# On Absolute Continuity of Conjugations between Circle Maps with Break Points

Authors: Habibulla Akhadkulov; Mohd Salmi Md Noorani
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

LetT1 and T2 be piecewise smooth circle homeomorphisms with break points and identical irrational rotation numbers. We provide one sufficient and necessary condition for the absolute continuity of conjugation map between T1 and T2.


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## Body

## 1. Introduction and Statement of Results

LetS1=ℝ/ℤ with clearly defined orientation, metric, Lebesgue measure, and the operation of addition be the unit circle. Let π:ℝ→S1 denote the corresponding projection mapping that “winds” a straight line on the circle. An arbitrary homeomorphism T that preserves the orientation of the unit circle S1 can “be lifted” on the straight line ℝ in the form of the homeomorphism LT:ℝ→ℝ with property LT(x+1)=LT(x)+1 that is connected with T by relation π∘LT=T∘π. This homeomorphism LT is called the lift of the homeomorphism T and is defined up to an integer term. The most important arithmetic characteristic of the homeomorphism T of the unit circle S1 is the rotation number(1.1)ρ(T)=limi→∞LTi(x)imod1,
where LT is the lift of T with S1 to ℝ. Here and below, for a given map F,Fi denotes its ith iteration. Poincaré proved that the above limit exists, does not depend on the initial point x∈ℝ of the lifted trajectory, and, up to additional of an integer, does not depend on the lift LT (see [1]). The rotation number ρ=ρ(T) is irrational if and only if the homeomorphism T has no periodic point. Hereafter, we will always assume that ρ is irrational and use its decomposition in an infinite continued fraction (see [2])
(1.2)ρ=1k1+1k2+1⋯kn+1⋯:=[k1,k2,…,kn,…].
The value of a “countable-floor” fraction is the limit of the sequence of rational convergentspn/qn=[k1,k2,…,kn]. The positive integers kn,n≥1, called incomplete multiples, are defined uniquely for irrational ρ. The mutually prime positive integers pn and qn satisfy the recurrent relations pn=knpn-1+pn-2 and qn=knqn-1+qn-2 for n≥1, where it is convenient to define p-1=0,q-1=1 and p0=1,q0=k1. Given a circle homeomorphism T with irrational rotation number ρ, one may consider a marked trajectory (i.e., the trajectory of a marked point) ξi=Tiξ0∈S1, where i≥0, and pick out of it the sequence of the dynamical convergentsξqn,n≥0, indexed by the denominators of consecutive rational convergents to ρ. We will also conventionally use ξq-1=ξ0-1. The well-understood arithmetical properties of rational convergents and the combinatorial equivalence between T and rigid rotationRρ:ξ→ξ+ρ mod 1 imply that the dynamical convergents approach the marked point, alternating their order in the following way:
(1.3)ξq-1<ξq1<ξq3<⋯<ξq2m+1<⋯<ξ0<⋯<ξq2m<⋯<ξq2<ξq0.
We define the nth fundamental intervalΔn(ξ0) as the circle arc [ξ0,Tqn(ξ0)] for even n and as [Tqn(ξ0),ξ0] for odd n. For the marked trajectory, we use the notation Δ0n=Δn(ξ0),Δin=Δn(ξi)=TiΔ0n. It is well known that the set Pn(ξ0,T)=Pn(T) of intervals with mutually disjoint interiors defined as
(1.4)Pn(T)={Δin-1,0≤i<qn;Δjn,0≤j<qn-1}
determines a partition of the circle for any n. The partition Pn(T) is called the nth dynamical partition of the point ξ0. Obviously the partition Pn+1(T) is a refinement of the partition Pn(T): indeed the intervals of order n are members of Pn+1(T) and each interval Δin-1∈Pn(T)0≤i<qn, is partitioned into kn+1+1 intervals belonging to Pn+1(T) such that
(1.5)Δin-1=Δin+1∪⋃s=0kn+1-1‍Δi+qn-1+sqnn.Class𝔅-homeomorphisms. These are orientation-preserving circle homeomorphisms T differentiable except in finite number break points at which left and right derivatives, denoted, respectively by DT- and DT+, exist, and such that(i)
there exist constants0<c1<c2<∞ with c1<DT(x)<c2 for all x∈S1∖BP(T), c1<DT-(xb)<c2 and c1<DT+(xb)<c2 for all xb∈BP(T), with BP(T) the set of break points of T on S1;(ii)
DT has bounded variation. The ratio σT(c):=(DT-(c))/(DT+(c)) is called the jump of T in c or the T-jump. General 𝔅-homeomorphisms with one break point was first studied by Khanin and Vul in [3]. Among other results it was proved by these authors that their renormalizations approximate fractional linear transformations. Let T be an orientation preserving C1-diffeomorphism of the circle. If the rotation number ρ is irrational and DT is of bounded variation then, by a well-known theorem of Denjoy, T is conjugate to the rigid rotation Rρ (see [1]). The conjugation means that there exists an essentially unique homeomorphism h of the circle such that T=h-1∘Rρ∘h. In this context, a natural question to ask is under what condition the conjugacy is smooth? Several authors, for example [4–6] have shown that if T is C2+α,α>0 and ρ satisfies certain diophantine condition then the conjugacy will be at least C1.The classical result of Denjoy can be easily extended to the case of𝔅-homeomorphisms. Next we consider the problem of the regularity of the conjugating map between two class 𝔅-homeomorphisms with one break point and coinciding irrational rotation numbers. The case of one break point with the same jump ratios, so called rigidity problem, was studied in detail by Teplinskii and Khanin in [7]. Let ρ=[k1,k2,…,kn,…] be the continued fraction expansion of the irrational rotation number ρ and define
(1.6)Mo={ρ:∃C>0,∀n∈ℕ,k2n-1≤C},Me={ρ:∃C>0,∀n∈ℕ,k2n≤C}.
The main result of [7] is as follows.Theorem 1.1.
LetTi∈C2+α(S1∖{bi}),i=1,2, α>0 be 𝔅-homeomorphisms with one break point that have the same jump ratio σ and the same irrational rotation number ρ∈(0,1). In addition, let one of the following restrictions be true: either σ>1 and ρ∈Me or σ<1 and ρ∈Mo. Then the map h conjugating the homeomorphisms T1 and T2 is a C1-diffeomorphism.In the case of different jump ratios, the following theorem was proved in [8] by Dzhalilov et al.Theorem 1.2.
LetTi∈C2+α(S1∖{bi}),i=1,2, α>0 be 𝔅-homeomorphisms with one break point that have different jump ratio and the same irrational rotation number ρ∈(0,1). Then the map h conjugating the homeomorphisms T1 and T2 is a singular function, that is, is continuous on S1 and Dh(x)=0 a.e. with respect to Lebesgue measure.LetT1 and T2 be 𝔅-homeomorphisms with identical irrational rotation number ρ. Now, we consider dynamical partitions Pn(ξ,T1)=Pn(T1) and Pn(h(ξ),T2)=Pn(T2) appropriate to the homeomorphisms T1 and T2. Denote by Δ^n intervals of partition of Pn(T2). Since the function h is a conjugation function between T1 and T2, so we have h(Δn)=Δ^n for any Δn∈Pn(T1). Denote by |A| the Lebesgue measure of the corresponding set of A⊂S1. Our purpose in this paper is to give some criteria for the absolute continuity of the conjugation map h. Our first main result is the following.Theorem 1.3.
Assume the rotation numberρ is irrational of bounded type. Suppose that there exist a sequence (τn) such that ∑n=1∞‍τn2<∞ with
(1.7)||Δ1||Δ2|-|Δ^1||Δ^2||≤τn
for each pair of adjacent intervals Δ1,Δ2∈Pn(T1) for all n>1. Then the conjugation map h is absolutely continuous function.In the proof of Theorem1.3, we will use the consideration of theory of martingales. The idea of using theory of martingales was established in [9] by Katznelson and Ornstein. Our second main result is the following.Theorem 1.4.
LetT1 and T2 be 𝔅-homeomorphisms with identical irrational rotation number ρ. If the conjugation map h is a absolutely continuous function, then for all δ>0, the sequence of Lebesgue measure of the set |{x:|logDT2qn(h(x))-logDT1qn(x)|≥δ}| tends to 0 when n goes to +∞.


## 2. The Denjoy Theory and Ergodicity of𝔅-Homeomorphisms

The assertions listed below, which are valid for any orientation-preserving homeomorphismT∈𝔅 with irrational rotation number ρ, constitute classical Denjoy theory. Their elementary proofs can be found in [10, 11].(a)
Generalized Denjoy estimate; letξ0∈S1 be a continuity point of DTqn, then the following inequality holds: e-v≤DTqn(ξ0)≤ev, where v=VarS1logDT.(b)
Exponential refinement; there exists a universal constantC1=C1(T) such that |Δ0n+m|≤C1λm|Δ0n|, where λ=(1+e-v)-1/2.(c)
Bounded geometry; let rotation numberρ is bounded type that is the coefficients in continued fraction expansion of ρ are bounded. Then there exist universal constants C2=C2(T), C3=C3(T) such that 0<C2<1 and C3≤1 with
(i)
each pair of adjacent intervals ofPn(T) are C2-comparable that is their ratio of lengths belongs to [C2,C2-1];(ii)
an intervalΔn+1 of Pn+1(T) is C3-comparable to the interval Δn of Pn(T) that contains it: C3|Δn|≤|Δn+1|.(d)
Generalized Finzi estimate; supposeξ∈S1,η∈Δn-1(ξ) and ξ,η are continuity points of DTqn. Then for any 0≤k<qn, the following inequality holds: |logDTk(ξ)-logDTk(η)|≤v.Let(S1,𝔊,μ) be a measure space and F:S1→S1 be a measurable map.Definition 2.1.
The setA∈𝔊 is said to be invariant with respect to the measurable F, if A=F-1A.Definition 2.2.
A measurable mapF:S1→S1 is said to be ergodic with respect measure μ if the measure μ(A) of any invariant set A equals 0 or 1.Letξ0∈S1, denote by Vn=Δn(ξ0)∪Δn-1(ξ0).Lemma 2.3.
LetT be a 𝔅-homeomorphism with irrational rotation number ρ. Suppose ξ∈Vn and ξ be a continuity point of DTqn. Then for any 0≤k<qn, the following inequality holds:
(2.1)e-v|Tk(Vn)||Vn|≤DTk(ξ)≤ev|Tk(Vn)||Vn|.Proof.
Let the system of intervalsℐ={I:I⊂Vn,andthemapDTqniscontinuousonI} be continuity intervals of DTqn. Let ξ∈Δn-1(ξ0). Then, by the mean value theorem, for any 0≤k<qn, we have
(2.2)|Tk(Δn-1(ξ0))|DTk(ξ)|Δn-1(ξ0)|=DTk(z1)|I1|+DTk(z2)|I2|+⋯+DTk(zd)|Id|DTk(ξ)|Δn-1(ξ0)|,
where zi∈Ii⊂Δn-1(ξ0) and Ii∈ℐ,1≤i≤d. If ξ∈Δn(ξ0) then we have
(2.3)|Tk(Δn(ξ0))|DTk(ξ)|Δn(ξ0)|=DTk(y1)|J1|+DTk(y2)|J2|+⋯+DTk(yt)|Jt|DTk(ξ)|Δn(ξ0)|,
where yi∈Ji⊂Δn(ξ0) and Ji∈ℐ,1≤i≤t. Apply generalized Finzi estimate to the right-hand side of relations (2.2) and (2.3), we get
(2.4)e-v≤|Tk(Δn-1(ξ0))|DTk(ξ)|Δn-1(ξ0)|≤ev,e-v≤|Tk(Δn(ξ0))|DTk(ξ)|Δn(ξ0)|≤ev.
Finally, we get
(2.5)e-v≤|Tk(Vn)|DTk(ξ)|Vn|≤ev.Lemma 2.4.
LetT be a 𝔅-homeomorphism of the circle S1 with irrational rotation number ρ, then T is ergodic with respect to Lebesgue measure.Proof.
Suppose that there exist an invariant setA of positive but not full Lebesgue measure |A|. Then by the Lebesgue Density Theorem, A has a density point z. We fix an arbitrary ϵ>0. By definition of density points, we can find a δ>0 such that for any interval [a,b] satisfying the conditions z∈[a,b],[a,b]⊂(z-δ,z+δ), we have |A∩[a,b]|≥(1-ϵ)|[a,b]|, or, in other words, |Ac∩[a,b]|<ϵ|[a,b]|, where Ac denotes the complement of A. Now, we choose such n that Vn=Δn(z)∪Δn-1(z)⊂(z-δ,z+δ). We can check that ⋃k=0qn-1‍Tk(Vn)=S1 and each point of the circle belongs to at most two intervals of this cover. Hence, the set Ac is invariant with respect to T, using the above lemma, we get
(2.6)|Ac|=∑k=0qn-1‍|Ac∩Tk(Vn)|=∑k=0qn-1‍∫Ac∩Vn‍DTk(x)dx≤ev|Ac∩Vn||Vn|∑k=0qn-1‍|Tk(Vn)|≤2evϵ.
Since ϵ was arbitrary, |Ac|=0. The theorem is proved.Lemma 2.5.
LetT1 and T2 are 𝔅-homeomorphisms with identical irrational rotation number. Then the conjugation map h between T1 and T2 is either absolutely continuous or singular function.Proof.
Consider two𝔅-homeomorphisms T1 and T2 of the circle S1 with identical irrational rotation number ρ. Let φ1 and φ2 be maps conjugating T1 and T2 with the rigid rotation Tρ, that is, φ1∘T1=Tρ∘φ1 and φ2∘T2=Tρ∘φ2. It is easy to check that the map h=φ2-1∘φ1 conjugates T1 and T2, that is
(2.7)h∘T1=T2∘h.
We know that conjugation function h is strictly increasing function on S1. Then Dh exists almost everywhere on S1. Denote by A={x:x∈S1,Dh(x)>0}. It is clear that the set A is mod 0 invariant with respect to T1. Since the class 𝔅-homeomorphism is ergodic with respect to the Lebesgue measure. Hence, the Lebesgue measure of set A is either null or full. If Lebesque measure of A is null then h is a singular function, if it is full then h is an absolutely continuous function.Remark 2.6.
LetT1 and T2 be 𝔅-homeomorphisms with identical irrational rotation number. Then conjugation map h-1 between T2 and T1 is either absolutely continuous or singular.


## 3. Martingales and Martingale Convergence Theorem

Our objective in this section is to develop the fundamentals of the theory of martingales and prepare for the main results and applications that will be presented in the subsequent sections.Definition 3.1.
Let(X,𝔉) be a measurable space. A sequence (𝔉m) of σ-algebras on X is said to be a filtration in 𝔉, if
(3.1)𝔉1⊆𝔉2⊆⋯⊆𝔉.Statement 3.2.
The sequence of algebras generated by dynamical partitions, which is also denoted by(Pm) (by abuse of notation) is a filtration in ℬ, where ℬ is a Borel σ-algebra on S1.Definition 3.3.
Let(ℛm) be a sequence of random variables on a measurable space (X,𝔉) and (𝔉m) a filtration in 𝔉. We say that (ℛm) is adapted to(𝔉m) if, for each positive integer m,ℛm is 𝔉m-measurable.Denote byE(ℛ∣𝔉) conditional expectation of random variables ℛ with respect to partition 𝔉.Definition 3.4.
Let(ℛm) be a sequence of random variables on a probability space (X,𝔉,P) and (𝔉m) a filtration in 𝔉. The sequence (ℛm) is said to be a martingale with respect to (𝔉m) if, for every positive integer m,(i)
(ℛm) is integrable;(ii)
(ℛm) is adapted to (𝔉m);(iii)
E(ℛm+1∣𝔉m)=ℛm.Lemma 3.5 (see [12]).
Let(ℛm) be a sequence of random variables on a probability space (X,𝔉,P). If supmE(|ℛm|p)<∞ for some p>1 and (ℛm) is a martingale, then there exists an integrable ℛ∈L1(X,𝔉) such that
(3.2)limm→∞ℛm=ℛ(a.eP),ℛm→ℛinL1-norm.Supposef is a homeomorphism (not necessary to be 𝔅-homeomorphism) of the circle S1. Using the homeomorphism f and sequence of dynamical partitions (Pm), we define the sequence of random variables on the circle which is generating a martingales. For any m≥1, we set
(3.3)ℛm(x)=|f(Δm)||Δm|,ifx∈Δm,Δm∈Pm.Lemma 3.6.
The sequence(ℛm) of random variables is a martingale with respect to (Pm).Proof.
To prove the martingale, it suffices to checkE(ℛm+1∣Pm)=ℛm, for any m≥1, because the sequence of random variables (ℛm) is sequence of step functions, so the sequence of step functions is integrable and adapted to (Pm). Denote by χI indicator function of interval I. Using definition of conditional expectation of random variables (ℛm) with respect to partition (Pm), we get
(3.4)E(ℛm+1∣Pm)=∑i=0qm-1-1‍E(ℛm+1∣Δim)χΔim+∑i=0qm-1‍E(ℛm+1∣Δim-1)χΔim-1.
Now, we calculate each sum of (3.4) separately. Note, that each interval of Pm order m is member of Pm+1 and each interval Δim-1∈Pm,0≤i<qm, is partitioned into km+1+1 intervals belonging to Pm+1 such that
(3.5)Δim-1=Δim+1∪⋃s=0km+1-1‍Δi+qm-1+sqmm.
Using this, we get
(3.6)E(ℛm+1∣Δim)=1|Δim|∫Δim‍ℛm+1(x)ℓ(dx)=1|Δim|∫Δim‍ℛm(x)ℓ(dx),(3.7)E(ℛm+1∣Δim-1)=1|Δim-1|∫Δim-1‍ℛm+1(x)ℓ(dx)=1|Δim-1|[∫Δim+1‍ℛm+1(x)ℓ(dx)]+1|Δim-1|[∑s=0km+1-1‍∫Δi+qm-1+sqmm‍ℛm+1(x)ℓ(dx)]=1|Δim-1|∫Δim-1‍ℛm(x)ℓ(dx).
Finally, summing (3.4), (3.6), and (3.7), we get
(3.8)E(ℛm+1∣Pm)=∑i=0qm-1-1‍ℛm(x)χΔim+∑i=0qm-1‍ℛm(x)χΔim-1=ℛm.The following inequality (sometimes called “parallelogram inequality”) is useful for estimating fractions, and we will use it in the proof of the next statement.Lemma 3.7.
Givena,b,c,d>0, the following inequalities hold
(3.9)min{ab,cd}≤a+cb+d≤max{ab,cd}.Proof.
Consider pointsA=(a,b),B=(c,d), and C=(a+c,b+d) on the plan xOy. The slope of the ray OC lies between slops of rays OA and OB.


## 4. Proof of Main Theorems

Leth be the conjugation homeomorphism between T1 and T2, that is, h∘T1=T2∘h. Without loss of generality, we assume h(0)=0. Consider dynamical partition Pm(T1). Define sequence of random variables (ℛm) on the S1 by this formula
(4.1)ℛm(x)=|h(Δm)||Δm|,ifx∈Δm,Δm∈Pm(T1).
Denote by Θm(x)=ℛm(x)-ℛm-1(x),m≥1 and ℛ0(x):≡0,x∈S1.Statement 4.1.
Let the sequence(τm) be defined in Theorem 1.3. Then there exists a universal constant C4=C4(T)>0 such that for all m≥1, the following inequality holds
(4.2)|Θm(x)|≤C4τm|ℛm-1(x)|,x∈S1.Proof.
It is clear that(4.3)|Θm(x)|=|ℛm-1(x)|||ℛm(x)||ℛm-1(x)|-1|.
Now, we estimate ℛm(x)/ℛm-1(x). Denote by ℛm(Δm):=ℛm(x),x∈Δm and Δm∈Pm(T1). Thus, we have
(4.4)|Δm-1|ℛm-1(Δm-1)=∑s=0km‍|Δm(s)|ℛm(Δm(s)),
where Δm(s)⊂Δm-1. Using Lemma 3.7, we get
(4.5)minℛm(Δm(s))≤ℛm-1(Δm-1)≤maxℛm(Δm(s)).
It is clear that for any 0≤s≤km holds
(4.6)minℛm(Δm(s))maxℛm(Δm(s))≤ℛm-1(Δm-1)ℛm(Δm(s))≤maxℛm(Δm(s))minℛm(Δm(s)).
Since, each pair of adjacent intervals of Pm(T1) are C2-comparable. By the assumption of Theorem 1.3, we get
(4.7)|ℛm(Δm(s+1))ℛm(Δm(s))-1|≤C2τm.
Hence, the rotation number ρ=[k1,k2,…,km,…] is of bounded type, and an easy trick gives us
(4.8)maxℛm(Δm(s))minℛm(Δm(s))≤(1+C2τm)K≤1+C4τm,
where K=sup(km) and C4=KC2. A similar lower bound holds true for minℛm(Δm(s)):maxℛm(Δm(s)). Therefore, we have
(4.9)1-C4τm≤ℛm-1(Δm-1)ℛm(Δm(s))≤1+C4τm,
for all 0≤s≤km.Proof of Theorem1.3.
For the proof of Theorem1.3, we use the above reasonings. By Lemma 3.6, the sequence (ℛm) of random variables is a martingale with respect to (Pm). We want to show that ℛm converges to Dh in the norm L1(S1,dℓ) when m→∞. By direct calculation, it is easy to see that Θm(x) and ℛm-1(x) is orthogonal, that is
(4.10)∫S1‍Θm(x)ℛm-1(x)dx=0.
Using the assertion of Statement 4.1, we get
(4.11)∥ℛm∥L22≤∥ℛm-1∥L22+∥Θm∥L22≤(1+C4τm2)∥ℛm-1∥L22.
Iterating the last relation, we have ∥ℛm∥L22≤∏j=1m‍(1+C4τj2). So far as the series ∑j=1∞‍τj2 converges. From this implies that the sequence of random variables (ℛm) is bounded in L2 norm. By Lemma 3.5, the sequence of random variables (ℛm) converges to some function ℛ in L1 norm. We prove that sequence of random variables (ℛm) converges to the Dh. Indeed, denote by αm and βm end points of interval Δm of dynamical partition Pm(T1). By definition of ℛm, we have
(4.12)|h(x)-∫0x‍ℛm(x)dx|≤|h(x)-h(αm)|+|h(Δm)||Δm||x-αm|≤2|h(Δm)|.
Moreover, using last inequality, we obtain
(4.13)|h(x)-∫0x‍ℛ(x)dx|≤|h(x)-∫0x‍ℛm(x)dx|+∫0x‍|ℛ(x)-ℛm(x)|dx≤2|h(Δm)|+∥ℛm-ℛ∥L1.
From this taking the limit when m→∞, we get h(x)=∫0x‍ℛ(x)dx. Since, ℛ∈L1(S1,dℓ), then h is absolutely continuous function and Dh(x)=ℛ(x) almost everywhere on S1. Thus, Theorem 1.3 is completely proved.Statement 4.2.
For allψ∈L1(S1,dℓ) hold this equality
(4.14)limm→∞∥ψ∘T1qm-ψ∥L1=0.Proof.
It is a well-known fact that the classC([a,b]) of continuous functions on [a,b] is dense (in ∥·∥L1) in L1([a,b],dℓ) (see [13]). From this fact it implies that if ψ∈L1(S1,dℓ), then for any ϵ>0 there exists a continuous function ψϵ∈C(S1) and ϕϵ∈L1(S1,dℓ) such that ψ=ψϵ+ϕϵ and ∥ϕϵ∥L1≤ϵ. Using this and Denjoy estimate, we obtain
(4.15)∥ψ∘T1qm-ψ∥L1≤∥ψϵ∘T1qm-ψϵ∥L1+(sup|DT1qm|-1+1)∥ϕϵ∥L1≤∥ψϵ∘T1qm-ψϵ∥L1+(1+ev)∥ϕϵ∥L1.
As ψϵ is uniformly continuous on S1 and by exponential refinement T1qm(x) uniformly tends to x, there exists a positive integer m0=m0(ϵ) such that for all m≥m0, the ∥ψϵ∘T1qm-ψϵ∥L1≤ϵ. Therefore, ∥ψ∘T1qm-ψ∥L1≤(2+ev)ϵ. Since ϵ>0 was arbitrary and sufficiently small.Proof of Theorem1.4.
Assume that conjugation maph is absolutely continuous, then Dh∈L1(S1,dℓ) and Dh>0 almost everywhere. For all positive integers n, the function Dh satisfies
(4.16)(Dh∘T1qn)DT1qn=D(T2qn∘h)Dha.e.
Taking the logarithm, we obtain
(4.17)log(Dh∘T1qn)-logDh=logD(T2qn∘h)-logDT1qn.
Denote by ψ=logDh, it is clear that ψ∈L1(S1,dℓ). Suppose, by contradiction, that there exists δ>0, such that the Lebesgue measure of the set Sδn={x:|logDT2qn(h(x))-logDT1qn(x)|≥δ} does not converge to 0 when n goes to infinity. Hence, for all positive integer n:
(4.18)∫S1‍|ψ(T1qn(x))-ψ(x)|dx≥∫Sδn‍|ψ(T1qn(x))-ψ(x)|dx≥δ|Sδn|.
But |Sδn| does not tend to 0 when n goes to +∞. Hence ∫S1‍|ψ(T1qn(x))-ψ(x)|dx does not tend to 0 when n goes to +∞, this contradicts Statement 4.2 and ends the proof of Theorem 1.4.



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