Non-standard couples of angles of rotations

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Non-standard couples of angles of rotations

Mostapha Benhenda

To cite this version:

Mostapha Benhenda. Non-standard couples of angles of rotations. 2012. �hal-00669028�

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Non-standard couples of angles of rotations

Mostapha Benhenda

^∗

February 10, 2012

Abstract

LetMbe a smooth compact connected manifold, on which there exists an effective smooth circle actionSt preserving a positive smooth volume. We show that onM, the smooth closure of the smooth volume-preserving conjugation class of some Liouville rotationsSα of angleαcontains a smooth volume-preserving diffeomorphismT that is metrically isomorphic to an irrational rotationRβon the circle, withα,±β, and withαandβchosen either rationally dependent or rationally independent. In particular, ifMis the closed annulus [0,1]×¹,Madmits a smooth ergodic pseudo-rotationTof angleαthat is metrically isomorphic to the rotationRβ. Moreover,Tis smoothly tangent toSαon the boundary ofM.

1 Introduction

Let = [0,1]×¹ be the closed annulus andT be a homeomorphism isotopic to the identity. Therotation setof T measures the asymptotic speeds of rotation of the orbits ofT around the annulus. It generalizes the notion of rotation number of a circle homeomorphism, introduced by Poincaré. T is an irrational pseudo-rotation if its rotation set is reduced to a single irrational numberα, called theangleofT. A broad question is raised by Béguin et al. [3]: what are the similarities between the dynamics of the rigid rotationS_αof angleαand the dynamics of an irrational pseudo-rotationT of angleα?

From a topological viewpoint, a similarity betweenS_αandT has been shown by Béguin et al. [3]: the rotationSα is in the closure of the conjugacy class ofT. Their result is analogous to a theorem by Kwapisz [11] on the torus²(in this case, the angle of a pseudo-rotation is an element of²). Jäger [9] and Wang [12] also investigated this broad question. However, there are also possible differences betweenSα andT. From a metric viewpoint, Anosov and Katok [1] constructed a smooth pseudo-rotation of that is metrically isomorphic to an ergodic translation of². Béguin et al. [2]

constructed on² a pseudo-rotation that is minimal, uniquely ergodic, but with positive entropy. In this paper, we construct a smooth pseudo-rotation of angleαthat is metrically isomorphic to an irrational rotationRβwithα,±β. This is a construction of a non-standard smooth realization, based on the method of approximation by successive conjugations (see [5] for a presentation), a method that is often fruitful in smooth realization problems.

∗Laboratoire d’Analyse, Geometrie et Applications, Paris 13 University, 99 Avenue J.B. Clement, 93430 Villetaneuse, France. Part of this work was accomplished during the TEMPUS Summer School on dynamical systems held at the PMF University of Tuzla, Bosnia and Herzegovina. I would like to thank Elvis Barakovic and Dino Bojadzija for hospitality in Tuzla, and Muharem Avdispahic, Huse Fatkic, Senada Kalabusic and Amil Pecenkovic for hospitality in Sarajevo. Contact: mostaphabenhenda@gmail.com

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We recall that asmooth realizationof an abstract system (X,f, ν) is a triplet (M,T, µ), whereM is a smooth compact manifold,µa smooth measure onM andT a smooth µ-preserving diffeomorphism of M, such that (M,T, µ) is metrically isomorphic to (X,f, ν) (when (M, µ) and (X, ν) are implied, we just say thatTis metrically isomorphic to f). Moreover, a smooth realization isnon-standardif MandXare not diffeomor- phic.

Suppose there exists an ergodic pseudo-rotationT of angleαthat is a non-standard smooth realization of a rotationRβon the circle. Then the couple (α, β) is called anon- standard couple of angles. In this paper, we show that there exists non-standard couple of angles (α, β), such thatα,±β, withαandβchosen either rationally dependent or rationally independent.

Anosov and Katok [1] showed the existence of an angleα such that (α, α) is a non-standard couple of angles. Fayad et al. [7] showed that for anyαLiouville, (α, α) is a non-standard couple of angles. The question arises about the existence of a non- standard couple of angles (α, β) withα,β.

It is worthy to recall that two ergodic rotationsRαandRβ on the circle are metrically isomorphic if and only ifβ=±α. Ifβ=α, the isomorphism is the identity, and if β=−α, an isomorphism is given by a symmetry of axis going through the center of the circle. Therefore, by applying the result of Fayad et al. [7], it becomes trivial to find a non-standard couple of angles (α,−α). Our result shows that if, instead of consider- ing metric automorphisms of the circle, we consider metric isomorphisms between the circle and the annulus, the situation becomes richer: we can haveα,±β, withαand βeither rationally dependent or rationally independent. However,αneeds to be Liou- ville. Indeed, a result by Herman (with a proof published by Fayad and Krikorian [6]) implies that if a smooth quasi-rotationT of the closed annulus has Diophantine angle (i.e. non-Liouville), thenT cannot be ergodic (and a fortiori,T cannot be metrically isomorphic to an ergodic rotation). However, the situation whereαis Liouville andβ is Diophantine, though not addressed in this paper, is not excluded yet. The existence of this situation would reply positively to the open question about the existence of a non-standard smooth realization of a Diophantine circle rotation [5].

More generally, let Mbe a smooth compact connected manifold of dimensiond, on which there exists an effective smooth circle actionStpreserving a positive smooth measureµ. Let A_α be the smooth conjugation class of the rotationSα, and ¯A_α its closure in the smooth topology. IfM =¹and ifαis Diophantine, then ¯A_α =A_αby Herman-Yoccoz theorem [14] (indeed, by continuity, the rotation number of a diffeo- morphismT ∈ A¯_αisα). On the other hand, whenαis Liouville, ¯A_α ,A_α. In this paper, ifMhas a dimensiond ≥2, then for some Liouvilleα, we show that ¯A_αcon- tains non-standard smooth realizations of circle rotationsRβ, withα,±β, and withα andβchosen either rationally dependent or rationally independent. In this case, (α, β) is still called anon-standard couple of angles. More precisely, we show the following theorem:

Theorem 1.1. Let M be a smooth compact connected manifold of dimension d ≥ 2, on which there exists an effective smooth circle action(St)t∈¹ preserving a positive smooth measureµ. For any u,v∈¹, for any >0, there exist(α, β)∈¹×¹in a -neighborhood of(u,v), T∈Diff^∞(M, µ), such that T ∈A¯_αand such that the rotation Rβ of angleβon¹is metrically isomorphic to T . Moreover,βcan be chosen either rationally dependent or rationally independent ofα.

Theorem 1.1 generalizes the particular caseM=[0,1]^d−1×¹:

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Theorem 1.2. Let d ≥ 2, M = [0,1]^d−1×¹,µthe Lebesgue measure. For t ∈¹, let St : M → M defined by St(x,s) = (x,s+t). For any u,v ∈ ¹, for any > 0, there exist(α, β)∈ ¹×¹in a-neighborhood of(u,v), T ∈Diff^∞(M, µ), such that for any j∈,(D^jT)_|∂M=(D^jS_α)_|∂Mand such that the rotation R_βof angleβon¹is metrically isomorphic to T . Moreover,βcan be chosen either rationally dependent or rationally independent ofα.

In the case of the closed annulusM =[0,1]×¹, we obtain:

Corollary 1.3. Let M = [0,1]×¹,µthe Lebesgue measure. For t ∈ ¹, let St : M → M defined by St(x,s) = (x,s+t). For any u,v ∈ ¹, for any > 0, there exist(α, β)∈¹×¹in a-neighborhood of(u,v), T ∈Diff^∞(M, µ)a pseudo-rotation of angleα, such that the rotation R_β of angleβon¹ is metrically isomorphic to T . Moreover,βcan be chosen either rationally dependent or rationally independent ofα.

To show these results, we suitably modify one of Anosov and Katok’s construc- tions. In [1], they constructed ergodic translations on the torus^h,h≥2, of coordinates (β1, ..., βh), translations that admit non-standard smooth realizations on [0,1]^d−1×¹, d ≥2, such thatT|∂M is a rotation of angleα. Moreover, in his construction,α ,βi, i=1, ...,h. In our paper [4], we show that oneβican be an arbitrarily chosen Liouville number. However, this construction does not apply directly to the one-dimensional case. This is why, to obtain our result, though we essentially follow [4], we still need some substantial modifications.

1.1 Definitions

Afinite measurable partitionξ¯ of a measured manifold (N, ν) is the equivalence class of a finite setξof disjoint measurable subsets ofN whose union isN, modulo sets ofν-measure zero. In most of this paper, we do not distinguish a partitionξwith its equivalent class ¯ξmodulo sets ofν-measure zero. In these cases, both are denotedξ.

Moreover, all partitions considered in this paper are representatives of a finite measurable partition. The distance between two finite measurable partitionsξandξ⁰is defined by:

d(ξ, ξ⁰)=inf X

c∈ξ,c⁰∈ξ⁰

ν(c∆c⁰)

A partitionξ⁰issubordinateto a partitionξif any element ofξis a union of elements ofξ⁰, modulo sets ofν-measure zero. In this case, ifB(ξ) denotes the completed algebra generated byξ, thenB(ξ)⊂ B(ξ⁰). The inclusion mapi: B(ξ)→ B(ξ⁰) will be denotedξ ,→ ξ⁰. This notation also means thatξ⁰issubordinatetoξ. A sequence of partitionsξnismonotonicif for anyn,ξn ,→ ξn+1. These definitions and properties are independent of the choice of the representativesξandξ⁰of the equivalence classes ξ¯and ¯ξ⁰.

A measure-preserving bijective bimeasurable mapT : (M₁, µ₁,B₁)→(M₂, µ₂,B₂) induces anisomorphismof measure algebras, still denotedT : (µ₁,B₁)→ (µ₂,B₂). If ξ₁, ξ₂ are partitions, and ifB₁ =B(ξ₁) andB₂ =B(ξ₂), we denoteT : ξ₁ → ξ₂this induced isomorphism of measure algebras. IfM₁ =M₂,µ₁=µ₂andB₁ =B₂, thenT

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is ameasure-preserving transformation. Its induced isomorphism is anautomorphism (see [8, p.43] and [13]).

Ametric isomorphism Lof measure-preserving transformationsT1 : (M₁, µ₁,B₁)→ (M₁, µ₁,B₁),T₂: (M₂, µ₂,B₂)→(M₂, µ₂,B₂) is a measure-preserving bijective bimeasurable map L : (M₁, µ₁,B₁) → (M₂, µ₂,B₂) such that LT₁ = T₂L a.e. For con- venience, when the measure is the Lebesgue measure and the algebra is the Bore- lian algebra, we omit to mention the measures and algebras, and we simply say that L: (M1,T1)→(M2,T2) is a metric isomorphism.

Let ¯ξbe a measurable partition andξa representative of this equivalent class modulo sets ofµ-measure zero. Forx∈ M, we denoteξ(x) the element of the partitionξ such thatx∈ξ(x). A sequences of partitionsξnof measurable setsgeneratesif there is a set of full measureFsuch that for anyx∈F,

{x}=F\

n≥1

ξn(x)

This property of generation is independent of the choice of the representativesξ_nof the equivalent class ¯ξ_nand therefore, we will say that the sequence of measurable partitions ¯ξ_ngenerates. LetM/ξdenote the equivalent class of the algebra generated byξ, modulo sets ofµ-measure zero.M/ξis independent of the choice of the representative ξof the equivalent class ¯ξ. If T : M1 → M2 is a measure-preserving map such that T(ξ1)=ξ2µ-almost everywhere, we can define a quotient map:T/ξ1:M/ξ1→M/ξ2. Aneffective actionof a groupGonM is an action such that there is a set of full measureF ⊂ Msuch that for anyx∈ F, there isg ∈Gsuch thatgx ,x. A smooth effective circle action (St)_t∈1 on M can be seen as a 1-periodic smooth flow (St)t≥0, we denoteA_α ={B⁻¹SαB,B∈Diff^∞(M, µ)}. WhenM =[0,1]^d−1×¹, we consider the periodic flowStdefined by:

St: [0,1]^d−1×¹ →[0,1]^d−1×¹ (x,s) 7→(x,t+s mod 1)

Fora,b∈¹, let [a,b[ be the positively oriented circular sector betweenaandb, with aincluded andbexcluded.

A sequenceT_nofµ-preserving mapsweakly convergestoT if, for any measurable setE,µ(T_nE∆E)→0, whereA∆B=(A−B)∪(B−A).

Forγ∈, we denote:|γ| mod 1=mink∈|k+γ|

Fort∈¹or, andA⊂I×¹, we denote

t+A={(x,t+s mod 1),(x,s)∈A}

SupposeM=[0,1]×¹is the closed annulus. Let ˜M =[0,1]×be the universal covering of M andp₂ : [0,1]× → the second coordinate projection. LetT be a homeomorphism of Misotopic to the identity and ˜T its lift to ˜M. Therotation set Rot( ˜T) of ˜T is defined by:

Rot( ˜T)=\

k≥0

[

n≥k

(p2( ˜T( ˜x))−p2( ˜x) n /x˜∈M˜

)

We let the rotation set ofT,Rot(T), be the equivalent class modulo 1 ofRot( ˜T).

IfRot(T)={α}is a singleton, and ifT is isotopic to the identity, thenT is apseudo- rotation. Note that, ifT_|∂M =S_α|∂M, thenT is isotopic to the identity. Indeed,

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t ∈ [0,1] 7→ Stα is a continuous path between the identity map andSα, and by Alexander’s trick, any homeomorphism equal to the identity on the boundary is isotopic to the identity. In this paper, all the diffeomorphisms that we construct are equal to a rotation on the boundary and therefore, they are all isotopic to the identity.

1.2 Basic steps of the proof

The metric isomorphism of theorem 1.2 is obtained as the limit of isomorphisms of finite algebras: indeed, we use the lemma [1, p.18]:

Lemma 1.4. Let M1and M2be Lebesgue spaces and letξ⁽ⁱ⁾_n (i=1,2) be monotonic and generating sequences of finite measurable partitions of Mi. Let Tn⁽ⁱ⁾be automorphisms of M_isuch that T_n⁽ⁱ⁾ξ_n⁽ⁱ⁾ =ξ⁽ⁱ⁾_n and T_n⁽ⁱ⁾ →T⁽ⁱ⁾in the weak topology. Suppose there are metric isomorphisms L_n:M₁/ξ_n⁽¹⁾→M₂/ξ⁽²⁾_n such that

L_nT_n⁽¹⁾/ξ⁽¹⁾_n =T_n⁽²⁾/ξ⁽²⁾_n L_n and

Ln+1ξ⁽¹⁾_n =ξ⁽²⁾_n then(M₁,T₁)and(M₂,T₂)are metrically isomorphic.

Said otherwise, if we have generating sequences of partitions and sequences of automorphisms T_n⁽ⁱ⁾ weakly converging towards T⁽ⁱ⁾, and if, for any integer n, the following diagram commutes:

ξ⁽¹⁾_n

T_n⁽¹⁾ 55 ^Lⁿ //

_

ξ_n⁽²⁾ _ ii ^Tn⁽²⁾

ξ⁽¹⁾_n₊₁

T_n+1⁽¹⁾ 22 ^Lⁿ⁺¹ //ξ⁽²⁾_n₊₁ll ^T_n+1⁽²⁾ then(M1,T1)and(M2,T2)are metrically isomorphic.

The proof of theorem 1.2 is in two steps. In the first step (lemma 1.5), we determine sufficient conditions on a sequence (R^pn

qnbn)_n≥0of periodic rotations of¹such that there exists sequences of finite partitions and automorphisms satisfying the assumptions of lemma 1.4 withM1 =¹,M2 =M,T_n⁽¹⁾ =R^pn

qnb_n,T_n⁽²⁾ =T_n, whereT_nis also smooth diffeomorphism, and such that the limitT in the smooth topology of the sequenceTn

is smooth, andT ∈A¯_αforα=limpn/qn.

In the second step (lemma 1.6), we construct sequences of integers satisfying the conditions of the first step, such that pn/qn → α,bnpn/qn → β, with (α, β) that can be chosen arbitrarily close to any (u,v) ∈ ¹×¹ , and with (α, β) either rationally dependent or rationally independent.

Lemma 1.5. There exists an explicit sequence of integers R₁(n)≥n, such that, if there exist increasing sequences of integers p_n,q_n,a_n,b_n∈^∗, and a sequence s_n∈^∗such that, for any integer n,

1. (primality) anbn−snqn=1.

2. (monotonicity) q_ndivides q_n₊₁and q_n<q_n₊₁.

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3. (isomorphism) qndivides an+1−an.

4. (convergence of the diffeomorphism, generation) 0<

pn+1

qn+1

−pn

qn

≤ 1

(bn+1qn)^R¹⁽ⁿ⁾ then all these assumptions imply that there areα, β∈¹such that

pn

qn

→^mod1α, pn

qn

b_n→^mod1β

and there is a smooth ergodic measure-preserving diffeomorphism T of M such that for any j ∈ ,(D^jT)|∂M = (D^jSα)|∂M and such that(¹,Rβ,Leb)is metrically isomorphic to(M,T, µ).

Lemma 1.6. For any u,v ∈ ¹, for any > 0, there exist (α, β) ∈ ¹×¹ in a- neighborhood of(u,v), such that there exist sequences of integers pn,qn,an,bn ∈^∗, sn∈^∗satisfying the assumptions of lemma 1.5, such that

pn

q_n →^mod1α, pn

q_nbn→^mod1β

Moreover,βcan be chosen either rationally dependent ofαor rationally independent ofα.

We divide the proof of lemma 1.5 in two main parts. In the first part of the proof, we elaborate sufficient conditions onBn ∈Diff^∞(M, µ), whereM =[0,1]^d−1×¹, so that ifTn = B⁻¹_n S^pn

qnBn weakly converges towards an automorphismT, then there exists a metric isomorphism between (¹,Rβ,Leb) and (M,T, µ). To that end, we apply lemma 1.4: we construct a monotonous and generating sequence of partitionsξ_n^∞ofMand a sequence of isomorphisms ¯K_n^∞ : ¹/ζn → M/ξ^∞_n, whereζn ={[i/qn,(i+1)/qn[,i = 0, ...,qn−1}, such that ¯K_n^∞R^pn

qn =TnK¯_n^∞and ¯K_n^∞₊_1|ζ

n =K¯_n^∞. In the construction of this isomorphism, assumption 3 is important. Moreover, we will see that the elements of ξ^∞_n are not the most elementary, because they must be chosen in a way that ensures the monotonicity of the sequence ¯K_n^∞. This condition of monotonicity induces com- binatorial constraints on the elements of the partitionξ^∞_n. Though it follows a similar scheme, the construction of the sequence ¯K_n^∞differs from [4], especially because the assumption 1 is new.

In the second part of the proof, we construct diffeomorphismsT_n = B⁻¹_n S ^pn

qnB_n onM stabilizingξ^∞_n, obtained by successive conjugations from the rotationS^pn

qn. The conjugacy Bn is constructed explicitly. In this second part, we essentially follow [4]

(which elaborated on [7]), except for the obtention of the generation of the sequence of partitions (ξ^∞_n)n≥1, for which we need to slightly modify the construction.

Another change with respect to [4] is in the construction of the limit anglesαand β, i.e. in the proof of lemma 1.6.

1.3 Construction of the limit angles α and β: proof of lemma 1.6.

1.3.1 The caseβ=pα

Letu,v∈ ¹and >0. Let p₀,q₀,b₀be positive integers such that gcd(b₀,q₀) =1, and such that:

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p0

q0

−u _{mod 1}

≤ 2,

p0b0

q0

−v _{mod 1}

≤ 2

By the Bezout theorem, there are integersa0,s0, witha0>0, such thata0b0−s0q0= 1.

Suppose we have definedpk,qk,ak,bk,sk, satisfying the assumptions of lemma 1.5, up to the rankk=n, and let us definepn+1,qn+1,an+1,bn+1,sn+1. (we will havesn=1 forn≥1). Letbn+1 =bn.

Letcnbe an integer sufficiently large so thatcn≥(b_n₊₁qn)^R¹⁽ⁿ⁾andcn≥bn+12ⁿ⁺¹/ (b_n = b0 is constant here, but this more general definition is used for the case (α, β) rationally independent). Let

a_n₊1=a_n+s_nc_nq_n Therefore, assumption 3 holds. Let also

qn+1=qnsn(1+cnbn) Therefore, assumption 2 holds. Moreover, we have:

an+1bn+1−qn+1 =1

Therefore, assumption 1 holds, with sn+1 =1. Moreover, let pn+1 = pn q_n+1

qn +1.

Sinceqn+1 ≥(bn+1qn)^R¹⁽ⁿ⁾, we have:

0<

p_n₊₁ qn+1

− p_n qn

= 1

qn+1

≤ 1

(bn+1qn)^R¹⁽ⁿ⁾ Therefore, assumption 4 holds. Moreover,

pn

qn

bn= p0

q0

b0+

n−1

X

k=0

pk+1

qk+1

bk+1−pk

qk

bk

!

= mod 1

p0

q0

b0+

n−1

X

k=0

pk+1

qk+1

− pk

qk

!

bk+1= p0

q0

b0+

n−1

X

k=0

bk+1

qk+1

Since 1/q_n₊₁≤/(2ⁿ⁺¹b_n₊₁), we get:

p_n qn

bn− p₀ q0

b0

≤ 2 Therefore,

pn

qn

bn→β with|β−u| ≤

Likewise,

pn

qn

→α with|α−v| ≤. Moreover, we haveβ=b0α.

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1.3.2 The case(α, β)rationally independent

The beginning of the construction is the same as in the caseβ = pα, except that we take:

b_n₊₁=b_n+q_n

qn+1=snqn(1+cnbn+cnqn+an) This ensures thatbn→+∞asn→+∞, and that

an+1bn+1−qn+1 =1

It only remains to show that the limit angles (α, β) are rationally independent. To that aim, it suffices to show that the translation of vector (α, β) on the torus ² is ergodic. We follow the proof of the ergodicity of the limit translation in [4], with a slight modification. We recall a theorem by Katok and Stepin [10]:

Theorem 1.7(Katok-Stepin [10]). Let U be an automorphism of a Lebesgue space (N, ν), let(U_n)n≥1be a sequence of measure-preserving transformations, and let(χ_n)n≥1

be a sequence of finite partitions of N with measurable elements. Suppose that:

• U_npermutes the elements ofχ_ncyclically.

• (χ_n)n≥1generates.

• P

c∈χnν(U(c)∆Un(c))=o(1/|χ_n|)(where|χ_n|is the cardinal ofχ_n).

then U is ergodic.

Letγ⁽ⁿ⁾ =(1,bn),gn=gcd(pn,qn). LetΓ⁽ⁿ⁾ ⊂²a fundamental domain of the flow (T^tγ⁽ⁿ⁾)t≥0on², whereT^tγ⁽ⁿ⁾ is the translation of vectortγ⁽ⁿ⁾. Note that the diameter of Γ⁽ⁿ⁾is less than 1/bn. Let

Γ0,n= [

0≤t<^gn

qn

T^tγ⁽ⁿ⁾Γ⁽ⁿ⁾ We have the lemma:

Lemma 1.8. Letζˆnbe the partition defined by:

ζˆn= (

Γi,n=Tⁱ^gnγ

(n)

qn Γ0,n,i=0, ...,qn

g_n −1 )

T^qn^pn^γ⁽ⁿ⁾ is a cyclic permutation onζˆ_n, andζˆ_ngenerates.

Proof. T^pn^qn^γ⁽ⁿ⁾is a cyclic permutation on ˆζnbecausegn=gcd(pn,qn).

To the vector space², we give the normk(x1,x2)k=max1≤i≤2|xi|and we consider its induced norm on².

Since

pn+1−qn+1

qn

pn =1 thenp_n₊₁and^q_qⁿ⁺¹

n are relatively prime. Sinceg_n₊₁dividesp_n₊₁andq_n₊₁, theng_n₊₁ dividesq_n. In particular,g_n₊₁ ≤q_n(this is the slight difference with the proof in [4]:

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in [4], we do not have: gcd(pn+1,^q_qⁿ⁺¹

n ) = 1. But on the other hand, in [4], we have:

gcd(p_n₊₁,^q_qⁿ⁺¹

n )=gcd(p_n₊₁,q_n₊₁). The important point is that in both cases,g_nis small enough).

Moreover, by assumption 3 of lemma 1.5, bn+1qn

qn+1

≤ 1

(bn+1qn)^R¹⁽ⁿ⁾⁻¹ →_n→₊_∞0 Therefore,

diam(Γ0,n)≤max 1 bn

,gnkγ⁽ⁿ⁾k qn

!

≤max 1 bn

,qn−1bn

qn

!

→_n→₊_∞0

It shows that ˆζngenerates.

It remains to estimateP

c∈ζˆ_nµ2

T^αc∆T^pn^qn^γ⁽ⁿ⁾c

, whereµ2 is the Lebesgue measure on². We have the lemma:

Lemma 1.9. We have:

X

c∈ζˆn

µ₂

T^(α,β)c∆T^pn^qn^γ⁽ⁿ⁾c

=o(g_n/q_n)=o(1/|ζˆ_n|)

Proof. We have:

X

c∈ζˆn

µ₂

=X

k≥n

X

c∈ζˆk

µ₂

T^pk+1^qk⁺¹^γ^(k+1)c∆T^pk^qk^γ^(k)c =X

k≥n

X

c∈ζˆk

µ₂

T^pk+1^qk⁺¹^γ^(k+1)⁻^qk^pk^γ^(k)c∆c

Letτ_nbe the (h−1)-volume of the border of an element of ˆζ_n. We have:

µ2

T^qk+1^pk⁺¹^γ^(k+1)⁻^pk^qk^γ^(k)c∆c

≤τk

pk+1

q_k₊1

γ^(k⁺¹⁾− pk

q_kγ^(k) =τk

pk+1

q_k₊1

− pk

q_k

! γ^(k⁺¹⁾

=τk

bk+1

q_k₊1

Moreover,

τ_n≤2 1 bn

+g_nb_n qn

!

≤2 1 bn

+q_n−1b_n qn

!

Therefore,

X

c∈ζˆn

µ₂

=o(g_n/q_n)

By combining lemmas 1.8 and 1.9, and by applying theorem 1.7, we obtain that the translation of vector (α, β) is ergodic with respect to the Lebesgue measure.

This completes the proof of lemma 1.6.

Let us make one remark. We were not able to show our theorem for anyαLiou- ville, because conditions 1-3 of lemma 1.5 introduce arithmetical constraints on the denominators of the convergents ofα. These conditions are analogous to those, in [4], which limit the set of possible translations of theh-dimensional torus,h≥2, that admit a non-standard smooth realization.

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A sufficient condition forαLiouville to belong to a non-standard couple of angles (α, β) withα ,±β, is the following: if, for the sequence p⁰_n,q⁰_n of convergents ofα, there exist positive integerscn,dn, withdn≤q^R(n)n for a fixed sequenceR(n), such that we can write:

q⁰_n₊₁=1+c_nb_n+d_na_n+c_nd_nq_n

then there isβ, ±αsuch that (α, β) is a non-standard couple of angles. (in this construction, we takeb_n₊₁=b_n+d_nq_n, withd_n∈)

The rest of the paper is dedicated to the proof of lemma 1.5. Part of lemma 1.5 is straightforward, namely, the convergence modulo 1 of ^p_qⁿ

n and_q^pⁿ

nbntowardsαandβ respectively:

Partial proof of lemma 1.5. By assumption 2, forn≥2,qn ≥2. By assumption 3, and sinceR1(n)≥n,pn/q_nis Cauchy, and converges.

To show the convergence modulo 1 of_q^pⁿ

nbn, we note that assumptions 1 at ranksn andn+1, and assumption 3 at ranknimply thatqndividesbn+1−bn. Indeed, let us writeb_n₊₁=b_n+k, withkinteger, and let us show thatq_ndividesk. By the assumption 3 at rankn,a_n₊₁=a_n+c_nq_n, withc_ninteger. Therefore,

1+sn+1qn+1=an+1bn+1=(an+cnqn)(bn+k)=anbn+ank+qn(cnbn+cnk) Therefore,

qn sn+1

qn+1

qn

−sn−cnbn−cnk

!

=ank

Thus,qndividesank. Sinceqnis relatively prime withan, thenqndividesk. There- fore,bn/qn=bn+1/qnmod 1. Therefore,

p_n₊₁b_n₊₁ qn+1

−p_nb_n qn

=^mod1

p_n₊₁ qn+1

− p_n qn

|bn+1| ≤ 1 (bn+1qn)^R¹⁽ⁿ⁾⁻¹ Since forn≥1,q_n≥2 andR₁(n)−1≥n−1, then the sequence_p_n_b_n

qn mod 1

n≥1

is Cauchy, and converges.

To show lemma 1.5, it remains to show that there is a smooth ergodic measure- preserving diffeomorphismT of M such thatT ∈ A¯_α and such that (¹,Rβ,Leb) is metrically isomorphic to (M,T, µ).

2 The metric isomorphism

In this section, our aim is to elaborate sufficient conditions onBn∈Diff^∞(M, µ), where M =[0,1]^d−1×¹, so that ifTn =B⁻¹_n S^pn

qnBnweakly converges towards an automor- phismT, then there exists a metric isomorphism between (¹,Rβ,Leb) and (M,T, µ).

To that end, we use lemma 1.4: we construct a monotonous and generating sequence of partitionsξ^∞_n of Mand a sequence of isomorphisms ¯K_n^∞: ¹/ζ_n → M/ξ^∞_n, where ζ_n = {[i/q_n,(i +1)/q_n[,i = 0, ...,qn −1}, such that ¯K_n^∞R^pn

qn = TnK¯_n^∞ and K¯^∞_n₊_1|ζ

n =K¯^∞_n.

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ζ_nis a partition of¹that is monotonic (becauseqndividesqn+1) and that generates (becauseqn→+∞). Letη_n={I×[j/q_n,(j+1)/q_n[,j=0, ...,qn−1}.η_nis a monotonic partition ofM.

The following lemma is straightforward, but important:

Lemma 2.1. Let a_nand q_ntwo relatively prime integers, and let Kn: ζn → ηn

h_i

q_n,ⁱ⁺_q¹

n

h 7→ I×h_ia_n

q_n,^ia_qⁿ⁺¹

n

h

Knis a metric isomorphism such that KnR1 qn =S^an

qnKn. In other words, the following diagram commutes:

ζ_n

R1

qn 88 ^Kⁿ //ηn San

gg qn

This lemma is related with two basic observations: the first is that bothR1

qn andS^an

qn

are isomorphic to cyclic permutations of{0, ...,qn−1}(this set is given the counting measure, i.e.µ(A)=#A); the second observation is that two cyclic permutations of the same order are always conjugated.

The following lemma combines lemma 2.1 with the facts thatζ_n,→ζ_n₊₁and η_n,→η_n₊₁:

Lemma 2.2. Let an,an+1,qn,qn+1 ∈ such thatgcd(an,qn) = gcd(an+1,qn+1) = 1, such that qn divides qn+1 and such that qndivides an+1−an. There exists a partition ηⁿ_n⁺¹ ,→ ηn+1 of M stable by S^an

qn, and there exists a metric isomorphism K_nⁿ⁺¹ : ζn → ηⁿ_n⁺¹ such that Kⁿ_n⁺¹ = Kn+1|ζ_n and such that K_nⁿ⁺¹R1

qn = S^an

qnK_nⁿ⁺¹. There exists also a metric isomorphism Cⁿ_n⁺¹ : ηn → ηⁿ_n⁺¹such that Cⁿ_n⁺¹S^an

qn = S^an

qnCⁿ_n⁺¹ and K_nⁿ⁺¹ = Cⁿ_n⁺¹Kn. Said otherwise, we have the following commutative diagram:

ζn R1

qn 88 ^Kⁿ //

Id

η_n gg ^S^an_qn

Cⁿ⁺¹_n

ζ_n

R1

qn 88 ^K

n+1n //

_

ηⁿ_n⁺ _¹ ll ^S^an_qn

ζ_n₊₁

R 1 qn+1 22

K_n+1 //ηn+1 S_an+1

hh qn+1

Proof. Since gcd(an+1,qn+1)=1, then by lemma 2.1,Kn+1is an isomorphism. More- over, sinceqndividesqn+1, thenζ_n,→ζ_n₊₁. Therefore, we can define the isomorphism Kⁿ_n⁺¹=Kn+1|ζn. Letηⁿ_n⁺¹=Kⁿ_n⁺¹(ζ_n). We haveηⁿ_n⁺¹,→η_n₊₁.

It remains to show thatK_nⁿ⁺¹R1 qn = S^an

qnKⁿ_n⁺¹ (it automatically implies thatηⁿ_n⁺¹ is stable byS^an

qn, and that there isC_nⁿ⁺¹ :η_n → ηⁿ_n⁺¹such thatCⁿ_n⁺¹S^an

qn =S^an

qnC_nⁿ⁺¹). Let 0≤i≤qn−1. We have:

Kⁿ_n⁺¹R1 qn

"

i q_n,i+1

q_n

"!

=Kn+1R1 qn







qn+1 qn −1

[

k=0

"

i q_n + k

q_n₊₁, i

q_n +k+1 q_n₊₁

"







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=Kn+1







qn+1qn −1

[

k=0

"

i+1 qn

+ k qn+1

,i+1 qn

+k+1 qn+1

"







=

qn+1qn −1

[

k=0

Kn+1

"1+i qn

+ k qn+1

,1+i qn

+k+1 qn+1

"!

=I×

qn+1qn −1

[

k=0

"

an+1

qn +an+1i

qn +an+1k qn+1

,an+1

qn +an+1i

qn +an+1k qn+1 + 1

qn+1

"

Sincean+1/qn=an/qnmod 1, we get:

Kⁿ_n⁺¹R¹

qn

"

i qn

,i+1 qn

"!

=I×

qn+1qn −1

[

k=0

"

an

qn +an+1i

qn +an+1k qn+1

,an

qn +an+1i

qn +an+1k qn+1 + 1

qn+1

"

Therefore,

K_nⁿ⁺¹R¹

qn

"

i qn

,i+1 qn

"!

=

qn+1qn −1

[

k=0

S^an

qn I×

"

an+1i qn

+an+1k qn+1

,an+1i qn

+an+1k qn+1

+ 1 qn+1

"!

=S^an

qn





 I×

qn+1 qn −1

[

k=0

"a_n₊₁i qn

+a_n₊₁k qn+1

,a_n₊₁i qn

+a_n₊₁k qn+1

+ 1 qn+1

"







=S^an

qnK_nⁿ⁺¹ " i

qn

,i+1 qn

"!

Let us denoteR⁽ⁿ⁾=K_nⁿ⁺¹h

0,_q¹

n

h. We also denoteRⁿ_i,n⁺¹ =Sian

qnR⁽ⁿ⁾,i =0, ...,qn−1. R⁽ⁿ⁾ is a fundamental domain of S^an

qn. Moreover, we have:

Cⁿ_n⁺¹:η_n → ηⁿ_n⁺¹

"

ian

q_n,ian+1 q_n

"

7→ Rⁿ_i,n⁺¹,i=0, ...,qn+1−1 Note also thatCⁿ_n⁺¹R^an

qn = R^an

qnC_nⁿ⁺¹. Moreover, by assumption 1 of lemma 1.5, a_nb_n/q_n=1/q_nmod 1. Therefore, we get:

C_nⁿ⁺¹R1

qn =Cⁿ_n⁺¹Ranbn qn =Ranbn

qn Cⁿ_n⁺¹=R1 qnC_nⁿ⁺¹

By iterating lemma 2.2, we get a corollary that is important for the construction of the isomorphism:

Corollary 2.3. For any m >n, there are partitionsη^m_n ,→ η^m_n₊₁ of M such thatη^m_n is stable by S 1

qn and there exists an isomorphism K^m_n :ζn→η^m_n such that K_n^mR1 qn =S^an

qnK^m_n and K_n^m=K_n^m₊_1|ηm

n.

Said otherwise, the following diagram commutes:

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ζ^m_n

R1

qn 77 ^K

nm //

_

η^m_n _hh ^S^an_qn

ζ_n₊₁

R 1 qn+1 22

K^m_n+1

//η^m_n₊₁ ^S^an+1

ll qn+1

Proof. The proof is similar to the corollary 3.2 in [4].

For anynfixed, the sequence of partitions (η^m_n)m≥nmust converge whenm→+∞, in order to obtain a full sequence of monotonic partitions. Moreover, the possible limit sequence (i.e. a possibleη^∞_n) must generate. Indeed, these assumptions are required to apply lemma 1.4. However, we can check that none of these assumptions are satis- fied, in general. Therefore, to obtain these assumptions, we pull back the partitionη^m_n by a suitable smooth measure-preserving diffeomorphismBm. The following lemma, already proved in [4], gives the conditions thatB_mmust satisfy:

Lemma 2.4([1],[4]). Let Bm∈Diff^∞(M, µ). Let Am+1=Bm+1B⁻¹_m. 1. If Am+1S ¹

qm =S ¹

qmAm+1and if X

m≥0

qmµ

∆0,q_m∆A⁻¹_m₊₁R^(m)

<+∞

then for any fixed n, when m → +∞, the sequence of partitionsξ^m_n = B⁻¹_mη^m_n converges. We denoteξ^∞_n the limit. The sequenceξ_n^∞is monotonous and Tn = B⁻¹_n S^pn

qnB_nstabilizes eachξ_n^∞.

2. If, moreover, the sequenceξn =B⁻¹_n ηngenerates, then so doesξ^∞_n.

C^m_m⁺¹ is not continuous in general, and A_m₊₁ is its differentiable approximation.

Lemma 2.4 is the reason why we need forMa manifold of dimensiond ≥2. Indeed, if we tookM=¹, we could not find a diffeomorphismB_msatisfying the assumptions of this lemma, except fora_n = 1 ora_n = q_n−1. The choicea_n = 1 gives that the rotationR_αon the circle is isomorphic to itself. The choicean =qn−1 gives thatR_α is isomorphic toR−α. The existence of these two isomorphisms are consistent with the fact, mentioned in the introduction, thatRαandRβare isomorphic, withαirrational, if and only ifα=±β.

By adding to lemma 2.4 the convergence of the sequenceTn, we obtain the required isomorphism:

Corollary 2.5. If both conditions 1. and 2. of lemma 2.4 hold, and if T_n =B⁻¹_n S ^pn

qnB_n weakly converges towards an automorphism T , then (¹,Rβ,Leb)and (M,T, µ)are metrically isomorphic.

Proof. By corollary 2.3,K_n^mR1 qn =S^an

qnK_n^m. By iteration, K_n^mRbn pn

qn =Sanbn pn qn

K_n^m Sinceanbn/qn=1/qnmod 1, then

K_n^mRbn pn

qn =S^pn

qnK_n^m