Circle Diffeomorphisms: Quasi-reducibility and Commuting Diffeomorphisms

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Circle Diffeomorphisms: Quasi-reducibility and Commuting Diffeomorphisms

Mostapha Benhenda

To cite this version:

Mostapha Benhenda. Circle Diffeomorphisms: Quasi-reducibility and Commuting Diffeomorphisms.

2011. �hal-00628298v4�

Circle Di ff eomorphisms: Quasi-reducibility and Commuting Di ff eomorphisms

Mostapha Benhenda

February 28, 2012

Abstract

In this article, we show two related results on circle diffeomorphisms. The first result is on quasi-reducibility: for a Baire-dense set ofα, for any diffeomorphism f of rotation numberα, it is possible to accumulateRαwith a sequencehnf h⁻¹_n , hnbeing a diffeomorphism. The second result is: for a Baire-dense set ofα, given two commuting diffeomorphisms fandg, such that fhasαfor rotation number, it is possible to approach each of them by commuting diffeomorphisms fnandgn

that are differentiably conjugated to rotations.

In particular, it implies that ifαis in this Baire-dense set, and ifβis an ir- rational number such that (α, β) are not simultaneously Diophantine, then the set of commuting diffeomorphisms (f,g) with singular conjugacy, and with rotation numbers (α, β) respectively, isC^∞-dense in the set of commuting diffeomorphisms with rotation numbers (α, β).

1 Introduction

It is well-known that there are circle diffeomorphisms with Liouville rotation num- bers (i.e. non-Diophantine) that are not smoothly conjugated to rotations [1, 7, 8, 9].

A natural question arises, namely, the problem of smooth quasi-reducibility: given a smooth diffeomorphism f of rotation numberα, is it possible to accumulate R_α in the C^∞-norm, with a sequence h⁻¹_n f hn, hnbeing a smooth diffeomorphism?In this case, we say that f is smoothlyquasi-reducibletoRα. Quasi-reducibility is a question that has been studied by Herman [7, pp.93-99], who showed that for anyC²-diffeomorphismf of irrational rotation numberα, it is possible to accumulateRαin theC^1+bv-norm, with a sequenceh⁻¹_n f hn,hnbeing aC²-diffeomorphism (i.e. h⁻¹_n f hn →Rαin theC¹-norm, and the total variation ofD(h⁻¹_n f hn−Rα) converges towards zero). Quasi-reducibility is also related to a problem solved by Yoccoz [10], who showed that it is possible to accumulate a smooth diffeomorphism f in theC^∞-norm with a sequencehnRαh⁻¹_n ,hn

being a smooth diffeomorphism. However, these two problems are not the same, and the method used by Yoccoz does not directly yield our result. In our case, we determine a Baire-dense set of rotation numbersαsuch that for any smooth diffeomorphism f of rotation numberα,f is smoothly quasi-reducible.

∗Laboratoire d’Analyse, Geometrie et Applications, Paris 13 University, 99 Avenue J.B. Clement, 93430 Villetaneuse, France. Contact: mostaphabenhenda@gmail.com. I would like to thank the two anonymous referees for their remarks, which helped improving the article.

Connected to the problem of quasi-reducibility is the following question, raised by Mather: given two commuting C^∞-diffeomorphisms f and g, is it possible to ap- proach each of them in the C^∞-norm by commuting smooth diffeomorphisms that are smoothly conjugated to rotations? In this paper, we determine a Baire-dense set of rotation numbersαsuch that if f andg are commutingC^∞-diffeomorphisms, with f of rotation numberα, then f andg are accumulated in theC^∞ norm by commuting C^∞-diffeomorphisms that areC^∞-conjugated to a rotation. This result is related to a theorem of Fayad and Khanin [6]. They showed that if (α, α⁰) are simultaneously Diophantine (i.e. there isCd > 0, β ≥ 0 such that for any p,p⁰ ∈ š, anyq ≥ 1, max(|α−p/q|,|α⁰−p⁰/q|)≥Cd/q^2+β. This set includes some pairs (α, α⁰) withαand α⁰Liouvillean), and if f andgare commutingC^∞-diffeomorphisms, with f andgof rotation numbersαandα⁰respectively, then f andgare smoothly linearizable. Fayad and Khanin’s result implies our result of quasi-reducibility in the particular case when the rotation numbers of f andgare simultaneously Diophantine. However, in general, our result is not implied by theirs. Indeed, our result holds for a set (α, α⁰) that is Baire- dense in’²(becauseαbelongs to a Baire-dense set of’andα⁰is arbitrary), whereas the set of simultaneously Diophantine numbers is not Baire-dense.¹

Moreover, for Diophantine rotation numbers, which are of full Lebesgue measure, the question of quasi-reducibility and Mather’s problem are trivial, because in this case, the diffeomorphism f is smoothly conjugated to a rotation. Therefore, these two ques- tions remain open for a meagre set of rotation numbers of zero Lebesgue measure.

In order to derive our results, we use estimates of the conjugacy to rotations of dif- feomorphisms having rotation numbers of Diophantine constant type. These estimates were obtained in [2].

The circle is denoted”¹. For r ∈ ’₊∪ {+∞}, we work in the universal cover D^r(”¹), which is the group of diffeomorphisms f of classC^rof the real line such that f −Idisš-periodic. Forα∈’, we denoteRα∈D^∞(”¹) the mapx7→x+α.

Let f ∈D⁰(”¹) be a homeomorphism andx∈’. The sequence ((fⁿ(x)−x)/n)n≥1

admits a limit independent ofx, denotedρ(f). This limit is called therotation number of f. This is a real number invariant by conjugacy.

Theorem 1.1. There is a Baire-dense set A1 ⊂ ’such that for any f ∈ D^∞(”¹)of rotation numberα∈ A₁, there is a sequence h_n ∈ D^∞(”¹)such that h⁻¹_n f h_n → R_αin the C^∞-topology.

Theorem 1.2. There is a Baire-dense set A2 ⊂ ’such that for any f ∈ D^∞(”¹)of rotation numberα ∈ A2and any g of class C^∞with f g =g f , f and g are accumu- lated in the C^∞-topology by commuting C^∞-diffeomorphisms that are C^∞-conjugated to rotations.

1The complementary in’²of simultaneously Diophantine numbers (notedS D^c) is Baire-dense. Indeed, we have:

S D^c=∩_k∈Ž∗∩_n∈Ž∗∪_q≥n(Aq,k×Aq,k) with:

Aq,k= (

α∈’/there is an integerp∈š, α−p

q < 1

q^k )

.

Aq,kis open (and so isAq,k×Aq,k), and for any integern,∪_q≥n(Aq,k×Aq,k) is dense, because it contains all pairs of rational numbers (ifα=p₁/q1andα⁰=p₂/q2, then (α, α⁰)∈(Ajq₁q₂,k×A_jq1q2,k) for anyj,k∈Ž^∗).

Therefore,S D^cis Baire-dense.

Remark1.3. The proof of theorem 1.1 also gives thathnRαh⁻¹_n → fin theC^∞-topology ifα∈A1.

Remark1.4. Combined with [6, p. 965], theorem 1.2 implies that ifα∈A2, and (α, β) are not simultaneously Diophantine, thenS_α,β, the set of couples (f,g) of smooth com- muting circle diffeomorphisms with singular conjugacies to R_α andR_β respectively, isC^∞-dense inF_α,β, the set of couples (f,g) of smooth commuting circle diffeomor- phisms with rotation numbersαandβrespectively.

Indeed, our result shows thatOα,β, the set of couples (f,g) of smooth commuting circle diffeomorphisms with smooth conjugacies toRαandRβrespectively, isC^∞-dense in Fα,β. Moreover, in [6, p.965], for (α, β) not simultaneously Diophantine, Fayad and Khanin described the construction of a couple (f,g) of smooth commuting circle diffeomorphisms with singular conjugacies toRαandRβrespectively. This construction relies on the method of successive conjugacies, which can be madeC^∞-dense inOα,β

[5].

Moreover, by slightly modifying [7, p.160, p.167], this implies that (O¹_α,β)^c, the set of couples (f,g) of smooth commuting circle diffeomorphisms with non-C¹ conjuga- cies to rotationsR_αandR_β, isC^∞-generic inF_α,β. See appendix A for a short proof.

2 Preliminaries

2.1 Basic properties

When the rotation numberαof f is irrational, and if f is of classC², Denjoy showed that f is topologically conjugated toR_α. However, this conjugacy is not always differ- entiable. It depends on the Diophantine properties of the rotation numberα.

Letα=a0+1/(a1+1/(a2+...)) be the development ofα∈’in continued fraction (see [4]). It is denotedα =[a0,a1,a2, ...]. Let p−2 = q−1 =0, p−1 =q−2 =1. For n≥0, we define integers pnandqnby:

pn=anpn−1+pn−2

qn=anqn−1+qn−2.

We haveq0 =1,qn ≥1 forn ≥1. The rationals pn/qnare called the convergents ofα. Remember thatqn+2 ≥2qn, forn≥ −1.

For any real numberβ≥0,α∈ ’−‘is Diophantine of orderβand constantCd

(a set denotedDC(Cd, β)) if there is a constantCd >0 such that for any p/q∈‘, we have:

α− p q

> Cd

q^2+β.

Each of the following relations characterizesDC(Cd, β) (see e.g. [11, pp.50-51]):

1. |α−pn/qn|>Cd/q²_n^+βfor anyn≥0 2. an+1<_C¹

dq^β_nfor anyn≥0 3. qn+1<_C¹

dq¹n^+βfor anyn≥0

4. α_n+1>Cdα¹_n^+βfor anyn≥0.

DC(C_d,0) is the set of irrational numbers ofconstant type C_d. The first derivative of f ∈D¹(”¹) is denotedD f.

2.2 Some useful lemmas

For anyninteger, letαn =[a0, ...,an,1, ...].

LetVα:N→Rdefined by:Vα(n)=max0≤i≤nai. Observe thatαn∈DC(1/Vα(n),0).

We will need the lemma:

Lemma 2.1. Letαbe an irrational number, qnits convergents andαn =[a0, ...,an,1, ...].

We have:

|α_n−α| ≤ 2 q²_n ≤ 4

2ⁿ.

Proof. Let ˜α_n =[a₀, ...,a_n,0, ...]. By induction, we can show that ˜α_n = p_n/q_n. More- over, ˜α_nis also then^thconvergent ofα_n. Therefore, by the best rational approximation theorem,|α−pn/qn| ≤1/q²_nand|αn−pn/qn| ≤1/q²_n. Moreover, sinceqn+2 ≥qn, then qn≥(

√ 2)ⁿ⁻¹.

We need the lemma:

Lemma 2.2. Letφ:Ž→’₊be such thatφ(n)→_n→+∞+∞. Let A={α∈’/V_α(n)< φ(n)for an infinity of n}.

Then A is Baire-dense.

Proof. First, we show that for any positive integersnandi,

Ai,n = {αsuch thatai < φ(n)} is open. Let u(x) = bxc, v(x) = ¹_x and w(x) = v(x)−u(v(x)). We have: ak+1 = v(w^k(x))−w^k+1(x). Since vis continuous andu is upper semi-continuous and non-negative, thenwis lower semi-continuous. Moreover, wis non-negative. Therefore,w^k andw^k+1 are also lower semi-continuous and non- negative. Since vis decreasing, then v◦w^k−w^k+1 is upper semi-continuous. We conclude thatAi,nis open.

Moreover, for anyp≥0,

∪_n≥p∩_i≤nAi,n

is dense. Indeed, sinceφ(n) → +∞, then it contains all numbers of constant type, which are dense. This set is also open and therefore,

A=∩_p≥0∪_n≥p∩_i≤nA_i,n is Baire-dense.

2.3 Notations

• For any real numbersaandb,a∨bdenotes max(a,b).

• Forφa realš-periodicC^rfunction, 0≤r<+∞, we define:

kφk_r =max

0≤j≤rmax

x∈’

|D^jφ(x)|.

Note that for f,g ∈ D^r(”¹), f −g isš-periodic, and for 1 ≤ j ≤ r, D^jf is

š-periodic. For f ∈D^r(”¹), we also define:

kfk_r=max kf−idk0,max

1≤j≤rkD^jfk₀

! .

Note that the notationkfk_ris not a norm when f ∈D^r(”¹), sinceD^r(”¹) is not a vector space.

• In all the paper,C denotes a constant depending onu. W(f) denotes the total variation of logD f, andS f denotes the Schwartzian derivative of f.

2.4 Estimates of the conjugacy

The following theorem gives an estimate of the linearization of a diffeomorphism hav- ing a rotation numbers of Diophantine constant type. This estimate, obtained in [2], is necessary to derive our results.

Theorem 2.3. Let l≥3be an integer andη >0. Let f ∈D^l(”¹)be of rotation number α, such thatαis of constant type Cd. There exists a diffeomorphism h ∈ D^l−1−η(”¹) conjugating f to Rα, and a function B of Cd,l, η,W(f),kS fk_l−3, which satisfy the esti- mate:

max 1

minDh,khkl−1−η

!

≤B(Cd,l, η,W(f),kS fk_l−3). (1) In particular, we remark that if fn is a sequence of diffeomorphisms of rotation numberαn, if the sequences W(fn) andkS fnk_l−3 are bounded (this will hold in our case, because we will take fn=λn+f for a properly chosenλn ∈’), ifVα(n)→+∞ and ifhnis the conjugacy to a rotation associated with fn, then there is a real function E(Vα(n)) such that, fornsufficiently large, we have:

max 1

minDhn

,khnk_l−1−η

!

≤E(Vα(n)).

3 Quasi-Reducibility

Theorem 3.1. Let l ≥ 3 be an integer, f ∈ D^l(”¹)be of rotation numberα ∈ ”¹. Letη >0be a real number. There exists a numerical sequence F(n), going to+∞as n→+∞, such that, if

lim infVα(n) F(n) =0

then there is a sequence hn of class C^l−1−ηsuch that h⁻¹_n f hn → Rα in the C^l−2−η- topology.

By applying lemma 2.2, we obtain the corollary:

Corollary 3.2. There is a Baire-dense set A₁ ⊂ ’ such that if l ≥ 3 is an integer, f ∈ D^l(”¹)of rotation numberα ∈ A1 andη > 0, then f is C^l−2−η-quasi-reducible:

there is a sequence hn∈D^l−1−η(”¹)such that h⁻¹_n f hn→R_αin the C^l−2−η-topology.

The idea of the proof of theorem 3.1 is the following. We observe that for any sequenceφ(n)→+∞, the set of numbersαsuch that for an infinity ofn,

sup_k≤nak≤φ(n), is Baire-dense (lemma 2.2).

The truncated sequence of constant type numbersαn =[a0, ...,an,1, ...] converges towardsαat a controlled speed:|α−αn| ≤4/2ⁿ(lemma 2.1).

Following an idea of Herman [7], we perturbate f toRλ_nf = f +λn of rotation numberαn, which is linearizable by a conjugacyhn(lemma 3.3). By writing:

h⁻¹_n f hn−Rα=h⁻¹_n f hn−h⁻¹_n Rλnf hn+Rαn−Rα

and by applying the Faa-di-Bruno formula, we obtain a control of the norm of h⁻¹_n f h_n −R_α in function of the norm ofh_n, and in function of|α−α_n|(lemma 3.4).

Moreover, we have an estimate of the norm ofh_nin function of sup_k≤na_k.

Thus, if we choose the speed of growth of the sequence sup_k≤na_ksufficiently small with respect to the speed of convergence ofαn towardsα, thenh⁻¹_n f hn converges to- wardsRα, and f is quasi-reducible.

Proof of theorem 1.1.We letη=l/3 in corollary 3.2. Since f is smooth, then there is a sequence (hn,l)n≥0∈D^∞(”¹) such that, for any integerl≥3 fixed,

kh⁻¹_n,lf hn,l−Rαk₂(^l₃⁻¹)→_n→+∞0.

In particular, there isn(l) such that:

kh⁻¹_n(l),lf hn(l),l−Rαk₂(₃^l−1)≤1 l.

Lethl=hn(l),l. Let >0, and letk>0 be an integer. There isl0 ≥0 such that for anyl≥l₀, we have: ≥1/l,k≤2_l

3−1 and:

kh⁻¹_l f hl−Rαk_k≤ kh⁻¹_l f hl−Rαk₂(^l3−1)≤1 l ≤.

Therefore,h⁻¹_l f hl →_l→+∞ R_α in theC^k-topology, for any k, and therefore, this convergence holds in theC^∞-topology.

3.1 The one-parameter family R

f

To prove theorem 3.1, we need to consider the one-parameter familyRλf =f +λ(see [7, p.31]). We have the lemma:

Lemma 3.3. Let l≥3be an integer, f ∈D^l(”¹),0< η≤l−3,α=ρ(f). Letα˜ be an irrational number of constant type. There existsλ₀ ∈ ’and a C^l−1−η-diffeomorphism h such that h⁻¹Rλ0f h=Rα˜. Moreover,

|λ₀|

minDh ≥ |α˜−α| ≥ |λ₀| kDhk0

.

Proof. Letµ(λ)=ρ(R_λf). µis continuous, non-decreasing andµ(’) =’(see [7, p.

31]). Therefore, there existsλ₀ ∈ ’such that ˜α = ρ(R_λ0f). Since ˜αis of constant type, there exists aC^l−1−η-diffeomorphismhsuch thath⁻¹Rλ0f h=Rα˜ and that satisfies estimate (1) of theorem 2.3. By the mean value theorem, for anyx, there isc(x) such that:

α˜+x−h⁻¹f h(x)=Rα˜(x)−h⁻¹f h(x)=h⁻¹Rλ₀f h(x)−h⁻¹f h(x)=D(h⁻¹)(c(x))λ0. By integrating this equation on an invariant measure ofh⁻¹f h, we get lemma 3.3.

Note that sinceh∈D¹(”¹), thenDh(x)>0 for anyx, and minDh>0.

3.2 The speed of approximation of R

The proof of theorem 3.1 is also based on the lemma:

Lemma 3.4. Let l≥3be an integer, f ∈ D^l(”¹),0 < η≤l−3,α =ρ(f). Letα˜ be an irrational number of constant type, and letλ₀ ∈’and h the C^l−1−η-diffeomorphism be given by lemma 3.3. Recall that C denotes a constant that only depends on u, 0≤u≤l−2−η. We have the estimate:

kh⁻¹f h−Rαk_u ≤Ckfk^C_ukhk^C_u+1 1

(minDh)^C|α˜−α|.

Before proving lemma 3.4, we show how theorem 3.1 is derived from it.

proof of theorem 3.1. Ifαis of constant type, then f is reducible and there is nothing to prove. Therefore, we can suppose thatV_α(n)→_n→+∞+∞. By applying theorem 2.3, there exists a real function ˜Fstrictly increasing withV_α(n), such that forαn, and for its associated diffeomorphismhngiven by lemma 3.3, we have, fornsufficiently large:

kh⁻¹_n f hn−Rαk_l−2−η≤exp

F˜(Vα(n))

|αn−α|.

LetF(n)=F˜⁻¹(n^1/2). By extracting, we can suppose that lim^V_F(n)^α(n) =0. Therefore, Vα(n)≤F(n) fornsufficiently large and therefore,

F˜(V_α(n))≤n^1/2. We get, fornsufficiently large,

kh⁻¹_n f hn−Rαk_l−2−η≤e^{−nlog 24} →_n→+∞0.

Hence theorem 3.1.

Now, we show lemma 3.4:

proof of lemma 3.4. We need the Faa-di-Bruno formula (see e.g. [3]):

Lemma 3.5. For every integer u≥0and functionsφandψof class C^u, we have:

D^uφ(ψ(x))=

u

X

j=0

D^jφ(ψ(x))Bu,j

Dψ(x),D²ψ(x), . . . ,D^(u−j+1)ψ(x) .

The Bu,jare the Bell polynomials, defined by Bu,0=1and, for j≥1:

B_u,j(x1,x2, . . . ,xu−j+1)=X u!

l1!l₂!· · ·lu−j+1! x1

1!

l₁x2

2!

l₂

· · · xu−j+1

(u−j+1)!

!l_u−j+1

. The sum extends over all sequences l1,l2,l3, ...,lu−j+1of non-negative integers such that:l1+l2+...= j and l1+2l2+3l3+...=u.

Therefore, for anyx, we have the estimate:

Bu,j

Dψ(x),D²ψ(x), . . . ,D^(u−j+1)ψ(x) ≤C

1∨ kψk_u^j

. (2)

Combining this estimate with lemma 3.5, we obtain the corollary:

Corollary 3.6. For every integer u≥0and functionsφandψof class C^u, we have:

kφ◦ψk_u≤Cmax

0≤j≤ukD^jφ◦ψk0 1∨ kψk^u_u.

We apply this corollary to estimatekh⁻¹k_u. We letφ(x) =1/xandψ= Dh◦h⁻¹. We observe thatD(h⁻¹)= _Dh◦h¹−1 =φ◦ψ. Since there isx0such thatDh(x0)=1, then kDhk0≥1 (and we also have 1≥minDh>0). Therefore, we get:

kD(h⁻¹)k_u≤Cmax

0≤j≤u

1 k Dh◦h^−1j+1

k₀kDh◦h⁻¹k^C_u. By corollary 3.6, we also have:

kDh◦h⁻¹k_u≤CkDhkukh⁻¹k^C_u. By combining these two estimates, we get:

kD(h⁻¹)ku ≤C 1

(minDh)^CkDhk^C_ukh⁻¹k^C_u. We iterate this estimate to estimatekh⁻¹k_u, foru≥1. We get:

kh⁻¹k_u+1 ≤C 1

(minDh)^Ckhk^C_u+1kh⁻¹k^C₁. (3) Now, we estimate theC^u-distance ofh⁻¹f htoR_α. Let ˜α, λ₀be as in lemma 3.3. We have:

h⁻¹f h−Rα=h⁻¹f h−h⁻¹Rλ₀f h+Rα˜ −Rα. Therefore,

kh⁻¹f h−R_αk_u≤ kh⁻¹f h−h⁻¹R_λ0f hk_u+|α˜−α|. (4)

On the other hand, by the Faa-di-Bruno formula, we have:

D^uh

h⁻¹f h−h⁻¹Rλ₀f hi (x)=

u

X

j=0

Bu,j

D(f h)(x), ...,D^u−j+1(f h)(x) hD^j(h⁻¹)(f h(x))−D^j(h⁻¹)(f h(x)+λ0)i

.

Since|D^j(h⁻¹)(f h(x))−D^j(h⁻¹)(f h(x)+λ₀)| ≤ kD^j+1(h⁻¹)k₀|λ₀|, then by applying estimate (2), we get:

kh⁻¹f h−h⁻¹Rλ₀f hku≤Ckf◦hk^C_ukh⁻¹k_u+1|λ0|.

By applying corollary 3.6, we get:

kh⁻¹f h−h⁻¹R_λ0f hk_u ≤Ckfk^C_ukhk^C_ukh⁻¹k_u+1|λ₀|.

By applying (3), we obtain:

kh⁻¹f h−h⁻¹Rλ₀f hk_u ≤Ckfk^C_ukhk^C_u 1

(minDh)^Ckhk^C_u+1kh⁻¹k^C₁|α˜−α|kDhk₀ kh⁻¹f h−h⁻¹Rλ₀f hku ≤Ckfk^C_ukhk^C_u+1 |α˜−α|

(minDh)^C. By estimate (4), we obtain:

kh⁻¹f h−R_αk_u ≤Ckfk^C_ukhk^C_u+1 1

(minDh)^C|α˜−α|. (5) Hence lemma 3.4.

4 Application to commuting di ff eomorphisms

Theorem 4.1. There exists a numerical sequence G(n), going to+∞as n→+∞, such that, for any l ≥3an integer, f ∈ D^l(”¹)of rotation numberα∈ ’,η >0and g of class C^lsuch that f g=g f , if

lim infV_α(n) G(n) =0

then there exists two sequences of diffeomorphisms fn and gn that are C^l−1−η- conjugated to rotations, such that fngn = gnfn, and with fn and gn converging re- spectively towards f and g in the C^l−2−η-norm.

Corollary 4.2. There is a Baire-dense set A2 ⊂ ’ such that if l ≥ 3 is an integer, f ∈ D^l(”¹)has a rotation numberα ∈ A₂, g is of class C^lsuch that f g = g f and η∈’₊, then there exists two sequences of diffeomorphisms f_nand g_nthat are C^l−1−η- conjugated to rotations, such that f_ng_n =g_nf_n and with f_nand g_nconverging respec- tively towards f and g in the C^l−2−η-norm.

We derive theorem 1.2 from corollary 4.2 by following the same argument as in the proof of theorem 1.1.

4.1 The speed of approximation of g by a linearizable and com- muting di ff eomorphism

To prove theorem 4.1, we consider (h_n)_n≥0, the sequence of conjugating diffeomor- phisms constructed in the proof of theorem 3.1, (λ_n)_n≥0 the associated sequence of real numbers such that f_n = R_λnf = h_nR_αnh⁻¹_n . We also consider g⁰_n = h⁻¹_n gh_n and g_n=h_nR_g⁰_n(0)h⁻¹_n . The diffeomorphisms f_nandg_ncommute, and f_n → f in theC^l−2−η- norm. To prove theorem 4.1, it suffices to show thatg_n →gin theC^l−2−η-norm. This convergence is based on the lemma:

Lemma 4.3. Let l ≥ 3be an integer, f ∈ D^l(”¹)of rotation numberα ∈ ’,η > 0, 0≤u≤l−2−η, and g∈D^l(”¹)be such that f g=g f . Let(q_t)_t≥0be the sequence of denominators of the convergents ofα, and let r≥0be an integer. Letα˜be an irrational number of constant type,λ₀ ∈ ’the associated number and h the associated C^l−1−η diffeomorphism given by lemma 3.3. Let f⁰ = h⁻¹f h and g⁰ = h⁻¹gh. We have the estimate:

kg−hR_g⁰₍₀₎h⁻¹k_u≤Ckhk^C_u+1kfk^C_ukgk^C_u+1 1 qr

+|α˜−α| (Ckhk_u+1kfk_u+1)^Cqr (minDh)^C

!!

. To show this lemma, the basic idea is the following: we approach modulo 1 points x∈Rbyp(x)αmod 1, where p(x)≤q_ris an integer, and where the integerrwill be fixed later. We have a control of|x−p(x)α|mod 1 in function ofq_r. Then, by using the assumption of commutationg⁰f^0p= f^0pg⁰, we can write:

g⁰(x)−Rg⁰(0)(x)=g⁰(x)−g⁰(pα)+g⁰(pα)−g⁰f^0p(0)+f^0pg⁰(0)−Rpα(g⁰(0))+Rg⁰(0)(pα)−Rg⁰(0)(x).

We use the distance of f^0ptoRpα, which depends onqrand the norm of f⁰−Rα. This distance has been estimated in the proof of the result of quasi-reductibility. We also useC^k analogues, k ≥ 2, of the mean value theorem, obtained with the Faa-di- Bruno formula. This allows to estimate the norm ofg−hRg⁰(0)h⁻¹ in function of the norm ofg⁰−Rg⁰(0).

To obtain theorem 4.1 from lemma 4.3, we take ˜α= α_n, and we consider the as- sociated sequences f_n,g_n,f_n⁰,g⁰_n,h_n. The integerq_r must be chosen sufficiently large with respect to the conjugacyh_n, so that|x−pα|mod 1 is sufficiently small. How- ever, this integerqr must not be too large, to keep the norm of f_n^0p−Rpαsufficiently small. This integerqris controlled with sup_k≤rak, which itself controls the norm ofhr. Thus, it suffices to properly choose the integerrin function ofn, in order to obtain the convergence ofgntowardsg.

Proof of theorem 4.1. Assuming lemma 4.3, we show theorem 4.1.

Let ˜α = α_nandh_n be the associated diffeomorphism given by lemma 3.3. Since V_α(n)→+∞, by applying the estimate for the conjugacyh_n, there exists ˜G(x) strictly increasing withxsuch that, fornsufficiently large:

kg−h_nR_g⁰_n(0)h⁻¹_n k_l−2−η≤e^{CG(V˜ α(n))}







 1 qr

+e^{CG(V˜ α(n))qr} 2ⁿ







. Moreover, sinceq_n=a_nqn−1+qn−2, andqn−2≤qn−1, then

(

√

2)ⁿ⁻¹≤qn≤

n

Y

k=1

(ak+1). (6)

Therefore, we get:

kg−h_nR_g⁰_n(0)h⁻¹_n k_l−2−η≤e^{CG(V˜ α(n))−12}(r−1) log 2+e^{CG(V˜ α(n))+CG(V˜ α(n))(Vα(r)+1)r−nlog 2}. (7) LetG(n) = G˜⁻¹((logn)^1/2). By extracting in the sequence Vα(n)/G(n), we can suppose that:

Vα(n) G(n) →0.

Therefore, fornsufficiently large, we have:

G(V˜ _α(n))≤(logn)^1/2.

Moreover, fornsufficiently large, we can take an integerrnsuch that:

(logn)^3/4 ≤rn≤(logn)^7/8. We get:

(Vα(rn)+1)^rn=e^{rnlog(Vα(rn)+1)}≤e^(logn)15/16.

The first term in estimate (7) tends towards 0. Moreover, since,fornsufficiently large,

(logn)^1/2e^(logn)15/16 ≤ n 2log 2 then the second term also tends towards 0. Hence theorem 4.1.

4.2 Higher-order analogous of the mean value theorem

Proof of lemma 4.3. We need two higher-order analogous of the mean value theorem.

The first one is:

Lemma 4.4. Let u≥0, s,t∈D^u(”¹). Letδ∈’. We have:

kst−R_δtk_u≤Cksk_u+1ks−R_δk_uktk^u_u.

Observe the presence of the termksku+1, which is absent in the mean value formula.

This is because of the estimate (2) on the Bell polynomial, in the Faa-di-Bruno formula.

Proof. Ifu=0, the estimate is trivial. We supposeu≥1. For anyx∈’, the Faa-di- Bruno formula gives:

D^u(st)(x)−D^u(R_δt)(x)=

u

X

j=0

(D^js)(t(x))−(D^jR_δ)(t(x)) B_u,j

Dt(x), ...,D^u−j+1t(x) .

Therefore, by estimate (2), and sincektk_u≥1,

|D^u(st)(x)−D^u(Rδt)(x)| ≤Cksk_u+1ks−Rδk_uktk^u_u. Hence lemma 4.4.

The second higher-order analogous of the mean value theorem is:

Lemma 4.5. Let u≥0, s∈D^u+1(”¹), t∈D^u(”¹),δ∈’. We have:

kst−sRδk_u≤Cksku+1ktk^u_ukt−Rδk_u.

Observe the presence of the termktku, which is absent in the mean value formula.

As in lemma 4.4, this is because of an estimate on the Bell polynomial, in the Faa-di- Bruno formula.

Proof. Ifu=0, the estimate holds. We supposeu≥1. We use the following lemma:

Lemma 4.6. Let u ≥ 1, j ≤ u be integers and a1, ...,au−j+1,x1, ...,xu−j+1 ≥ 0. Let x≥max{|xk| ∨1; 1≤k≤u−j+1}and let a≥max{|ak|; 1≤k≤u−j+1}. Let Bu,j

be a Bell polynomial. We have:

|Bu,j(x1+a1, ...,xu−j+1+au−j+1)−Bu,j(x1, ...,xu−j+1)| ≤Ca(x+a)^u. Proof. Letp≥1 andl1, ...,lpbe integers. Then we have:

(x1+a1)^l1...(xp+ap)^lp−x^l₁¹...x^l_p^p=

p

X

i=1

x^l₁¹...x^l_i−1ⁱ⁻¹(xi+ai)^li...(xp+ap)^lp−x^l₁¹...x^l_iⁱ(xi+1+ai+1)^li+1...(xp+ap)^lp

(x₁+a₁)^l1...(x_p+a_p)^lp−x^l₁¹...x^l_p^p=

p

X

i=1

x^l₁¹...x^l_i−1ⁱ⁻¹(x_i+1+a_i+1)^li+1...(x_p+a_p)^lph

(x_i+a_i)^li−x^l_iⁱi (with the conventionsx^l₁¹...x^l₀⁰ =1 andx^l_p^p+1₊₁...x^l_p^p=1).

Since (xi+ai)^li−x^l_iⁱ≤li|ai|(|xi|+|ai|)^li−1≤lia(|xi|+a)^li−1, 1≤li≤uandx+a≥1 (becausex≥1), we obtain:

|B_u,j(x₁+a₁, ...,xu−j+1+au−j+1)−B_u,j(x₁, ...,xu−j+1)| ≤a(u−j+1)uB_u,j(x+a, ...,x+a).

By the formula giving the Bell polynomials, we have:

Bu,j(x+a, ...,x+a)≤C(x+a)^u.

To show lemma 4.5, For any 0≤v≤u, we write:

D^v(st)(x)−D^v(sRδ)(x)=

v

X

j=0

D^js(t(x))h Bv,j

Dt(x), ...,D^v−j+1t(x)

−Bv,j

DRδ(x), ...,D^v−j+1Rδ(x)i + hD^js(t(x))−D^js(R_δ(x))i

Bv,j

DRδ(x), ...,D^v−j+1Rδ(x) .

We apply lemma 4.6 witha=kt−Rδk_uandx=kRδk_u≥1. Sincet∈D^u(”¹), then ktk_u≥1. We get:

Bv,j

Dt(x), ...,D^v−j+1t(x)

−Bv,j

DRδ(x), ...,D^v−j+1Rδ(x)

≤Ckt−Rδk_u(1+kt−Rδk_u)^u

B_v,j

Dt(x), ...,D^v−j+1t(x)

−B_v,j

DR_δ(x), ...,D^v−j+1R_δ(x)

≤Ckt−R_δk_u(2+ktk_u)^u≤Ckt−R_δk_uktk^u_u.

4.3 Successive estimates

To prove lemma 4.3, we also need these successive estimates:

Lemma 4.7. Let l ≥ 3be an integer, f ∈ D^l(”¹)of rotation numberα ∈ ’,η > 0, 0≤u≤l−2−η, and g∈D^l(”¹)be such that f g=g f . Let(q_t)_t≥0be the sequence of denominators of the convergents ofα. Letα˜be an irrational number of constant type, λ₀ ∈ ’the associated number and h the associated C^l−1−η diffeomorphism given by lemma 3.3. Let f⁰=h⁻¹f h and g⁰=h⁻¹gh. We have the estimates:

A1,u=kh⁻¹k_u≤Ckhk^C_u 1

(minDh)^C (8)

A_2,u=kf⁰k_u≤CA_1,ukfk^C_ukhk^C_u (9) A3,u(m)=kf^0mk_u ≤C^mA^mC_2,u (10) A_4,u =kf⁰−R_αk_u≤Ckhk^C_u+1kfk^C_u 1

(minDh)^C|α˜−α| (11) A5,u(m)=kf^0m−Rmαk_u≤mCA4,uA^C_2,u max

k≤m−1A3,u+1(k) (12)

A6,u=kg⁰k_u≤CA1,ukgk^C_ukhk^C_u (13) and for any integer r≥0, we have:

A7,u=kg⁰−Rg⁰(0)k_u ≤A_6,u+1+1 qr +max

m≤2q_r

A6,u+1A^C_3,u(m)A5,u(m)+A^C_6,uA3,u+1(m)A5,u(m) (14) A8,u=kg⁰h⁻¹−Rαh⁻¹k_u≤CA6,u+1A7,uA^C_1,u (15) A_9,u=khg⁰h⁻¹−hRg⁰(0)h⁻¹k_u≤Ckgk^C_uA_8,uA^C_1,ukhku+1. (16) The crucial estimate is (14), which is obtained by approaching modulo 1 eachx∈’ by am(x)α, withm(x) ≤ q_r. Ifq_r increases,x−m(x)αis smaller modulo 1, but the bound onA_3,u(m(x)) andA_5,u(m(x)) increases. In the proof of theorem 4.1, we make a proper choice ofr(andq_r).

estimate (11) corresponds to estimate (5) of the proof of the result of quasi-reducibility.

The other estimates, namely, estimates (8),(9),(10), (12),(13), (15) and (16) are derived from applications of the Faa-di-Bruno formula: either corollary 3.6, lemma 4.4 or lemma 4.5.

Proof of lemma 4.7. ForA1,u, by estimate (3), we have:

kh⁻¹k_u ≤Ckhk^C_u 1 (minDh)^C. Hence estimate (8).

ForA_2,u, by applying corollary 3.6 twice, we have,

kf⁰k_u≤CA1,ukfk^C_ukhk^C_u. Hence estimate (9).

ForA_3,u, by applying corollary 3.6 again, we have, for anym, kf^0m+1k_u ≤Ckf^0mk_ukf⁰k^C_u

and therefore, by iteration, we get:

kf^0mk_u≤C^mkf⁰k^mC_u . Hence (10).

estimate (11) is a direct application of estimate (5).

For estimate (12), we observe that for any 0≤v≤u:

D^vf^0m−D^vRmα=D^v

m−1

X

k=0

f^0m−kRkα−f^0m−k−1R(k+1)α

D^vf^0m−D^vRmα=

m−1

X

k=0

D^v

f^0m−k−1f⁰

Rkα−D^v

f^0m−k−1Rα Rkα.

By applying lemma 4.5, and by noting that for anyk,kf^0m−k−1k_u+1≤max_{0≤k≤m−1}kf^0kk_u+1, we get:

kf^0m−Rmαk_u≤mCkf⁰k^C_u max

0≤k≤m−1

kf^0kk_u+1kf⁰−Rαk_u. Hence (12).

ForA6,u, estimate (13) is the same as (9):

kg⁰k_u≤Ckh⁻¹k_ukgk^C_ukhk^C_u. Hence (13).

ForA7,u, letm≥0 andu≥v≥1. For anyx,D^vRα(x)=R1

0 D^vg⁰(y)dy. Therefore,

|D^vg⁰(x)−D^vR_α(x)|=

D^vg⁰(x)− Z 1

0

D^vg⁰(y)dy

=

Z 1 0

D^vg⁰(x)−D^vg⁰(y) dy

≤ max

x,y∈[0,1]|D^vg⁰(x)−D^vg⁰(y)|.

On the other hand, we have:

D^vg⁰(x)−D^vg⁰(y)=D^vg⁰(x)−D^vg⁰(y+mα)+D^vg⁰(R_mα(y))−D^v(g⁰f^0m(y))+ D^v(f^0mg⁰(y))−D^vg⁰(y).

Moreover, we have:

|D^vg⁰(x)−D^vg⁰(y+mα)| ≤ |D^u+1g⁰|₀|x−y−mα|.

By lemma 4.5, we also have:

|D^vg⁰(Rmα(y))−D^v(g⁰f^0m(y))| ≤Ckg⁰k_u+1kf^0mk^C_ukf^0m−Rmαk_u. Finally, by lemma 4.4, we have:

|D^v(f^0mg⁰(y))−D^v(Rmαg⁰(y))| ≤Ckf^0mk_u+1kf^0m−Rmαk_ukg⁰k^C_u.

SinceRmαg⁰(y)=g⁰(y)+mα, andv≥1, thenD^v(Rmαg⁰(y))=D^v(Rmαg⁰(y)). There- fore, the same estimate holds for|D^v(f^0mg⁰(y))−D^v(g⁰(y))|.

By combining these estimates, we obtain:

|D^vg⁰(x)−D^vg⁰(y)| ≤ kg⁰k_u+1|x−y−mα|+Ckg⁰k_u+1kf^0mk^C_ukf^0m−R_mαk_u+ Ckf^0mk_u+1kf^0m−Rmαk_ukg⁰k^C_u. Moreover, for anyr≥0, anyx,y∈’, there is an integerm(x,y)≤2qr, there are real numbersx⁰,y⁰such thatx⁰−x∈š,y⁰−y∈šand such that|x⁰−y⁰−m(x,y)α| ≤1/q_r. Sincev ≥ 1, then|D^vg⁰(x)−D^vg⁰(y)| = |D^vg⁰(x⁰)−D^vg⁰(y⁰)|. We apply the former estimate withx⁰andy⁰and we get:

max

1≤v≤u

D^vg⁰−D^vRg⁰(0)

₀≤ A6,u+1+1 qr

+max

m≤2q_r

A6,u+1A^C_3,u(m)A_5,u(m)+A^C_6,uA3,u+1(m)A_5,u(m) . Ifv=0, we note that for anyr≥0, anyx∈’, there is an integerm(x)≤q_rand a real numberx⁰∈’such thatx⁰−x∈š, and such that|x⁰−m(x)α| ≤1/q_r. Moreover, we have:g⁰(x)−R_g⁰₍₀₎(x)=g⁰(x⁰)−R_g⁰₍₀₎(x⁰), and

g⁰(x⁰)−Rg⁰(0)(x⁰)=g⁰(x⁰)−g⁰(mα)+g⁰(mα)−g⁰f^0m(0)+f^0mg⁰(0)−Rmα(g⁰(0))+Rg⁰(0)(mα)−Rg⁰(0)(x⁰).

Hence estimate (14).

ForA_8,u, estimate (15) follows immediately from lemma 4.4.

ForA_9,u, letx∈’. Let 0≤v≤u. By the Faa-di-Bruno formula:

D^v hg⁰h⁻¹

(x)−D^v

hRg⁰(0)h⁻¹ (x)=

v

X

j=0

D^jh(g⁰h⁻¹(x))B_v,j D

g⁰h⁻¹

(x), ...,D^v−j+1

g⁰h⁻¹(x)

−

D^jh(g⁰h⁻¹(x))Bv,j D

Rg⁰(0)h⁻¹

(x), ...,D^v−j+1

Rg⁰(0)h⁻¹(x)