Estimates of the Linearization of Circle Diffeomorphisms

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Estimates of the Linearization of Circle Diffeomorphisms

Mostapha Benhenda

To cite this version:

Mostapha Benhenda. Estimates of the Linearization of Circle Diffeomorphisms. 2011. �hal-00628297�

Estimates of the Linearization of Circle Di ff eomorphisms

Mostapha Benhenda

September 30, 2011

Abstract

A celebrated theorem by Herman and Yoccoz asserts that if the rotation number αof aC^∞-diffeomorphism of the circle f satisfies a Diophantine condition, then fisC^∞-conjugated to a rotation. In this paper, we establish explicit relationships between theC^knorms of this conjugacy and the Diophantine condition onα. To obtain these estimates, we follow a suitably modified version of Yoccoz’s proof.

1 Introduction

In his seminal work, M. Herman [5] shows the existence of a set Aof Diophantine numbers of full Lebesgue measure such that for any rotation numberα ∈ Aof a cir- cle diffeomorphism f of classC^ω (resp. C^∞), there is a C^ω-diffeomorphism (resp.

C^∞-diffeomorphism)hsuch thath f h⁻¹ =Rα. In theC^∞case, J. C. Yoccoz [14] ex- tended this result to all Diophantine rotation numbers. Results in analytic class and in finite differentiability class subsequently enriched the global theory of circle dif- feomorphisms [11, 9, 8, 13, 7, 15, 4, 10]. In the perturbative theory, KAM theorems usually provide a bound on the norm of the conjugacy that involves the norm of the perturbation and the Diophantine constants of the numberα(see [5, 12, 3] for exam- ple). We place ourselves in the global setting, we compute a bound on the norms of this conjugacyhin function ofk,|D f|₀,W(f),|S f|_k−3,βandCd.

To obtain these estimates, we follow a suitably modified version of Yoccoz’s proof.

Indeed, Yoccoz’s proof needs to be modified because a priori, it does not exclude the fact that the set:

EX =n

|Dh|₀/∃f ∈Diff^k₊(”¹), f =h⁻¹Rαh, α∈DC(β,Cd),max (k, β,Cd,|D f|₀,W(f),|S f|_k−3)≤Xo could be unbounded for any fixedX>0.

These estimates have natural applications to the global study of circle diffeomor- phisms with Liouville rotation number: in [2], they allow to show the following results:

1) Given a diffeomorphismf of rotation numberα, for a Baire-dense set ofα, it is pos- sible to accumulateRαwith a sequencehnf h⁻¹_n ,hn being a diffeomorphism. 2) Given two commuting diffeomorphisms f andg, with the rotation numberαof f belonging to a specified Baire-dense set, it is possible to approach each of them by commuting diffeomorphismsfnandgnthat are differentiably conjugated to rotations.

∗Laboratoire d’Analyse, Geometrie et Applications, Paris 13 University, 99 Avenue J.B. Clement, 93430 Villetaneuse, France. Contact: mostaphabenhenda@gmail.com

1.1 Notations

We follow the notations of [14].

• The circle is noted”¹. The group of orientation-preserving circle diffeomor- phisms of classC^r is denoted Diff^r₊(”¹). The group ofš-periodic diffeomor- phisms of classC^r of the real line is notedC^r(”¹). We often work in the uni- versal coverD^r(”¹), which is the group of diffeomorphisms f of classC^rof the real line such that f −Id ∈ C^r(”¹). Note that if f ∈ D^r(”¹) andr ≥ 1, then D f ∈C^r−1(”¹).

• The derivative of f ∈ D¹(”¹) is notedD f. The Schwartzian derivativeS f of f ∈D³(”¹) is defined by:

S f =D²logD f −1

2(DlogD f)²

• The total variation of the logarithm of the first derivative of f is:

W(f)= sup

a₀≤...≤a_n n

X

i=0

|logD f(ai+1)−logD f(ai)|

• For any continuous andš-periodic functionφ, let:

|φ|₀ =kφk₀ =sup

x∈’

|φ(x)|

• Let 0< γ⁰<1.φ∈C⁰(”¹) is Holder of orderγ⁰if:

|φ|γ⁰=sup

x,y

|φ(x)−φ(y)|

|x−y|^γ0 <+∞

Letγ≥1 be a real number. In all the paper, we writeγ=r+γ⁰withr∈Žand 0≤γ⁰<1.

• A functionφ∈C^r(”¹) is said to be of classC^γifD^rφ∈C^γ0(”¹). The space of these functions is notedC^γ(”¹) and is given the norm:

kφkγ=max max

0≤j≤rkD^jφk0,|D^rφ|γ⁰

!

Ifγ=0 orγ≥1, theC^γ-norm ofφis indifferently denotedkφk_γ or|φ|_γ. Thus, when possible, we favor the simpler notation|φ|_γ.

• Forα∈ ’(respectively,α∈ ”¹), we denoteRα ∈ D^∞(”¹) (respectively,Rα ∈ Diff^∞₊(”¹)). the mapx7→x+α.

• An irrational numberα ∈ DC(C_d, β) satisfies a Diophantine condition of order β≥0 and constantC_d>0 if for any rational numberp/q, we have:

α− p q

≥ C_d q^2+β Moreover, ifβ=0, thenαis of constant typeC .

• Letα₋₂ = α, α₋₁ = 1. Forn ≥ 0, we define a real numberα_n (theGauss sequenceofα) and an integeranby the relations 0< α_n< α_n−1and

αn−2=anαn−1+αn

• In the following statements,Ci[a,b, ...] denotes a positive numerical function of real variablesa,b, ..., with an explicit formula that we compute.

C[a,b, ...] denotes a numerical function ofa,b, ..., with an explicit formula that we do not compute.

• We use the notationsa∧b =a^b,e⁽ⁿ⁾∧xthen^th- iterate of x7→ expx,bxcfor the largest integer such thatbxc ≤ x, anddxefor the smallest integer such that dxe ≥x.

We recall Yoccoz’s theorem [14]:

Theorem 1.1. Let k ≥ 3an integer and f ∈ D^k(”¹). We suppose that the rotation numberαof f is Diophantine of orderβ. If k>2β+1, there exists a diffeomorphism h∈D¹(”¹)conjugating f to Rα. Moreover, for anyη >0, h is of class C^{k−1−β−η}.

1.2 Statement of the results

1.2.1 C¹estimations

Theorem 1.2. Let f ∈ D³(”¹)of rotation numberα, such thatαis of constant type Cd. There exists a diffeomorphism h∈D¹(”¹)conjugating f to Rα, which satisfies the estimation:

|Dh|0≤e∧ C1[W(f),|S f|₀] Cd

!

The expression of C1is given page 10.

More generally, for a Diophantine rotation numberα∈DC(Cd, β), we have:

Theorem 1.3. Let k≥3be an integer and f ∈D^k(”¹). Letα∈DC(Cd, β)be the rota- tion number of f . If k>2β+1, there exists a diffeomorphism h∈D¹(”¹)conjugating

f to Rα, which satisfies the estimation:

|Dh|₀≤C₂[k, β,C_d,|D f|₀,W(f),|S f|_k−3] (1) The expression of C2is given page 23.

Moreover, if k≥3β+9/2, we have:

|Dh|0≤e⁽³⁾∧

C3[β]C4[C_d]C5[|D f|₀,W(f),|S f|₀]C6[|S f|_d3β+3/2e]

(2) The expressions of C3,C4,C5,C₆are given page 28.

Letδ=k−2β−1. Whenδ→0, we have:

|Dh|₀≤e⁽³⁾∧ 1

δ²C7[k,Cd,|D f|₀,W(f),|S f|₀]+C[δ]

δ² C[k,Cd,|D f|₀,W(f),|S f|₀,|S f|_k−3]

!

(3) where C[δ]→_δ→00. The expression of C₇is given page 30.

Remark 1.4. Katznelson and Ornstein [9] showed that the assumptionk >2β+1 in Yoccoz’s theorem is not optimal (instead it isk> β+2). Therefore, the divergence of the bound given by estimation (3) is because we compute the bound of the conjugacy by following the Herman-Yoccoz method.

Remark1.5. Letα_nbe the Gauss sequence associated withα. Yoccoz’s proof already gives the following result: ifk≥3β+9/2 and if, for anyn≥0,

αn+1

α_n ≥C8[n,k,W(f),|S f|_k−3] (4) then:

|Dh|₀ ≤exp

C₉[k,W(f),|S f|_k−3]^C10(β)

|D f|²₀

The expressions ofC8,C9,C10are given page 30.

1.2.2 C^uestimations

Theorem 1.6. Let k≥3an integer,η >0and f ∈ D^k(”¹). Letα∈DC(Cd, β)be the rotation number of f . If k >2β+1, there exists a diffeomorphism h∈ D^{k−1−β−η}(”¹) conjugating f to Rα, which satisfies the estimation:

kDhk_{k−2−β−η} ≤e(dlog((k−2−β)/η)/log(1+1/(2β+3))e)∧(C₁₁[η,k, β,Cd,|D f|₀,W(f),|S f|_k−3]) (5) The expression of C₁₁is given page 48.

Moreover, if k≥3β+9/2, we have:

kDhk k

2(β+2)−¹₂ ≤e∧

C12[k]e⁽²⁾∧(2+C3[β]C4[Cd]C5[|D f|₀,W(f),|S f|₀]C6[|S f|_k−3]) (6) The expression of C12is given page 46.

Ifαis of constant type, for any k>3, we have:

kDhkk

4−¹₂ ≤e∧







C13[k]

"

C14[W(f),|S f|_k−3]+C1[W(f),|S f|₀] Cd

#4





 (7) The expressions of C13and C14are given page 47.

2 Preliminaries

Let f ∈ D⁰(”¹) be a homeomorphism and x ∈ ’. When n tends towards infinity, (fⁿ(x)−x)/nadmits a limit independent of x, notedρ(f). We call it thetranslation numberof f. Two lifts of f ∈Diff⁰₊(”¹) only differ by a constant integer, so this is also the case for their translation numbers. We call the class ofρ(f) modštherotation numberof f. We still denote itρ(f). It is invariant by conjugacy. Let f ∈ D²(”¹).

Whenα=ρ(f) is irrational, Denjoy showed that f is topologically conjugated toRα. However, this conjugacy is not always differentiable (see [1, 5, 7, 15]). The regularity

of this conjugacy depends on the Diophantine properties of the rotation numberα(see Yoccoz’s theorem 1.1).

Letαbe an irrational number. Let the distance ofαto the closest integer be:

||α||=inf

p∈š|α−p|

Forn≥1,an ≥1. Letα=a0+1/(a1+1/(a2+...)) be the development ofαin continued fraction. We denote itα=[a0,a1,a2, ...]. Letp−2=q−1=0,p−1=q−2=1.

Forn≥0, letpnandqnbe:

pn=anpn−1+pn−2

q_n=a_nqn−1+qn−2

We haveq0 =1,qn ≥1 forn ≥1. The rationals pn/qnare called the convergents ofα. They satisfy the following properties:

1. αn =(−1)ⁿ(qnα−pn) 2. α_n =||qnα||, forn≥1

3. 1/(q_n+1+qn)< α_n<1/q_n+1forn≥0.

4. α_n+2< ¹₂α_n,q_n+2≥2q_n, forn≥ −1

We recall thatDC(Cd, β) denotes the set of Diophantine numbers of constantsβand Cd. One of the following relations characterizesDC(Cd, β):

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. αn+1>Cdα¹_n^+βfor anyn≥0

In all the paper, we denoteC_d⁰ =1/Cd.

• Letmn(x)=f^qn(x)−x,n≥1,x∈”¹, letMn=sup_x∈”1|f^qn(x)−x|and mn=inf_x∈”1|f^qn(x)−x|.

• For anyφ, ψ∈C^γ(”¹), we have:

|φψ|γ≤ kφk0|ψ|γ+|φ|γkψk0 (8) kφψkγ≤ kφk0|ψ|γ+kφkγkψk0 (9)

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

In the rest of the paper, for any integeri,C_i^f denotes a constant depending only onW(f) and |S f|₀ (i.e. C_i^f is a numerical function of these variables). C_i^f,kdenotes a constant depending only onk,W(f),|S f|₀and|S f|_k−3. C_idenotes a constant that might depend onk,W(f),|S f|₀,|S f|_k−3and alsoβandC_d.

3 C

estimations: constant type

3.1 A 2-parameters family of homographies

In this subsection, we show the existence of a lower bound on the norm of the conjugacy in function ofCdin the particular case of a 2-parameters family of homographies. We also establish an upper bound on theC¹norm of the conjugacy for this family. These bounds are similar to what is given by the local KAM theory. However, these bounds are very specific to this setting. Our general bounds given in theorems 1.2 and 1.3 are much larger.

Proposition 3.1. Let f :{z∈ƒ/|z|=1} → {z∈ƒ/|z|=1}defined by f(z)=h⁻¹Rθh(z), with Rθ(z)=e^iθz and h is a homography defined by:

h(z)= z−a az−1

Let2>a>1, let Cdsuch that C_d⁻¹ ≥6is a positive integer; and0< θ=2πC_d ≤ π/3, (therefore,θ/(2π)=[0,C_d⁻¹,1]is of constant type Cd). Let f˜:”¹ →”¹the circle diffeomorphism induced by f andh the conjugacy induced by h. We have the following˜ estimation:

2

πC₁₅(|Df˜(0)|,|D²f˜(0)|)/C_d≤ |Dh|˜₀≤9C₁₅(|Df˜(0)|,|D²f˜(0)|)/C_d

Proof. For anyφ∈ ’/š, we can writeh(e^iφ)=e^ih(φ)˜ . By differentiating this expres- sion, we have:

Dh(φ)˜ =e^iφDh(e^iφ) h(e^iφ) and

Dh(z)=(a−1)(a+1) (az−1)² Therefore

Dh(φ)˜ =e^iφ a²−1 (ae^iφ−1)(e^iφ−a)

|Dh(φ)|˜ reaches its maximum forφ=0, and|Dh˜|₀= ^a_a−1⁺¹. Moreover, we have:

D²f˜(φ)

Df˜(φ) =i+ie^iφD²f(e^iφ)

D f(e^iφ) −ie^iφD f(e^iφ) f(e^iφ) Since ^D_D²_f(φ)^f_˜^˜(φ) ∈’andDf˜(φ)∈’, we have:

D²f(1) D f(1)

=







 D²f˜(0)

Df˜(0)

!² +

Df˜(0)−12







1/2

=8C15(|Df˜(0)|,|D²f˜(0)|) Therefore, in order to get the proposition, it suffices to show:

1 4πC

D²f(1) D f(1)

≤ |Dh|0≤ 9 8C

D²f(1) D f(1)

Let us write

f(z)= (e^iθ−a²)z−a(e^iθ−1)

a(e^iθ−1)z−(a²e^iθ−1) = bz−c cz+d We have

D f(z)= db+c² (cz+d)² and

D²f(z)=−2 D f(z) z+d/c Moreover,

D f(1)=(1+a)²e^iθ (ae^iθ+1)² and

D²f(1)=−2D f(1) a(e^iθ−1) (1−a)(1+ae^iθ) We have

D²f(1) D f(1)

=2

a 1+ae^iθ

|e^iθ−1| a−1 Since|e^iθ−1| ≥sinθ≥²_πθ(because 0≤θ≤π/2), then:

D²f(1) D f(1)

≥4 π

a 1+a

θ a−1 Therefore,

D²f(1) D f(1)

≥ 4 π

a

(1+a)²|Dh|₀θ i.e.

9π 4θ

D²f(1) D f(1)

≥ |Dh|0

Hence the first part of the inequality.

On the other hand, sinceθ≤π/3, then|1+ae^iθ| ≥1+acosθ≥1+a/2≥¹

2(a+1).

Furthermore,|e^iθ−1|² =2−2 cosθ=2−2(cos²θ/2−sin²θ/2)=4 sin²θ/2≤θ². Therefore,

D²f(1) D f(1)

≤ θ a−1

4a

a+1 ≤ 4θ

a−1 =4θ|Dh|₀ a+1 i.e.

1 2

D²f(1) θD f(1)

≤ |Dh|0

Hence the second part of the inequality.

3.2 Proof of theorem 1.2

The proof of theorem 1.2 is divided in three steps. The first step is based on the im- proved Denjoy inequality, which estimates theC⁰-norm of logD f^ql. In the second step, we extend this estimation to logD f^N for any integerN. To do this, following Denjoy and Herman, we writeN=PS

s=0b_sq_s, withb_sintegers satisfying 0≤b_s≤q_s+1/q_sand we apply the chain rule. In the third step, we derive aC⁰-estimation of the derivative Dhof the conjugacyh.

The first step is based on the Denjoy inequality:

Proposition 3.2. Let f ∈Diff³₊(”¹)and x∈”¹. We have:

|logD f^ql(x)| ≤W(f)

Proposition 3.2 is used to obtain an improved version of Denjoy inequality [14, p.342]:

Lemma 3.3. Let f ∈Diff³₊(”¹). We have:

|logD f^ql|₀≤C₁₆^f M_l^1/2

|D f^ql−1|0≤C₁₇^f M_l^1/2 Moreover, we can take:

C₁₆^f =2

√

2(2e^W(f)+1)e^W(f)(|S f|₀)^1/2 and

C₁₇^f =6

√

2e^3W(f)|S f|^1/2

0

In the second step, we estimateDlogD f^N independently ofN. This step is based on the following lemma:

Lemma 3.4. Let f ∈Diff³₊(”¹)and M_l=sup_x∈”1|f^ql(x)−x|. We have:

X

l≥0

pMl≤ 1 q

C₁₉^f −C₁₉^f with

C₁₉^f = 1 p

1+e^−C22f

(10) and:

C₂₂^f =6

√ 2e^2W(f)

|S f|^1/2

0 ∨1

(11) Proof. To obtain this lemma, we need the claim:

Claim 3.5. Let f ∈Diff²₊(”¹)of rotation numberα, and let p_n/q_nbe the convergents ofα. Then for all x∈”¹, we have:

[x,f^2ql+2(x)]⊂[x,f^ql(x)]

Proof. By topological conjugation, it suffices to examine the case of a rotation of angle α. It is also sufficient to takex=0.

By absurd, if the lemma was false, then we would have the following cyclic order on”¹: −ql+2α ≤(ql+2−ql)α≤ 0 ≤(ql−ql+2)α≤ql+2α. In particular, (ql+2−ql)α would be closer to 0 thanql+2α, which would contradict the fact that

kql+2αk=inf{kqαk/0<q≤ql+2}.

For any intervalIof the circle, if|I|denotes the length ofI, lemma 3.3 implies the estimation:

|f^ql+2(I)|

|I| ≥e^{−C22fMl+21/2}

Let x ∈ ”¹ such that M_l+2 = f^ql+2(x)−xand let I = [x,f^ql+2(x)]. The former estimation implies

|f^2ql+2(x)−f^ql+2(x)| ≥e^{−C22fM1/2l+2}M_l+2 By applying claim 3.5, and sinceM_n≤1, we obtain:

Mn+2+e^−C22f Mn+2≤Mn+2+e^{−C22f Mn+21/2}Mn+2≤Mn

Therefore, for anyl≥0,

Ml≤(C₁₉^f )^l−1 (12)

with

C₁₉^f = 1 p

1+e^−C22f Estimation (12) above gives:

X

l≥0

pMl≤ 1 q

C₁₉^f 1 1−

q C₁₉^f

≤ 1

q

C₁₉^f −C₁₉^f Hence lemma 3.4.

Now, letN be an integer. Following Denjoy, sinceαis of constant type, we can writeN =Ps

l=0blql, withblintegers satisfying 0 ≤bl ≤ql+1/ql≤C⁻¹_d . By the chain rule and by lemma 3.3, since for ally∈”¹,D f^N(y)>0, then :

|logD(f^N)(y)|=|logD(f^{Pl=0s blql})(y)|=|Ps l=0

Pbs

i=0logD f^ql◦ f^iql(y)|

≤sup_0≤l≤sblPs

l=0|log|D(f^ql)|₀| ≤C⁻¹_d C₂₂^f P

l≥0M^1/2_l

By taking the upper bound ony ∈ ”¹ andN ≥ 0, we obtain an estimation of sup_N≥0|logD(f^N)|.

We turn to the third step: we relate the norms ofDhandD f^N. By [14],hisC¹and conjugates f to a rotation. Therefore, we have:

logDh−logDh◦ f =logD f hence, for allninteger:

logDh−logDh◦ fⁿ=logD(fⁿ)

Since there is a pointzin the circle such thatDh(z)=1, we then have:

|logDh◦ fⁿ(z)|=|logD(fⁿ)(z)| ≤sup

i≥0

|logD(fⁱ)|₀

Moreover, since (fⁿ(z))n≥0is dense in the circle, and sinceDhis continuous, then we obtain:

|logDh|₀≤sup

i≥0

|logD(fⁱ)|₀ We conclude:

|Dh|₀≤exp C⁻¹_d C₂₂^f q

e^{C22f max(M01/2,M1/21 )}+1(p M0+p

M1)

!

(13) Finally, since max(M^1/2₀ ,M^1/2₁ )≤1, we obtain:

|Dh|₀≤exp C₁^f/C_d

whereC₁^f =2C₂₂^f p

e^C22f +1. We recall that:

C₂₂^f =6

√ 2e^2W(f)

|S f|^1/2₀ ∨1 Hence the theorem.

Corollary 3.6. Since_min¹

”1Dh ≤exp

sup_i≥0|logD(fⁱ)|0

, the proof above also provides an estimation on_min¹

”1Dh:

1

min_”1Dh ≤exp C₁^f/Cd

4 C

estimations: non-constant type

We have maxn≥0|D fⁿ|₀≤maxn≥0Mn/mn, by [14, p. 348]. Therefore, in order to prove theorem 1.3, we can estimateMn/mn. To that end, we proceed in two steps: first, we establish some preliminary results. The most important result is corollary 4.6, which gives an estimation ofMn+1/M_n in function ofMn,α_n+1/α_nand a constantC₂₈^f,k. This estimation is already given in [14, p. 345], but we still recall the steps to reach it, because we need to estimate the constantC₂₈^f,kin function ofk,W(f),|S f|₀and|S f|_k−3. In the second step, we establish an estimation of theC¹-conjugacy, based on a mod- ification of the proof given in [14]. The main idea is to establish an alternative between

two possible situations for the sequencesMnandα_n: the "favorable" situation (Rn) and the "unfavorable" situation (R⁰_n) (proposition 4.10). The "unfavorable" situation only occurs a finite number of times, due to the Diophantine condition onα(propositions 4.12 and 4.14).

In the "favorable" situation (R_n), we can estimateM_n+1/α_n+1in function ofM_n/α_n (see estimation (26)) and likewise, we can estimateα_n+1/m_n+1 in function ofα_n/m_n. Therefore, we can estimate Mn/mn in function of Mn₄/mn₄, where n4 is the integer such that for anyn ≥ n4, the favorable case occurs (see proposition 4.19). We relate M_n4/m_n4 toD f|

αn24

0 (proposition 4.17), and we compute a bound onα_n4 (proposition 4.15). Yoccoz’s proof needs to be modified because in its original version, it does not allow to compute a bound onα_n4.

4.1 Preliminary results

First, we recall the following lemmas, which are in [14] (lemmas 3,4 and 5):

Lemma 4.1. For l≥1and x∈”¹, we have:

q_n+1−1

X

i=0

D fⁱ(x)l

≤C₂₃^f M^l−1_n m_n(x)^l with C₂₃^f (l)=e^lW(f).

Remark4.2. This lemma is obtained by applying Denjoy inequality.

Lemma 4.3. Let f ∈Diff^k₊(”¹), k≥3. For any x∈”¹, any n∈Ž, any0≤ p≤q_n+1, we have:

|S f^p|₀≤C₂₄^f Mn

m²_n

|S f^p(x)| ≤C₂₄^f Mn

mn(x)²

|DlogD f^p|₀≤C₂₅^f M^1/2n

mn

|DlogD f^p(x)| ≤C₂₆^f M_n^1/2 mn(x) with:

• C₂₄^f =|S f|₀e^2W(f)

• C₂₅^f = p

2|S f|₀e^W(f)

• C₂₆^f =9p

2|S f|₀e^4W(f)

Lemma 4.4. For1≤r≤k−1, n≥0,0≤p≤qn+1, x∈”¹, we have:

|D^rlogD f^p(x)| ≤C₂₇^f (r)







 M^1/2n

mn(x)









r

(14) with

C₂₇^f (1)=C₂₆^f , C₂₇^f (2)=82|S f|₀e^8W(f) and, for r≥3:

C₂₇^f (r)=h

82(2r)^2r(1∨ |S f|_r−2)²e^(r+8)W(f)ir!

In particular,

C₂₇^f,k:=C₂₇^f (k−1)≤h

100(2k−2)^2k−2(1∨ |S f|_k−3)²e^(k+7)W(f)i^(k−1)!

Proof of lemma 4.4. The proof follows the line of [14], lemma 5: see appendix 6.1.

The important preliminary result, corollary 4.6, is obtained from the following proposition. It is obtained by computing the constants in proposition 2 of [14]:

Proposition 4.5. Let

C₂₈^f,k=(k+3)^(k+3)!e^(k+2)!W(f)(max(1,|S f|_k−3))^k! (15) For any x∈”¹, we have:

mn+1(x)−αn+1

α_n mn(x)

≤C₂₈^f,kh

M_n^(k−1)/2mn(x)+M_n^1/2mn+1(x)i

(16) Corollary 4.6.

Mn+1≤Mn α_n+1

α_n +C₂₈^f,kM^(k−1)/2_n

1−C₂₈^f,kM_n^1/2 (17)

m_n+1≥m_n

α_n+1

α_n −C₂₈^f,kM_n^(k−1)/2 1+C₂₈^f,kM^1/2_n

The proof of proposition 4.5 combines the following three lemmas [14, pp. 343- 344] (lemmas 6, 7 and 8):

Lemma 4.7. For any x∈”¹, there exists y∈[x,f^qn(x)],z∈[f^qn+1(x),x]such that mn+1(y)= αn+1

αn

mn(z)

Lemma 4.8. Suppose that mn+1 is monotonous on an interval Iz =(z,f^q(z)), z∈”¹. Then, for any x∈”¹, for any y∈I_x(I_x=(x,f^q(x))), we have:

mn+1(y) mn+1(x)−1

≤C₂₉^f,kM_n^1/2 with

C₂₉^f,k=2⁹(k+2)e^(11+k/2)W(f)(C₁₇^f )²C₂₆^f

Lemma 4.9. If mn+1 is not monotonous on any interval of the form Iz = (z,f^q(z)), z∈”¹, then for any x∈”¹, y∈Ix, we have:

|m_n+1(y)−m_n+1(x)| ≤C₃₀^f,kM^(k−1)/2_n m_n(x) with

C₃₀^f,k=(C₂₇^f (k−1))e^W(f) e^(k/2+2)W(f)(1+e^W(f))²e^(k/2+2)W(f)−1 e^W(f)−1

!^k−1

Proof of proposition 4.5. Let us recall the proof of proposition 4.5 from these three lemmas. (see [14, p.344]). Letx∈”¹andy∈I_x,z∈[f^qn+1(x),x] the points given by lemma 4.7. By combining lemmas 4.8 and 4.9, we obtain:

|m_n+1(y)−m_n+1(x)| ≤ max

C₂₉^f,k,C₃₀^f,k M^1/2_n m_n+1(x)+M^(k−1)/2_n m_n(x) Moreover, by lemma 3.3, we have:

|mn(z)−mn(x)| ≤C₁₇^f M_n^1/2|z−x| ≤C₁₇^f M_n^1/2mn+1(x) By applying lemma 4.7, and sinceαn+1/αn≤1, we get:

mn+1(x)−αn+1

α_n mn(x)

≤

mn+1(x)−αn+1

α_n mn(z) +αn+1

α_n |mn(z)−mn(x)|

mn+1(x)−α_n+1 αn

mn(x)

≤ |mn+1(y)−mn+1(x)|+|mn(z)−mn(x)|

Therefore, we have:

mn+1(x)−α_n+1 αn

mn(x)

≤C₃₁^f,k

M_n^1/2mn+1(x)+M^(k−1)/2_n mn(x) withC₃₁^f,k=max(C₂₉^f,k,C₃₀^f,k)+C₁₇^f .

Finally, let us estimateC₃₁^f,k. Sincek≥3, then:

[4(k/2+1)(200k)]²≤(k+3)^(k+3)(k+2)k/2 and therefore,

2^2(k−1)(k/2+1)^k−1(200k)^{2(k+1)(k−1)!}≤(k+3)^(k+3)!/2 Therefore, we have:

C₃₀^f,k+C₁₇^f ≤(k+3)^(k+3)!e^(k+2)!W(f)(max(1,|S f|_k−3))^k!

Sincek≥3, we also have:

C₂₉^f,k+C₁₇^f ≤(k+3)^(k+3)!e^(k+2)!W(f)(max(1,|S f|_k−3))^k!

Therefore,C₃₁^f,k ≤C₂₈^f,k=(k+3)^(k+3)!e^(k+2)!W(f)(max(1,|S f|_k−3))^k!. Hence proposi- tion 4.5.

4.2 Estimation of the C

-conjugacy. Proof of estimation (1).

We choose an integern1such that for anyn≥n1, we have:

C₂₈^f,kM_n^1/2 ≤C₂₈^f,k(C₁₉^f )ⁿ⁻¹² <1/2 (18) We take:

n1=













−log

2C₂₈^f,k/(C₁₉^f )^1/2 log

(C₁₉^f )^1/2













We choose a parameterθsuch that (k+1)/2−θ > (1+β+θ)(1+θ) (for the interpretation of this parameterθ, see the remark after proposition 4.10). We take:

θ=min







1/2, 3+β 4

! 





−1+ 1+2(k−2β−1) (3+β)²

!1/2













 (19) (in the proof of estimation (2), we takeθ=1/2 instead).

We recall that for x ≥ 0, 1+x ≤ e^x and for 0 ≤ x ≤ 1/2, log (1/(1−x)) ≤ x/(1−x)≤2x. We apply estimation (18), we use the definition ofn1and the fact that θ≤1/2. We get:

+∞

Y

n=n1

1+M^θ_n

≤exp











+∞

X

n=n1

M^θ_n











≤exp









1 2C₂₈^f,k(1−(C₁₉^f )^θ)









+∞

Y

n=n1







 1 1−C₂₈^f,kM_n^1/2









≤exp











+∞

X

n=n1

2C₂₈^f,kM_n^1/2











≤exp







 1 1−(C₁₉^f )^1/2









Therefore,

+∞

Y

n=n1









1+M_n^θ 1−C₂₈^f,kMn^1/2









≤exp







 2 1−(C₁₉^f )^θ







=C32 (20) Let:

C33=max

(4C₂₈^f,k)(1+β+θ)(1+θ)−1¹ ,C32

(21) For any

n≥

−log 2(C33)²

logC₁₉^f +1=C34 (22)

we have:

Mn≤(C₁₉^f )ⁿ⁻¹≤ 1

2C²₃₃ (23)

We use this estimation in the second step of the proof, to which we come now:

Let

n2 =max(n₁,n˜2) (24) where ˜n₂is the integer defined by

C34+ 4

log 2log(1/Cd)+1≤n˜2<C34+ 4

log 2log(1/Cd)+2 (25) Having defined the integern2, we can present the alternative between the "favor- able" case (Rn) and the "unfavorable" case (R⁰_n).

Proposition 4.10. Let an₂ = 1/((C33)²). Let1 ≥ ηn ≥ 0 be a sequence such that α_n=α^1−η_n+1ⁿ. For any n≥n2, we can define a sequence an,1/((C₃₃)²)≤an≤1/C₃₃and a sequenceρ_n<1such that Mn=anα^ρ_nⁿ. The sequence anis defined by:

if

(Rn) C₂₈^f,kM^(k_n^+1)/2−θ≤Mn

α_n+1 αn

then an+1=an

1+M^θ_n 1−C₂₈^f,kM^1/2_n and if

(R⁰_n) C₂₈^f,kM_n^(k+1)/2−θ>Mn

αn+1

αn

then an+1=an

Moreover, if(Rn)holds, thenρn+1≥ρn+ηn(1−ρn);

and if(R⁰_n)holds, thenρn+1≥((k+1)/2−θ)(1−ηn)ρn. In particular, the sequence (ρn)n≥n₂is increasing.

The threshold between the alternatives (Rn) and (R⁰_n) is controlled with a parameter θ, which could be freely chosen such thatθ >0 and (k+1)/2−θ≥(1+β+θ)(1+θ).

Whenθincreases, the numbern3of occurrences of (R⁰_n) increases. Whenn3increases, all other quantities being equal, the bound on the norm of the conjugacy increases.

Moreover, ifθgets too large, we can no longer show thatn₃ is finite (see proposition 4.14), and therefore, we can no longer estimate the norm of the conjugacy.

On the other hand, whenθis smaller,C32 increases. It increases the numbern2

above which we consider the alternatives (Rn) and (R⁰_n).C35 increases too (see propo- sition 4.19). WhenC32 andC35 increase, all other quantities being equal, the bound on the norm of the conjugacy increases. Moreover, whenθ → 0,C32 → +∞, which makes this bound on the conjugacy diverge.

Thus, the variation ofθhas contradictory influences on the bound of the norm of the conjugacy, and there is a choice ofθthat optimizes this bound. However, in this paper, we do not seek this optimalθ, since it would complicate further the expression of the final estimate. Instead, in estimation (2), we fixθ=1/2, which allows simplifying the expression of the estimate. In estimation (3), we takeθ → 0, which also allows simplifying the estimate.

Proof of proposition 4.10: For anyn≥n₂, sincen₂≥n₁, an₂= 1

C²₃₃ ≤an≤an₂ +∞

Y

n=n1









1+M^θ_n 1−C₂₈^f,kM^1/2_n









≤C32

C₃₃² ≤ 1 C₃₃ and since

α^ρ_nⁿ >anα^ρ_nⁿ=Mn≥α_n thenρ_n<1.

Second, if (R_n) holds, then by applying corollary 4.6, we have:

Mn+1≤ 1+M_n^θ 1−C₂₈^f,kM^1/2_n Mn

α_n+1 αn

(26) Therefore,

Mn+1=an+1α^ρ_nⁿ⁺¹₊₁ ≤an+1αn+1α^ρ_nⁿ⁻¹=an+1αn+1α^(1−η_n+1^n)(ρn−1) and then:

ρn+1−1≥(1−ηn)(ρn−1) hence the estimation:

ρn+1≥ρn+ηn(1−ρn)

If (R⁰_n) holds, sinceC₂₈^f,kM^1/2n ≤1/2, then by applying corollary 4.6, we obtain:

Mn+1≤4C₂₈^f,kM^(k_n^+1)/2−θ Moreover, sincean≤1/C₃₃<1, then:

a^(k_n^+1)/2−θ≤a⁽¹_n^{+β+θ)(1+θ)}=ana⁽¹_n^{+β+θ)(1+θ)−1}≤ an

C⁽¹₃₃^{+β+θ)(1+θ)−1}

≤ an

4C₂₈^f,k Therefore, by combining these two estimations, we obtain:

a_n+1α^ρ_nⁿ⁺¹₊₁ =M_n+1≤4C₂₈^f,kM_n^(k+1)/2−θ≤4C₂₈^f,ka^(k_n^+1)/2−θα^ρ_n^{n((k+1)/2−θ)}≤a_nα^ρ_n^{n((k+1)/2−θ)} Moreover, sincean+1=an, then

1≤α^(ρn((k+1)/2−θ))(1−η_n)−ρ_n+1 n+1

hence the estimation:

ρ_n+1≥(ρ_n((k+1)/2−θ))(1−η_n)

The reader can notice that until now, we have not used the Diophantine condition onαyet. Now, we introduce this condition in order to estimateρn₂from below (propo- sition 4.11), and in order to determine a bound ρabove which (Rn) always occurs (proposition 4.12).

Proposition 4.11. Ifβ >0, we have the estimation:

ρn₂≥ log 2

((1+β)ⁿ²⁺¹−1) log(1/Cd)/β Ifβ=0, we have the estimation:

ρ_n2≥ log 2 (n₂+1) log(1/C_d)

Proof. Sinceαis Diophantine, we have:αn+1≥Cdα¹_n^+β. Therefore, forβ >0, log 1

αn+1

!

+log(1/Cd)

β ≤(1+β) log (1/αn)+log(1/Cd) β

!

and sinceα−1=1, then by iteration, for anyn≥0, log (1/αn)≤

(1+β)ⁿ⁺¹−1log(1/Cd) β Ifβ=0, we have:

log (1/α_n)≤(n+1) log(1/C_d)

Moreover, sinceρn₂ =−log(Mn₂/an₂)/log(1/αn₂) andMn₂/an₂ ≤1/2, then we get proposition 4.11.

Proposition 4.12. Letβ₁=β+_(n^{2 log(1/C}_{2−1) log 2}^d). If

ρ_n≥ β₁

(k−1)/2−θ =ρ (27)

then(Rn)occurs.

Remark4.13. Note thatρ <1, because (k+1)/2−θ≥(1+β+θ)(1+θ) andβ1 ≤β+1/2.

Proof. Sinceα_n≤(1/2)ⁿ⁻¹² , then

0< logC_d

logα_n ≤ −logC_d

n−1

2 log 2 (28)

Furthermore, sinceαn+1=α_n^1−ηn1 ≥Cdα¹_n^+β, then 1

1−ηn

logαn≥logCd+(1+β) logαn

and since logαn<1 forn≥0, then by (28), 1

1−η_n −1≤β+logCd

logα_n ≤β+log(1/Cd)

n−1 2 log 2 Therefore, if estimation (27) holds, then

k−1 2 −θ!

ρ_n+1− 1 1−ηn

≥0