Renormalization for piecewise smooth homeomorphisms on the circle

Ann. I. H. Poincaré – AN 30 (2013) 441–462

www.elsevier.com/locate/anihpc

Renormalization for piecewise smooth homeomorphisms on the circle

Kleyber Cunha

, Daniel Smania

aDepartamento de Matemática, Universidade Federal da Bahia, Av. Ademar de Barros s/n, CEP 40170-110, Salvador, BA, Brazil bDepartamento de Matemática, ICMC/USP-Campus de São Carlos, Caixa Postal 668, CEP 13560-970, São Carlos, SP, Brazil

Received 9 December 2011; received in revised form 12 September 2012; accepted 26 September 2012 Available online 3 October 2012

Abstract

In this work we study the renormalization operator acting on piecewise smooth homeomorphisms on the circle, that turns out to be essentially the study of Rauzy–Veech renormalizations of generalized interval exchange maps with genus one. In particular we show that renormalizations of such maps with zero mean nonlinearity and satisfying certain smoothness and combinatorial assumptions converge to the set of piecewise affine interval exchange maps.

Crown Copyright_©2012 Published by Elsevier Masson SAS. All rights reserved.

MSC:37E10; 37E05; 37E20; 37C05; 37B10

Keywords:Renormalization; Interval exchange transformations; Rauzy–Veech induction; Universality; Homeomorphism on the circle;

Convergence

1. Introduction and results

One of the most studied classes of dynamical systems are orientation-preserving diffeomorphisms of the circle.

It may be classified according to their rotation numberρ(f )which, roughly speaking, measures the average rate of rotation of orbits around the circle. Whenρ(f )∈Qthenf has a periodic point and all other orbits will converge to some periodic orbit both in the future and in the past. Ifρ(f )is irrational thenf has not periodic point and its dynamics depends on the smoothness of f. Denjoy result ensures that iff is C² then it is conjugate to the rigid rotation of angleρ(f ).In this context, it is a natural question to ask under what conditions the conjugacy is smooth.

Several authors, Herman[4], Yoccoz[17], Khanin and Sina˘ı[7,14], Katznelson and Ornstein[6], have shown that if f isC³orC^2+ν andρ(f )satisfies certain Diophantine condition then the conjugacy will be at leastC¹.

A natural generalization of diffeomorphisms of the circle are diffeomorphisms with breaks, i.e.,f has jumps in the first derivative on finitely many points. In this setting Khanin and Vul[9]show that for diffeomorphisms with one

* Corresponding author.

E-mail addresses:kleyber@ufba.br(K. Cunha),smania@icmc.usp.br(D. Smania).

URLs:http://www.sd.mat.ufba.br/~kleyber/(K. Cunha),http://www.icmc.usp.br/~smania/(D. Smania).

1 K.C. was partially supported by FAPESP 07/01045-7.

2 D.S. was partially supported by FAPESP 2008/02841-4 and 2010/08654-1, CNPq 310964/2006-7 and 303669/2009-8.

0294-1449/$ – see front matter Crown Copyright©2012 Published by Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.09.004

break the renormalization operator converges to a two-dimensional space of the fractional linear transformation. Our first main result generalizes results of Khanin and Vul[9]for finitely many break points. A key combinatorial method in our proof is to consider a piecewise smooth homeomorphism on the circle as a generalized interval exchange transformation.

Let I be an interval and letA be a finite set (thealphabet) withd 2 elements and P= {I_α: α∈A}be an A-indexed partition of I into subintervals.³ We say that the triple (f,A,P), where f:I →I is a bijection, is a generalized interval exchange transformation with d intervals (g.i.e.m. with d intervals, for short), iff|I_α is an orientation-preserving homeomorphism for eachα∈A. Most of the time we will abuse the notation saying thatf is a g.i.e.m. with d intervals. The order of the subintervals in the domain and image constitute the combinatorial data forf,which will be defined explicitly in the next section.

Whenf|Iα is a translation we say whichf is a standard i.e.m. Standard i.e.m. arise naturally as Poincaré return maps of measured foliations and geodesic flows on translation surfaces. But they are also interesting examples of simple dynamical systems with very rich dynamics and have been extensively studied for their own sake. When d =2,by identifying the endpoints ofI,standard i.e.m. correspond to rotations of the circle and generalized i.e.m.

correspond to circle homeomorphisms.

In another article, Khanin and Sina˘ı[7]show a new proof of M. Herman’s theorem. From the viewpoint of the renormalization group approach they show the convergence of the renormalizations of a circle diffeomorphism to the linear fixed point of the renormalization operator for diffeomorphisms of the circle. We use a similar approach to study generalized interval exchange maps of genus one.

1.1. Renormalization: Rauzy–Veech induction

To describe the combinatorial assumptions of our results, we need to introduce the Rauzy–Veech scheme. This is a renormalization scheme. Renormalization group techniques are a very powerful tool in one-dimensional dynamics.

For example see Khanin and Vul[9]for circle homeomorphisms and de Melo and van Strien[3]for unimodal maps.

Following the algorithm of Rauzy [12] and Veech[15], for every i.e.m.f without connections, we define the Rauzy–Veech induction by considering the first return mapsf_n off on a decreasing sequence of intervalsIⁿ,with the same left endpoint asI.The mapfnis again generalized i.e.m. with the same alphabetAbut the combinatorial data may be different.

Given two intervalsJ andU, we will writeJ < U if their interiors are disjoint andx < y for everyx∈J and y∈U. This defines a partial order in the set of all intervals. Denote the length of an intervalJby|J|.

Letf:I⁰→I⁰be a g.i.e.m. with alphabetAandπ_j:A→ {1, . . . , d}, withj=0,1, be bijections such that I_α< I_β

iffπ0(α) < π0(β)and f (Iα) < f (Iβ) iffπ1(α) < π1(β).

The pairπ=π(f )=(π0, π1)is called the combinatorial data associated to the g.i.e.m.f. We call

p=π1◦π₀⁻¹:{1, . . . , d} → {1, . . . , d} (1) themonodromy invariantof the pairπ=(π₀, π₁).When appropriate we will use the notationp=(p(1) p(2) . . . p(d)) for the data combinatorial off.We also assume that the pairπ=(π0, π1)is irreducible, i.e., for allj∈ {1, . . . , d−1}

we haveπ1◦π₀⁻¹({1, . . . , j})= {1, . . . , j}.

For eachε∈ {0,1}, defineα(ε)=π_ε⁻¹(d). If|I_α(0)| = |f (I_α(1))|we say thatf is Rauzy–Veech renormalizable (or simply renormalizable, from now on). If|I_α(0)|>|f (I_α(1))|we say that the letterα(0)is the winner and the letter α(1)is the loser. We say thatf is type 0 renormalizable and we can define a mapR(f )ˆ as the first return map off to the interval

I¹=I\f (I_α(1)).

3 All the subintervals will be bounded, close on the left and open on the right.

Otherwise|I_α(0)|<|f (I_α(1))|, the letter α(1)is the winner and the letterα(0)is the loser, we say thatf is type 1 renormalizable and we can define a mapR(f )ˆ as the first return map off to the interval

I¹=I\I_α(0).

We want to seeR(f ) as a g.i.e.m. To this end we need to associate to this map an A-indexed partition of its domain. We do this in the following way. The subintervals of theA-partitionP¹ofI¹are denoted byI_α¹.Iff has type 0, I_α¹=Iα. If α=α(0), I_α(0)¹ =Iα(0)\f (Iα(1))and whenf has type 1,I_α¹=Iα if α=α(1), α(0), I_α(1)¹ = f⁻¹(f (Iα(1))\Iα(0))andI_α(0)¹ =Iα(1)\I_α(1)¹ .It is easy to see that in both cases (type 0 and 1) we have

R(f )(x)=

f²(x), ifx∈I_α(1¹ _−ε), f (x), otherwise

and(R(f ),A,P¹)is a g.i.e.m., called the Rauzy–Veech renormalization(or simply renormalization, for short) off. Iff is renormalizable with typeε∈ {0,1}then the combinatorial dataπ¹=(π₀¹, π₁¹)ofR(f )is given by

π_ε¹:=π_ε and

π₁¹_−ε(α)=

π_1−ε(α), ifπ_1−ε(α)π_1−ε(α(ε)), π_1−ε(α)+1, ifπ_1−ε(α(ε)) < π_1−ε(α) < d, π_1−ε(α(ε))+1, ifπ_1−ε(α)=d.

Sinceπ¹depends only onπand the typeε, we denoter_ε(π )=π¹.

A g.i.e.m. is infinitely renormalizable ifRⁿ(f )is well defined, for everyn∈N. For every interval of the form J= [a, b)we denote∂J= {a}.We say that a g.i.e.m.f has no connection if

f^m(∂I_α)=∂I_β for allm1, α, β∈Awithπ₀(β)=1.

This property is invariant under iteration ofR.Keane[5]show that no connection condition is a necessary condition forf to be infinitely renormalizable.

Let εn be the type of the n-th renormalization, αn(εn) be the winner andαn(1−εn) be the loser of then-th renormalization.

We say that infinitely renormalizable g.i.e.m.f hask-bounded combinatoricsif for eachnandβ, γ ∈Athere existn1, p0, with|n−n1|< kand|n−n1−p|< k, such thatα_n1(ε_n1)=β,α_n1+p(1−ε_n1+p)=γ and

α_n1+i(1−ε_n1+p)=α_n1+i+1(ε_n1+i) for every 0i < p.

We say that a g.i.e.m.f:I →I hasgenus oneby Veech[16](or belongs to therotation classby Nogueira and Rudolph[11]) iff has at most two discontinuities. Note that every g.i.e.m. with either two or three intervals has genus one. Iff is renormalizable and has genus one, it is easy to see thatR(f )has genus one.

Given two infinitely renormalizable g.i.e.m.f andg, defined with the same alphabetA, we say thatf andghave thesame combinatoricsifπ(f )=π(g)and then-th renormalization off andghave the same type, for everyn∈N. It follows thatπⁿ(f )=πⁿ(g)for everyn, whereπⁿ(f )is the combinatorial data of then-th renormalization off. Definition 1.1.LetB²_k^+ν,k∈Nandν >0, be the set of g.i.e.m.f:I→I such that

(i) for eachα∈Awe can extendf toIα as an orientation-preserving diffeomorphism of classC^2+ν; (ii) the g.i.e.m.f hask-bounded combinatorics;

(iii) the mapf has genus one and has no connection.

LetH be a non-degenerate interval, let g:H →Rbe a diffeomorphism and let J⊂H be an interval. We de- fine the Zoom of g in H, denoted by ZH(g), the transformation ZH(g)=A₁◦g◦A₂, where A₁ and A₂ are

orientation-preserving affine maps, which send [0,1] into H and g(H ) into [0,1] respectively. Consider the set C²([0,1],R)of allC²functionsg:[0,1] →Rwith the usual norm

d_C2(f, g):=

2 i=0

sup

x∈[0,1]

D⁽ⁱ⁾f (x)−D⁽ⁱ⁾g(x),

whereD⁽ⁱ⁾f andD⁽ⁱ⁾gdenote thei-th derivative off andgrespectively.

Denote byMthe set of Möbius transformationsM:[0,1] → [0,1]such thatM(0)=0 andM(1)=1.Note that Mis a one-dimensional real Lie group. Indeed any elementM∈Mhas the form

M=MN(x)= xe^−2N 1+x(e^−2N −1)

(2) for someN∈RandM_N1◦M_N2=M_N1+N2. MoreoverM_Nis the unique Möbius transformationMwhich sends[0,1]

onto[0,1], M(0)=0, M(1)=1,and 1

0

D²M(x)

DM(x) dx=N.

1.2. Main results

Theorem 1.Letf ∈B²_k^+ν.Then there areC=C(f ) >0and0< λ=λ(k) <1such that d_C2

ZI_αⁿ

Rⁿ(f ) , M_Nⁿ

α

Cλⁿ

for allα∈A.Here N_αⁿ=

I_αⁿ

D²Rⁿ(f )(x) DRⁿ(f )(x) dx.

In particular d_C²

ZI_αⁿ

Rⁿ(f ) ,M

Cλⁿ.

We can say more about the mean nonlinearitiesN_αⁿ. Denote byq_αⁿ∈Nthe first return time of the intervalI_αⁿto the intervalIⁿ,i.e.,Rˆⁿ(f )|I_αⁿ=f^qαn, for someq_αⁿ∈N.

Theorem 2. Letf∈B²_k^+ν.Then there areC=C(f ) >0and0< λ=λ(k) <1such that N_αⁿ−

q_αⁿ

i=1|fⁱ(I_αⁿ)|

|I|

D²f (x) Df (x) dx

Cλ^√n. (3) In particular if

[0,1]

D²f (x) Df (x) dx=0 then|N_αⁿ|< Cλ^√n.

The rate of convergence obtained in (3) is enough for our purposes. In the case of circle diffeomorphisms Khanin and Teplinsky[8]obtained an exponential rate using a different approach.

Our third result is an almost direct consequence of Theorems 1 and 2.

Theorem 3.Letf∈B²_k^+νsuch that

[0,1]

D²f (x)

Df (x) dx=0.

Then there areC=C(k) >0and0< λ=λ(k) <1such that ZI_αⁿ

Rⁿ(f )

−Id

C²C·λ^√n for allα∈A.

The structure of this paper is as follows. In Section2we describe general results on compositions of diffeomor- phisms of classC^2+ν. In Section3we study renormalization of generalized interval exchange maps of genus one and prove Theorem1. In Section4we codify the dynamics off using a specially crafted symbolic dynamics to obtain finer geometric properties of the partitions associated with renormalizations off and we finally prove Theorems2 and3.

This is the first of a series of two papers based on the Ph.D. Thesis of the first author Cunha[1]. In the second work[2]we continue our study of the renormalization operator for generalized interval exchange transformations of genus one and its consequences, particularly the rigidity (universality) phenomena in the setting of piecewise smooth homeomorphisms on the circle.

2. Comparing compositions ofC^2+νmaps with Möbius maps

In this section, we show some results about composition ofC^2+ν-diffeomorphisms, comparing these composi- tions with Möbius maps. Letf:[a, b] → [f (a), f (b)] be aC² orientation-preserving diffeomorphism. Define the nonlinearity functionnf:[a, b] →Rby

n_f(x)=D²f (x) Df (x) =D

lnDf (x) . Notice that

n_f◦g(x)=n_f g(x)

Dg(x)+n_f(x),

consequently iff_iareC²-diffeomorphisms such thatf =f_n◦ · · · ◦f₁is defined in[a, b]we have

[a,b]

D²f (x) Df (x) dx=

n

i=1

fⁱ⁻¹[a,b]

D²f_i(x)

Df_i(x) dx. (4)

If[a, b] = [0,1]we define Nf =

[0,1]

D²f (x) Df (x) dx.

The nonlinearityn_f definesf up to its domain and image. Indeed, given a continuous functionn:[0,1] →Rthere is uniqueC²-diffeomorphismf:[0,1] → [0,1]such thatf (0)=0,f (1)=1 andn_f =n. Indeed, see Martens[10]

f (x)= _x

0 exp(_z

0n(y) dy) dz 1

0exp(z

0n(y) dy) dz

. (5)

Letf:[0,1] → [f (0), f (1)]be aC²orientation-preserving diffeomorphism. If[a, b] ⊂ [0,1], letf˜=Z_[a,b](f )be the Zoom off in[a, b].Then

n_{Z(f )}(x)=(b−a)·n_f

a+x(b−a)

. (6)

Suppose

n_f(x)−n_f(y)C₀· |x−y|^ν (7)

forx, y∈ [0,1]and

|n_f|_C⁰_[0,1]:= sup

x∈[0,1]

n_f(x)C₁. Then by (7) and (6) we have that

n_{Z(f )}(x)−n_{Z(f )}(y)C0·δ^1+ν and |n_{Z(f )}|C⁰[0,1]C1·δ, (8) withx, y∈ [0,1]andδ=b−a.Note that

N_{Z(f )}= 1 0

(b−a)·n_f

a+x(b−a) dx=

[a,b]

D²f (x)

Df (x) dx. (9)

Proposition 2.1.Letf:[0,1] → [f (0), f (1)]be an orientation-preserving diffeomorphism of classC^2+ν,[a, b] ⊂ [0,1]and definef˜=Z_[a,b]f. Then

d_C2(f , M˜ _N˜

f)=O δ^1+ν

, whereδ=b−a.

Before we prove Proposition2.1we prove the following lemma:

Lemma 2.2.LetN∈R. Letf_N:[0,1] → [0,1]be a diffeomorphism such thatn_f(x)=Nfor allx∈ [0,1],f (0)=0, f (1)=1. Then

d_C2(f_N, M_N)=O N²

. Proof. By (5) we have

f_N(x)= x

0 e^{N z}dz ₁

0 e^{N z}dz =e^{N x}−1 e^N−1 . Therefore,

f_N(x)−M_N(x)= e^{N x}−1

e^N−1 − xe^−2N 1+x(e^−N2 −1)

=

N x+^N2₂^x2 +O(N³)

N+^N₂² +O(N³) − x(1−^N₂ +O(N²)) 1+x(−^N₂ +O(N²))

= x

1+N x

2 −N 2

+O

N²

−x

1+N x 2 −N

2

+O N²

=O N²

, Df_N(x)−DM_N(x)=

N e^{N x}

e^N−1− e^−2N [1+x(e^−2N −1)]²

=

(1+N x+O(N²)) (1+^N₂ +O(N²)) −e^−N2

1+N x

2 +O N²²

=

(1+N x)

1−N 2

+O

N²

−e^−N2

1+N x+O N²

= 1−N

2 +N x+O N²

−1−N

2 +N x+O N²

=O N²

,

D²fN(x)−D²MN(x)= N²e^{N x}

e^N−1 −−2e^−2N(e^−N2 −1) [1+x(e^−N2 −1)]³

= N+O

N²

− N+O

N² 1+N x

2 +O N²³

=O N²

. 2

Proof of Proposition2.1. By Eq. (8) we have

|N_f˜||n_f|_C⁰_[0,1]δ.

To simplify the notation, denoteN˜ =N_f˜. First note that d_C2(f , M˜ _N˜)d_C2(f , f˜ _N˜)+d_C2(f_N˜, M_N˜).

In view of Lemma 2.2, we only need to estimate the first term of the right-hand side. For this note that N˜ = ₁

0n_f˜(s) ds=n_f˜(θ )for someθ∈ [0,1].If n_f(x)−n_f(y)C₀|x−y|^ν,

then by Eq. (8) we have|n_f˜(x)− ˜N| = |n_f˜(x)−n_f˜(θ )|C₀·δ^1+ν,son_f˜(x)= ˜N+O(δ^1+ν).Then by (5) we obtain

˜ f (x)=x

1−N˜

2 +N˜ 2x+O

δ^1+ν

; (10)

Df (x)˜ =1+ ˜N x−N˜ 2 +O

δ^1+ν

; (11)

D²f (x)˜ = ˜N+O δ^1+ν

. (12)

Using the estimates forf_N˜, Df_N˜ andD²f_N˜ similar to those in the proof of Lemma2.2, the proof is complete. 2 From now on let f_i:[0,1] → [0,1], i∈ N, be orientation-preserving diffeomorphisms of class C^2+ν, with f_i(0)=0,f_i(1)=1, and such that there existC₀, C₁>0 satisfying

n_fi(x)−n_fi(y)C0· |x−y|^ν (13) and

|nf_i|C⁰[0,1]C1

for everyi∈N. Let[a_i, b_i] ⊂ [0,1],δ_i=b_i−a_i, andf˜_i=Z_[ai,bi](f_i), M_i=M_N˜

fi, f˜₁ⁿ= ˜f_n◦ ˜f_n−1◦ · · · ◦ ˜f₁ and M₁ⁿ=M_n◦M_n−1◦ · · · ◦M₁.

The following proposition is the main result of this section. It compares the compositions off˜i’s andMi’s.

Proposition 2.3.(See also[9].) Letf_i be as above. Then for everyC₂>0 there existsC₃>0 with the following property. If ⁿ_i=1δ_iC₂,then

f˜₁ⁿ−M₁ⁿ

C²C₃·

1maxjnδ_j ν

.

The proof of Theorem 1 in Khanin and Vul[9]is the main motivation to Proposition2.3. Since Proposition2.3 is not stated explicitly in the paper cited above in its full generality, we include the full argument for the sake of completeness. Before we prove this proposition, we need some lemmas.

Lemma 2.4.There isC₄=C₄(C₁, C₂) >0such that e^−C4Df˜₁ⁿ(x)e^C4,

for allx∈ [0,1]and for alln0.

Proof.

lnDf˜₁ⁿ(x)

Df˜₁ⁿ(y)=lnDf˜n(f˜₁ⁿ⁻¹(x))·Df˜n−1(f˜₁ⁿ⁻²(x))· · ·Df˜1(x) Df˜n(f˜₁ⁿ⁻¹(y))·Df˜n−1(f˜₁ⁿ⁻²(y))· · ·Df˜1(y)

= n

j=1

lnDf˜_jf˜₁^j−1(x)

−lnDf˜_jf˜₁^j−1(y)

= n

j=1 f˜₁^j−1(x)

˜ f₁^j−1(y)

D²f˜_j(s) Df˜_j(s) ds

= n

j=1

D²f˜_j(z_j−1) Df˜j(zj−1)

f˜₁^j−1(x)− ˜f₁^j−1(y), for somez_j−1∈ [ ˜f₁^j−1(y),f˜₁^j−1(x)].

Therefore by (8) we have lnDf˜₁ⁿ(x)

Df˜₁ⁿ(y)

n

j=1

D²f˜_j(z_j−1) Df˜j(zj−1)

C₁· n

j=1

δ_jC₁C₂=C₄.

Takingy∈ [0,1]such thatDf˜₁ⁿ(y)=1 we have the result. 2 Lemma 2.5.There isC₅=C₅(C₁, C₂) >0such that

D²f˜₁ⁿ(x)C₅,

for allx∈ [0,1]and for alln0.

Proof. Note that D²f˜₁ⁿ(x)=

D²f˜₁ⁿ(x) Df˜₁ⁿ(x)

·Df˜₁ⁿ(x)

=

n

j=1

D²f˜_j(f˜₁^j−1(x))

Df˜j(f˜₁^j−1(x)) ·Df˜₁^j−1(x)

·Df˜₁ⁿ(x) e^2C4

n

j=1

D²f˜_j(f˜₁^j−1(x)) Df˜_j(f˜₁^j−1(x))

e^2C4·C₁·

n

j=1

δ_je^2C4·C₁C₂=C₅. 2

Lemma 2.6.There areC6=C6(C1, C2), C7=(C1, C2) >0such that e^−C6DM₁ⁿ(x)e^C6, D²M₁ⁿ(x)C7, D³M₁ⁿ(x)C8

for allx∈ [0,1]and for alln0.

Proof. SinceMis a commutative Lie group, we haveM₁ⁿ=M_N, whereN= ⁿ_i=1N_f˜

i. By Eq. (8) we have|N_f˜

i|<

C₁δ_i, so

|N|C1·C2.

One can easily use Eq. (2) to obtain estimates forDⁱM₁ⁿ,i=1,2,3. 2 Proof of Proposition2.3. We write

f˜₁ⁿ−M₁ⁿ= n

i=1

M_iⁿ₊₁◦ ˜f₁ⁱ−M_iⁿ◦ ˜f₁ⁱ⁻¹, whereM_nⁿ₊₁=f₁⁰=Id.Then

f˜₁ⁿ(x)−M₁ⁿ(x) n

i=1

M_iⁿ₊₁f˜_i◦ ˜f₁ⁱ⁻¹

(x)−M_iⁿ₊₁

M_i◦ ˜f₁ⁱ⁻¹(x) e^C6·

n

i=1

f˜_i◦ ˜f₁ⁱ⁻¹(x)−M_i◦ ˜f₁ⁱ⁻¹(x) e^C6·C·

n

i=1

δ¹_i^+ν e^C6·C·

1maxin

δ_i ν

i

δ_i e^C6·C·C₂·

max

1in

δ_i ν

, whereC >0 is the constant given by Proposition2.1;

Df˜₁ⁿ(x)−DM₁ⁿ(x)=

n

i=1

DM_iⁿ₊₁f˜_i◦ ˜f₁ⁱ⁻¹(x)

·Df˜_if˜₁ⁱ⁻¹(x)

−DM_iⁿ₊₁

Mi◦ ˜f₁ⁱ⁻¹(x)

·DMi

f˜₁ⁱ⁻¹(x)

Df˜₁ⁱ⁻¹(x) e^C4·

n

i=1

DM_iⁿ₊₁f˜_i◦ ˜f₁ⁱ⁻¹(x)

·Df˜_if˜₁ⁱ⁻¹(x)

−DM_iⁿ₊₁

Mi◦ ˜f₁ⁱ⁻¹(x)

·DMi

f˜₁ⁱ⁻¹(x).

Now, add and subtract the termDM_iⁿ₊₁(f˜_i◦ ˜f₁ⁱ⁻¹(x))·DM_i(f˜₁ⁱ⁻¹(x))in the above expression to obtain Df˜₁ⁿ(x)−DM₁ⁿ(x)e^C4·e^C6·

n

i=1

Df˜_if˜₁ⁱ⁻¹(x)

−DM_if˜₁ⁱ⁻¹(x) +e^C4·e^C6·

n

i=1

DM_iⁿ₊₁f˜_i◦ ˜f₁ⁱ⁻¹(x)

−DM_iⁿ₊₁

M_i◦ ˜f₁ⁱ⁻¹(x) e^C4·e^C6·C·

n

i=1

δ_i^1+ν+e^C4·e^C6·C₇·C· n

i=1

δ¹_i^+ν e^C4·e^C6·(1+C7)·C·C2·

1maxinδi

ν

. Now note that

D²f˜₁ⁿ(x)−D²M₁ⁿ(x)=(I)+(II)+(III),

where (I)=

n

i=1

D²M_iⁿ₊₁f˜_i◦ ˜f₁ⁱ⁻¹(x)

·

Df˜_if˜₁ⁱ⁻¹(x)2

·

Df˜₁ⁱ⁻¹(x)2

− n

i=1

D²M_iⁿ₊₁

M_i◦ ˜f₁ⁱ⁻¹(x)

·

DM_if˜₁ⁱ⁻¹(x)2

·

Df˜₁ⁱ⁻¹(x)2

,

(II)= n

i=1

DM_iⁿ₊₁f˜_i◦ ˜f₁ⁱ⁻¹(x)

·D²f˜_if˜₁ⁱ⁻¹(x)

·

Df˜₁ⁱ⁻¹(x)2

− n

i=1

DM_iⁿ₊₁

Mi◦ ˜f₁ⁱ⁻¹(x)

·D²Mi

f˜₁ⁱ⁻¹(x)

·

Df˜₁ⁱ⁻¹(x)2

and

(III)= n

i=1

DM_iⁿ₊₁f˜i◦ ˜f₁ⁱ⁻¹(x)

·Df˜i

f˜₁ⁱ⁻¹(x)

·D²f˜₁ⁱ⁻¹(x)

− n

i=1

DM_iⁿ₊₁

Mi◦ ˜f₁ⁱ⁻¹(x)

·DMi

f˜₁ⁱ⁻¹(x)

·D²f˜₁ⁱ⁻¹(x).

In(I)we first add and subtract the term n

i=1

D²M_iⁿ₊₁f˜_i◦ ˜f₁ⁱ⁻¹(x)

·

DM_if˜₁ⁱ⁻¹(x)2

·

Df˜₁ⁱ⁻¹(x)2

and then we use Lemmas2.4and2.6with estimates for the first derivative off˜₁ⁿandM₁ⁿ,to obtain (I)2·max{C9, C10} ·

max

1inδi

ν

, (14)

whereC9=C2·C7·C·e^2C4(e^C4+e^C6)andC10=C·C2·C8·e^2C4·e^C6. In(II), we first add and subtract the term

n

i=1

DM_iⁿ₊₁

Mi◦ ˜f₁ⁱ⁻¹(x)

·D²f˜i

f˜₁ⁱ⁻¹(x)

·

Df˜₁ⁱ⁻¹(x)2

and then we use Lemmas2.4and2.6with estimates for the first derivative off˜₁ⁿandM₁ⁿ,to obtain (II)2·max{C₁₁, C₁₂} ·

1maxin

δ_i ν

, (15)

whereC11=C·C2·C5·C7·e^2C4 andC12=C·C2·e^2C4+C6. Finally we add and subtract the expression

n

i=1

DM_iⁿ₊₁f˜_i◦ ˜f₁ⁱ⁻¹(x)

·DM_if˜₁ⁱ⁻¹(x)

·D²f˜₁ⁱ⁻¹(x)

in(III)and use again Lemmas2.4and 2.6, obtaining (III)2·max{C₁₃, C₁₄} ·

max

1in

δ_i ν

, (16)

whereC13=C·C2·C5·e^C6andC14=C·C2·C5·C7·e^C6.

TakingC3=6·max{C9, C10, C11, C12, C13, C14}we get the result. 2