Renormalization of Hamiltonians for Diophantine frequency ...of Diophantine frequency vectors and obtain some bounds that will be used in section 7 to prove the convergence of the

Renormalization of Hamiltonians for Diophantine frequency vectorsand KAM tori

Saša Kocić

Department of Physics, University of Texas at Austin,

1 University Station C1600, Austin, TX 78712, USA

E-mail: [email protected]

(Dated: July 31, 2005)

We construct a rigorous renormalization scheme for two degree-of-freedom analytic Hamiltonians,associated with Diophantine frequency vectors. We prove the existence of an attracting, integrablelimit set of the renormalization. As an application of this renormalization scheme, we give a proofof a KAM theorem. We construct analytic invariant tori with Diophantine frequency vectors fornear-integrable Hamiltonians in the domain of attraction of this limit set.

1. INTRODUCTION AND SUMMARY OF THE RESULTS

Half a century ago, Kolmogorov [24] stated a theorem concerning the persistence of quasi-periodic motion (invariant tori) with sufficiently incommensurate frequency vectors in thephase space of a (non-degenerate) integrable Hamiltonian system under small analytic per-turbations. Several years later, a detailed proof of the theorem was given by Arnol’d [1] andthe requirement of analyticity was weakened by Moser [34]. It was the beginning of a series ofrigorous results and methods that are collectively referred to as KAM theory (see also [26]).

Almost at the same time, the simple and novel idea of renormalization originated in the-oretical physics. It was introduced first in quantum field theory by Stueckelberg and Pe-terman [38] and later in statistical mechanics by Kadanoff [17]. In the theory of dynamicalsystems, renormalization group techniques were first introduced by Feigenbaum [7, 8] andCoullet and Tresser [5] in studying the universality of period-doubling sequences in one-parameter families of one-dimensional maps. Since then, renormalization group ideas havebeen widely applied in investigation of a variety of dynamical phenomena. In discrete-timedynamical systems those include self-similarity of period-doubling bifurcations in two andhigher-dimensional mappings, bifurcations in circle maps and bifurcations of invariant KAMcurves in area-preserving maps [9, 16, 18, 19, 25, 29–33, 35–37].

In the framework of the breakup of invariant tori in Hamiltonian flows, renormalizationgroup ideas were first introduced by Escande and Doveil [6]. Some related ideas concerningthe renormalization of Hamiltonian flows, due to MacKay and others, can be found in [14, 33].A rigorous renormalization scheme for analytic Hamiltonians, associated to frequency vectorsof a specific type, was formulated by Koch [20, 21]. References such as [10, 11, 27] containsome later related work.

Here, we construct a rigorous renormalization scheme for analytic Hamiltonian functions oftwo degree-of-freedom systems which applies to the problem of stability of invariant tori withDiophantine frequency vectors. The renormalization scheme is an extension of Koch’s renor-malization, from quadratic irrational to a set of full Lebesgue measure Diophantine frequencyratios, in the case of two degree-of-freedom Hamiltonians. Recently, such a generalizationwas done by Lopes-Dias for vector fields on a torus of dimension two [28]. It is also relatedto the approximate renormalization schemes of MacKay [31] and Chandre and Moussa [3] inthe framework of the break-up of invariant tori in Hamiltonian systems (see also [4]).

Further, we apply our renormalization scheme to give a proof of a KAM theorem. We con-struct analytic invariant tori with Diophantine frequency vectors for near-integrable Hamil-tonians. A similar method has been used in [23] to prove the existence of non-differentiableinvariant tori with golden mean winding number for Hamiltonians attracted to the criticalfixed point of the renormalization [22]. A different method for construction of invariant tori,based on the analogy with quantum field theory and the summation of the (generically di-vergent) Lindstedt series, was used by Gallavotti et al [12, 13] and Bricmont et al [2]. Arelated result for discrete-time systems, concerning the existence of invariant graphs for pairsof commuting maps, for Diophantine rotation numbers, has been obtained by Haydn [16].

2 Saša Kocić

We define a sequence of renormalization operators Rn, n ∈ N0 = N∪ {0}, between Banachspaces of analytic two degree-of-freedom Hamiltonians. The Hamiltonians are functions oncomplex neighborhoods of T2 × {0}, where T2 = R2/(2πZ)2 and 0 is the zero vector in R2.We call these neighborhoods the phase space. We refer to the coordinates of this space, q andp, as angles and momenta, respectively.

Each of the Banach spaces, indexed by n, contains an integrable Hamiltonian H0n with aninvariant torus of a given frequency ωn ∈ R2\{0} at p = 0 and frequencies of nearby toritwisted in the direction of Ωn ∈ R2\{0}. The vectors ωn and Ωn are uniquely determinedgiven a pair of vectors (ω,Ω) = (ω0,Ω0). The domain of the n

th-step renormalization operatorRn will be restricted to a small neighborhood of the integrable Hamiltonian H0n, keeping theanalysis within the scope of classical KAM theory. The nth-step renormalization operatorRn is a function from that neighborhood into another that maps a Hamiltonian Hn, close toH0n, into Hn+1 = Rn(Hn), close to H0n+1 = Rn(H0n). On the winding ratio αn = ωn2/ωn1,ωn1 6= 0, of the frequency vector ωn = (ωn1, ωn2)∗, the operator Rn acts as a shift of itscontinued fraction expansion. The renormalization of a Hamiltonian H0 consists of successiveapplication of the operators Rn, n ∈ N0. Given a Hamiltonian H0, we call the sequence ofHamiltonians Hn, n ∈ N0, consisting of H0 and its images Hn+1 = Rn ◦ · · · ◦ R0(H0), theorbit of the Hamiltonian H0.

Renormalization techniques in dynamical systems are designed to study systems on progres-sively smaller spatial scales and longer time scales. The renormalization scheme for Hamiltoni-ans is essentially a transformation of classical Hamiltonian functions generated by a scaling ofthe phase space, modulo a set of transformations that preserve the topological characteristicsof the orbits of the Hamiltonian flows. This set includes canonical transformations homotopicto the identity, scaling of the momenta, time and energy. The nth-step renormalization opera-tor basically consists of time-rescaling, composed with a nonlinear diffeomorphism homotopicto the identity, and a linear scaling of the (lifted) phase space.

The linear scaling transformation is intended to enlarge a region around the orbits of theintegrable Hamiltonian flow while keeping the periodicity of the angle coordinates. It is acomposition of a linear scaling of the momentum space and a linear canonical transformationof the phase space generated by a point transformation in GL(2,Z).

The growth of the Fourier modes of the Hamiltonian in the direction of the dominant flow isprevented by eliminating these modes in each step. This is achieved by a nonlinear canonicaltransformation homotopic to the identity. The process of elimination (of “irrelevant” modesof a Hamiltonian) and rescaling (of the Fourier lattice) is similar in spirit to block spintransformations in statistical mechanics - a standard tool in the theory of critical phenomena.

Additionally, time-rescaling, a nonlinear scaling of the momenta and a translation in mo-mentum space are included. The latter transformation prevents the renormalization operatorsfrom having an expanding eigendirection. These transformations are also homotopic to theidentity and do not change the winding number of the orbits of the Hamiltonian flow.

Main results: The results of this paper concerning the constructed renormalization schemefor analytic Hamiltonians and its applications are:

(i) The nth-step renormalization operator Rn, n ∈ N0, is a well-defined analytic mapon an open ball of analytic Hamiltonians around H0n for a generic frequency vectorωn ∈ R2\{0} (Theorem 4.4).

(ii) For a set of Diophantine frequency vectors ω0 of full Lebesgue measure, the renormal-ization orbits of all Hamiltonians in a neighborhood of an integrable Hamiltonian H00associated to ω0, approach the orbit of the integrable Hamiltonian H

00 (Theorem 7.2).

(iii) Every Hamiltonian H0 sufficiently close to such a H00 has an invariant torus on which

the motion is conjugate to a linear flow of Diophantine frequency vector ω0 (Theo-rem 9.5).

(iv) The invariant torus is an analytic function (Theorem 10.4).

The above-mentioned set of Diophantine frequency vectors for which the convergence resulthas been obtained and invariant tori have been constructed contains those with Diophantineexponent β < (

√161− 11)/10. Though this set is not optimal, it is of full Lebesgue measure.

Renormalization scheme for Hamiltonians and KAM tori 3

We believe that this is not a restriction of the renormalization method and that the set canbe extended to other Diophantine frequency vectors.

The paper is organized as follows. In the next section, we define the Banach spaces ofHamiltonians that will be renormalized. In section 3, we describe the performed scaling ofthe phase space. Section 4 contains the construction of the nth-step renormalization operator.The existence of a canonical transformation that eliminates the non-resonant modes of anear-integrable Hamiltonian is proved in section 5. In section 6, we recall the definitionof Diophantine frequency vectors and obtain some bounds that will be used in section 7to prove the convergence of the renormalization dynamics to an integrable limit set for ωsatisfying a Diophantine condition. The remaining sections contain an application of thepreviously constructed renormalization scheme. Section 8 contains the definition of invarianttori. In section 9, we construct the invariant tori with Diophantine frequency vectors fornear-integrable Hamiltonians. Finally, in section 10, we prove that the constructed invarianttori can be extended to analytic functions.

2. THE SPACES OF HAMILTONIANS

Let us start by defining the spaces of Hamiltonians that we will consider. Given n ∈ N0,let ωn,Ωn ∈ R2\{0} be two vectors not parallel to each other. We introduce the normalizedvector ω̂n = ωn/‖ωn‖. Here and in what follows ‖ · ‖ denotes `1-norm of a vector. Defineω′n and Ω

′n in R2, by the following relations: ω′n · Ωn = 0, ω′n · ω̂n = 1, Ω′n · ω̂n = 0, and

Ω′n · Ωn = 1.

Definition 2.1 Given a pair of positive numbers ρ = (ρ1, ρ2), define

Dn,1(ρ1) = {q ∈ C2 : |Imω′n · q| < ρ1, |Im Ω′n · q| < ρ1} ,Dn,2(ρ2) = {p ∈ C2 : |ω̂n · p| < ρ2, |Ωn · p| < ρ2} ,

(1)

and let Dn(ρ) = Dn,1(ρ1)×Dn,2(ρ2).

The Hamiltonians are analytic functions Hn : Dn(ρ)→ C, 2π-periodic in both q-variables.These functions can be expanded in Fourier-Taylor series

Hn(q, p) =∑

(ν,k)∈I

(Hn)ν,k(ω̂n · p)k1(Ωn · p)k2eiq·ν , (2)

where k = (k1, k2) and I = Z2 ×N20. We will refer to each term in this sum as a mode of theHamiltonian Hn.

Definition 2.2 Given ρ > 0, componentwise, define An(ρ) to be the Banach space of func-tions Hn that are analytic on Dn(ρ), extend continuously to the boundary of Dn(ρ), and havefinite norm

‖Hn‖n,ρ =∑

(ν,k)∈I

|(Hn)ν,k|ρ‖k‖2 e

ρ1(|ω̂n·ν|+|Ωn·ν|) . (3)

Let us also define the projection operators Pkn on An(ρ) by

PknHn = (Hn)0,k(ω̂n · p)k1(Ωn · p)k2 , (4)

and let E =∑k∈N20

Pkn be the projection operator onto the subspace of q-independent Hamil-tonians. Define the functionals pkn : An(ρ)→ C, by pknHn = (Hn)0,k.

The analysis of this paper is focused on Hamiltonians of the form Hn = H0n + hn, where

H0n = ωn · p+1

2(Ωn · p)2, (5)

4 Saša Kocić

is an integrable Hamiltonian and hn ∈ An(ρ) is a perturbation. The integrable Hamiltoni-ans H0n are degenerate, in the sense that they do not satisfy Kolmogorov’s non-degeneracycondition, but they do have a twist in the Ωn direction, i.e. they satisfy

det

[∂2H0n∂pi∂pj

]= 0,

∣∣∣∣ ∂2H0n∂(Ωn · p)2∣∣∣∣ = 1 6= 0. (6)

Hamiltonians satisfying this weaker (than Kolmogorov’s original) non-degeneracy conditionhave been included in the improved versions of the KAM theory.

The sequence of renormalization transformations is associated to the sequence of vectorpairs (ωn,Ωn), n ∈ N0. This sequence has been constructed from a pair of vectors ω,Ω ∈R2\{0}. We assume that ω ∈ R2 is of the form ω = `(1, α)∗, where ` ∈ R+ and α > 1 is anirrational number. Here “ ∗ ” stands for “transpose”.

The unique continued fraction expansion [15] of such an irrational α ∈ R is given by

α = a0 +1

a1 +1

a2+...

, (7)

where an ∈ Z+, n ≥ 0, are called partial quotients. We also write α = [a0, a1, . . . ]. Thevectors ωn are constructed from the continued fraction expansion of the winding ratio α ofthe frequency vector ω. We define ωn = `(1, αn)

∗, where αn = [an, an+1, . . . ].We choose a vector Ω ∈ R2 which is suitable in the sense of the following Definition.

Definition 2.3 Ω ∈ R2 will be called suitable if it is of the form Ω = (Ω1,Ω2)∗, with |Ω1| ≥|Ω2| ≥ 0, Ω1Ω2 ≤ 0 and ‖Ω‖ = 1.

Notice that a suitable vector Ω can not be arbitrary close to the direction of ω. The smaller(positive) angle between these vectors is larger than π/4 and smaller than 3π/4.

3. SCALING OF THE PHASE SPACE

The renormalization scheme for d degree-of-freedom Hamiltonians introduced by Koch isassociated to a frequency vector ω whose components span an algebraic number field of degreed ∈ N. For two degree-of-freedom systems, that scheme can be applied to Hamiltoniansassociated to a frequency vector with a quadratic irrational slope. In that case, one canfind a hyperbolic matrix T ∈ GL(2,Z) with determinant ±1, for which ω is an expandingeigenvector (Lemma 4.1 in [20]). That matrix is then used to perform linear scaling of thephase space at each renormalization step.

In order to construct a renormalization scheme which applies to a larger set of frequencyvectors, we use a different linear scaling transformation at each step. We generate the nth-stepscaling transformation of the phase space using the matrix Tn ∈ GL(2,Z) defined by

Tn =

[0 11 an

]. (8)

More precisely, at the nth renormalization step, the scaling of the (lifted) phase space isperformed with a linear map T −1n on C2 × C2, defined by Tn(q, p) = (Tnq, µnT ∗n

−1p), whereµn = α

−1n+1‖T−1n Ωn‖−2 is a positive number. This map generates the transformation of

Hamiltonians Hn 7→ Hn ◦ Tn. Notice that the vectors ωn could also be defined recursivelyusing this matrix, via ωn+1 = αn+1T

−1n ωn, given ω = ω0. Similarly, given a suitable vector

Ω = Ω0, the vectors Ωn are defined recursively by Ωn = T−1n−1Ωn−1/‖T

−1n−1Ωn−1‖, for n ∈ N.

Thus, under this scaling, the integrable Hamiltonian H0n is mapped into α−1n+1µnH

0n+1, n ∈ N0.

A time rescaling, then, normalizes the latter to H0n+1.The matrix Tn has the following properties. If α > 1, then, for all n ≥ 0, an ≥ 1 and Tn is

a hyperbolic matrix with det(Tn) = −1 and eigenvalues λn and −1/λn, where

λn =an +

√a2n + 4

2. (9)


The expanding and contracting eigendirections corresponding to these eigenvalues are givenby (1, λn)

∗ and (1,−1/λn)∗, respectively. The expanding eigenvector is close to ωn, in thesense that the absolute value of the angle between it and ωn is smaller than π/4.

The map ωn 7→ ωn+1 is related with the Gauss map of the fractional part of αn. Givenα ∈ R, let [α] be the integer part of α, i.e. [α] = max{k ∈ Z : k ≤ α}. Also, let {α} = α− [α]be the fractional part of α.

For x > 0, the Gauss map is defined as

G : x 7→{

1

x

}. (10)

If we define xn = αn − an, for n ≥ 0, we obtain xn+1 = G(xn). Thus, the shift[an, an+1, . . . ] 7→ [an+1, an+2, . . . ] of the continued fraction expansion of αn corresponds tothe Gauss map of its fractional part xn.

The transformation αn 7→ αn+1 = (αn−an)−1 is a special case of a modular transformation,

α 7→ T2,1 + αT2,2T1,1 + αT1,2

, (11)

generated by the action of an integer matrix T = (Ti,j) ∈ GL(2,Z) of determinant ±1.Numbers related by such a transformation are called equivalent. Two numbers are equivalentif and only if they have the same tail in their continued fraction expansions (Theorem 175in [15]). The matrix Tn generates an equivalence relation between αn+1 and αn.

The linear coordinate change T −1n , maps the orbit of H0n with rotation number αn into theorbit of H0n+1 with the rotation number αn+1. We have made the particular choice of Tn inorder to provide the desired “scale change” in the sense explained below.

Let us consider the sequence of periodic orbits approximating an invariant torus withfrequency ratio α. Recursively, define the sequence of convergent matrices associated to α,

Pn =

[qn−1 pn−1qn pn

], (12)

for n ≥ 0, via Pn = TnPn−1, with P−1 being the identity matrix in GL(2,Z). Thus, qn andpn are determined by the following recursion relations

qn = anqn−1 + qn−2,

pn = anpn−1 + pn−2, (13)

and q−2 = 1, p−2 = 0, q−1 = 0, p−1 = 1. The convergent matrices define a sequence ofconvergents pn/qn = [a0, . . . , an] that approaches α as n→∞.

The values of qn and pn are associated to α. In the following, we will stress this fact bywriting explicitly qn(α) and pn(α). Consider the sequence of periodic orbits of frequency

ratios pk(αn)qk(αn) . The matrix Tn produces the following “scale change”(−an 1

1 0

)(qk(αn)pk(αn)

)=

(qk−1(αn+1)pk−1(αn+1)

), (14)

motivating its use. A periodic orbit labelled by k that approximates an invariant torus withfrequency ratio αn of H

0n is mapped into a periodic orbit labelled by k− 1 that approximates

an invariant torus with frequency ratio αn+1 of H0n+1.

The composition Hn ◦Tn represents a singular operator, as the matrix Tn has an expandingeigendirection. The composition is, however, harmless when the domain of the operator isrestricted to Hamiltonians which contain only the modes for which ‖k‖ is large or ν is almostperpendicular to ω. These modes are called resonant. These are the modes that producesmall denominators in KAM theory.

Definition 3.1 Given σ, κ > 0 and vectors ωn,Ωn ∈ R2, we define the non-resonant indexset,

I−

n = {(ν, k) ∈ I : |ωn · ν| > σ|Ωn · ν|, |ωn · ν| > κ‖k‖}. (15)

6 Saša Kocić

The resonant index set is defined as its complement, I+

n = I\I−

n . The corresponding projection

operators on An(ρ), I−

n and I+

n = I− I−

n , are defined by setting

(I−

nHn)(q, p) =∑

(ν,k)∈I−n

(Hn)ν,k(ω̂n · p)k1(Ωn · p)k2eiq·ν . (16)

Hamiltonians consisting only of resonant modes, will be called resonant. On the space ofresonant Hamiltonians, the map Hn 7→ Hn ◦ Tn is analyticity improving.

Proposition 3.2 Let 0 < ρ′ < ρ with 3ρ′ > 2ρ, componentwise. If σ and κ are sufficiently

small constants, then every Hamiltonian Hn ∈ I+

nAn(ρ′), n ∈ N0, has an analytic extension toTnDn+1(ρ). The linear map from I

+

nAn(ρ′) to An+1(ρ), given by : Hn 7→ Hn ◦Tn, is compact.

Proof: As

Hn ◦ Tn(q, p) =∑

(ν,k)∈I+n

(Hn)ν,k(µnT−1n ω̂n · p)k1(µnT−1n Ωn · p)k2eiq·T

∗ν , (17)

one can obtain

‖Hn ◦ Tn‖n+1,ρ =∑

(ν,k)∈I+n

|(Hn)ν,k|(µn‖ωn+1‖αn+1‖ωn‖

ρ2

)k1(µn‖T−1n Ωn‖ρ2)k2

· eρ1

(αn+1‖ωn+1‖

|ωn·ν|+ |Ωn·ν|‖T−1n Ωn‖

)

=∑

(ν,k)∈I+n

|(Hn)ν,k|(µn‖ωn+1‖ρ2αn+1‖ωn‖ρ′2

)k1 (µn‖T−1n Ωn‖ρ2ρ′2

)k2ρ′‖k‖2

· e(ρ1αn+1‖ωn‖‖ωn+1‖

−ρ′1)|ω̂n·ν|+

(ρ1

‖T−1n Ωn‖−ρ′1

)|Ωn·ν|

eρ′1(|ω̂n·ν|+|Ωn·ν|)

≤ ‖Hn‖n,ρ′ ,

(18)

provided that all of the modes contract. The terms with ν obeying the inequality |ωn · ν| ≤σ|Ωn · ν| contract for sufficiently small σ satisfying the second part of the double inequality

σ αn+1‖ωn+1‖ +1

‖T−1n Ωn‖

1 + σ‖ωn‖≤ σ`

+2

3<ρ′1ρ1. (19)

Given σ > 0, the first part is satisfied for any n ∈ N0. The other conditions are triviallysatisfied as µn‖T−1n Ωn‖ ≤ 2/3.

The modes indexed by (ν, k) satisfying |ωn ·ν| ≤ κ‖k‖ also contract if κ satisfies the secondpart of the double inequality

µn‖T−1n Ωn‖eκ

(ρ1αn+1‖ωn+1‖

− ρ′1

‖ωn‖

)≤ 2

3eκρ1/` <

ρ′2ρ2. (20)

Given κ > 0, the first part is satisfied for any n ∈ N0.These estimates show that Hn◦Tn is analytic in Dn+1(ρ). Now, given σ, κ > 0 satisfying the

second parts of the double inequalities (19) and (20), one can find r > ρ satisfying 3ρ′ > 2rcomponentwise, such that Hn ◦ Tn is also analytic and bounded in Dn+1(r). The assertionnow follows from the fact that the inclusion map from An+1(r) to An+1(ρ) is compact. �

4. nth-STEP RENORMALIZATION OPERATOR

We will restrict the domain of the nth-step renormalization operator to resonant Hamiltoni-ans. The composition of a resonant Hamiltonian with Tn produces, in general, non-resonant


modes. In the nth renormalization step, we completely eliminate these modes such thatthe renormalized Hamiltonians are also resonant. We also include a translation in the vari-able Ωn+1 · p, to prevent the nth-step renormalization operator from having an expandingeigendirection. The existence of the non-zero quadratic (in the components of p) part of theintegrable Hamiltonian H0n is essential for the construction of such a transformation. Finally,we perform an additional, nonlinear, scaling of the action (momentum) variables and time,in order to fix the coefficients of the (ωn+1 · p) and (Ωn+1 · p)2 modes of the renormalizedHamiltonians to 1 and 1/2, respectively.

Definition 4.1 The nth-step renormalization operator Rn is defined (formally) on an openball in I+nAn(ρ′), with ρ′ > 0, componentwise, by the following action on a HamiltonianHn ∈ I

+

nAn(ρ′),

Rn(Hn) =θ′nµ′n

(Hn ◦ Λn − P(0,0)n+1Hn ◦ Λn), (21)

where Λn = Tn ◦ VH′n+1 ◦ UH′′n+1 ◦ SH̃n+1 . Here H′n+1 = (θn/µn)Hn ◦ Tn, with θn =

αn+1, H′′n+1 = H

′n+1 ◦ VH′n+1 and H̃n+1 = H

′′n+1 ◦ UH′′n+1 . The transformation VH′n+1 :

Dn+1(%) → Dn+1(ρ), where 3/2ρ′ > ρ > % > %′ > ρ′, componentwise, represents thetranslation VH′n+1(q, p) = (q, p − vH′n+1Ω

′n+1), with vH′n+1 ∈ C determined by the equation

p(0,1)n+1H

′n+1 ◦VH′n+1 = 0. The map UH′′n+1 : Dn+1(%

′)→ Dn+1(%) is a canonical transformationthat satisfies I−n+1 (H ′′n+1◦UH′′n+1) = 0. The transformation SH̃n+1 : Dn+1(ρ

′)→ Dn+1(%′) rep-resents a scaling SH̃n+1(q, p) = (q, zH̃n+1p) of the action variables. The scaling parameters θ

′n

and µ′n take the values θ′n = θn ·τH̃n+1 and µ

′n = µn ·zH̃n+1 . The parameters zH̃n+1 , τH̃n+1 ∈ C

are determined such that p(0,2)n+1Hn+1 = 1/2 and p

(1,0)n+1Hn+1 = 1, where Hn+1 = Rn(Hn).

In the following, we show that the translations and scalings included in the nth-step renor-malization operator are well-defined on a sufficiently small open ball around H0n+1. Notice

first that H0n+1 = H̃0n+1 = H

0′

n+1 = H0′′

n+1 = Rn(H0n). For n ∈ N0, b > 0 and ρ > 0, com-ponentwise, define Bn,ρ(b) to be the open ball of radius b, in An(ρ), centered at H0n. Definealso B+n,ρ(b) to be the open ball in I

+

nAn(ρ) of radius b, centered at H0n.

Proposition 4.2 Given ρ2 > %2 > 0 and ρ1 = %1 > 0, the following holds for a suffi-ciently small constant b > 0. For every Hamiltonian Hn+1 ∈ Bn+1,ρ(b), n ∈ N0, there existsvHn+1 ∈ C, such that the translation map VHn+1 : Dn+1(%) → Dn+1(ρ), is well-defined byVHn+1(q, p) = (q, p − vHn+1Ω′n+1), where p

(0,1)n+1Hn+1 ◦ VHn+1 = 0. The derivative of the map

VHn+1 : An+1(ρ)→ An+1(%), defined by VHn+1(Hn+1) = Hn+1 ◦VHn+1 , at H0n+1, is the linearmap DVn+1(H0n+1) = I− P

(0,1)n+1 .

Proof: Define the function F : An+1(ρ) × C → C, by setting F (Hn+1, v) = p(0,1)n+1Hn+1 ◦ V ,where V (q, p) = (q, p− vΩ′n+1). The implicit equation F (H0n+1, v) = 0 has a unique solutionv = vH0n+1 = 0. Moreover, D2F (H

0n+1, v)|v=0 = −1 6= 0. We can use this fact to solve

the implicit equation, F (Hn+1, v) = 0, for a given Hamiltonian Hn+1 in a ball Bn+1,ρ(b) ofsufficiently small, n-independent radius b > 0.

The problem of the existence of a solution v = vHn+1 of the implicit equation F (Hn+1, v) =0, for a given Hamiltonian Hn+1 ∈ Bn+1,ρ(b), is equivalent to the problem of the existenceof a fixed point of the function GHn+1 : v 7→ v + F (Hn+1, v). Notice that |GHn+1(0)| =|(hn+1)0,(0,1)| < b/ρ2. We will show that, given λ > 1, for sufficiently small b > 0, GHn+1 isa contraction on a ball of radius λb/ρ2.

Let |v| ≤ λb/ρ2, λ > 1. The norms of GHn+1(v) and G′Hn+1(v) can be bounded by

|GHn+1(v)| ≤∞∑k2=1

|(hn+1)0,(0,k2)|k2|v|k2−1 <

b

ρ2(1− λb/ρ22)2= b̃

8 Saša Kocić

and

|G′Hn+1(v)| = |1 +D2F (Hn+1, v)| ≤∞∑k2=2

|(hn+1)0,(0,k2)|k2(k2 − 1)|v|k2−2 <

2b

ρ22(1− λb/ρ22)3,

respectively. If b > 0 is sufficiently small, then

λb/ρ2 < ρ2 − %2 ,1

(1− λb/ρ22)2< λ and

2b

ρ22(1− λb/ρ22)3< 1− 1

λ< 1 . (22)

These bounds show that for sufficiently small b > 0, GHn+1 is a contraction on a closed ball of

radius b̃, and thus, has a unique fixed point in that ball. As the bounds (22) are independentof n, so is b. The first of the bounds (22) shows that the translation map is well-defined fromDn+1(%) to Dn+1(ρ) and that Hn+1 ◦ VHn+1 belongs to An+1(%). �

The map H ′′n+1 7→ H ′′n+1◦UH′′n+1 is well-defined on a sufficiently small ball centered at H0n+1.

Theorem 5.6, proved in the next section, guarantees that given % > %′ > 0, componentwise,for every Hamiltonian H ′′n+1 ∈ An+1(%), sufficiently close to H0n+1, there exists a canonicalmap UH′′n+1 : Dn+1(%

′)→ Dn+1(%), that satisfies the equation I−

n+1 (H′′n+1 ◦ UH′′n+1) = 0.

Proposition 4.3 Let %′2 > ρ′2 > 0 and %

′1 = ρ

′1 > 0. For sufficiently small constant b > 0 and

for every Hamiltonian Hn+1 ∈ Bn+1,%′(b), n ∈ N0, there exist zHn+1 , τHn+1 ∈ C such that themap SHn+1 : Hn+1 7→ (τHn+1/zHn+1)(Hn+1 ◦SHn+1−P

(0,0)n+1Hn+1 ◦SHn+1), with SHn+1(q, p) =

(q, zHn+1p), is well-defined from An+1(%′) to An+1(ρ′), and satisfies p(0,2)n+1 SHn+1(Hn+1) = 1/2

and p(1,0)n+1 SHn+1(Hn+1) = 1. The map SHn+1 maps Dn+1(ρ′) into Dn+1(%′). The derivative

of the map SHn+1 at the point H0n+1 is given by DSn+1(H

0n+1) = I− P

(0,0)n+1 − P

(1,0)n+1 − P

(0,2)n+1 .

Proof: Let Hn+1 ∈ Bn+1,%′(b). For sufficiently small b > 0, the scaling parameters τHn+1and zHn+1 that satisfy the equations p

(0,2)n+1 SHn+1(Hn+1) = 1/2 and p

(1,0)n+1 SHn+1(Hn+1) = 1,

are given by τHn+1 = 1/(1+(hn+1)0,(1,0)) and zHn+1 = (1+(hn+1)0,(1,0))/(1+2(hn+1)0,(0,2)).We have the bound

|zHn+1 | ≤ |zHn+1 − 1|+ 1 ≤|(hn+1)0,(1,0) − 2(hn+1)0,(0,2)|

1− 2|(hn+1)0,(0,2)|+ 1 ≤ 1 + b/%

′2

1− 2b/%′22<%′2ρ′2,

where the last inequality is satisfied for sufficiently small n-independent constant b > 0. Thescaling map is well-defined from Dn+1(ρ′) to Dn+1(%′) and the resulting Hamiltonian belongsto An+1(ρ′). �

We can now show that the nth-step renormalization operator is well-defined on an openball around H0n.

Theorem 4.4 Given ρ′1 > 0, for sufficiently small σ, κ > 0 and ρ′2 > 0 satisfying 3ρ

′2/2 <

σ < `/3, there exists a constant C ′ > 0, such that the nth-step renormalization operator Rnis a well-defined analytic map from an open ball B+n,ρ′(ζn), of radius ζn = C

′/(αnαn+1)2, into

I+n+1An+1(ρ′). Also, ‖Rn(H0n + hn)−H0n+1‖n+1,ρ′ ≤ ζ−1n ‖hn‖n,ρ′ and

‖Rn(H0n + hn)−H0n+1 −DRn(H0n)hn‖n+1,ρ′ ≤ [ζn(ζn − ‖hn‖n,ρ′)]−1‖hn‖2n,ρ′ . (23)

Proof: Let 3ρ′/2 > 3ρ′/(2 + 9ρ′2/(2`)) > ρ > % > %′ > ρ′, componentwise. The bound (18)

implies that there exists a constant b1 > 0, such that (θn/µn)B+n,ρ′(ζn) ◦ Tn ⊂ Bn+1,ρ(b1C ′),

where ζn = C′/(αnαn+1)

2 and C ′ > 0. Proposition 4.2 guarantees that for sufficiently smallC ′ > 0,

{H ′′n+1 : H ′′n+1 = H ′n+1 ◦ VH′n+1 , H′n+1 ∈ Bn+1,ρ(b1C ′)} ⊂ Bn+1,%(b1b2C ′),

with b2 > 0. If C′ > 0 is chosen sufficiently small, there exists (by Theorem 5.6) a canon-

ical transformation UH′′n+1 for Hamiltonians H′′n+1 in a neighborhood of H

0n+1 containing

Bn+1,%(b1b2C′), that satisfies the equation I−n+1 (H ′′n+1 ◦ UH′′n+1) = 0. Furthermore,

{H̃n+1 : H̃n+1 = H ′′n+1 ◦ UH′′n+1 , H′′n+1 ∈ Bn+1,%(b1b2C ′)} ⊂ B+n+1,%′(b1b2b3C

′),


where b3 > 0 is a constant (dependent on σ, κ and %). Finally, Proposition 4.3 guaranteesthat for sufficiently small C ′ > 0,

{SH̃n+1(H̃n+1) : H̃n+1 ∈ B+n+1,%′(b1b2b3C

′)} ⊂ B+n+1,ρ′(b1b2b3b4C′),

with b4 > 0. This shows that for sufficiently small C′ > 0, the nth-step renormalization

operator is well-defined from B+n,ρ′(ζn) to I+

n+1An+1(ρ′). As the composition of analyticmaps, it is an analytic map itself.

Define the map g : z 7→ Rn(H0n + zhn) − H0n+1, where hn ∈ I+

nAn(ρ′) is given suchthat Hn = H

0n + hn ∈ B+n,ρ′(ζn). This map is analytic from an open ball in C, of radius

ζn/‖hn‖n,ρ′ > 1, into I+

n+1An+1(ρ′). As g(0) = 0, we have, by the maximum principle, thebound

‖g(1)‖n+1,ρ′ ≤ sup‖h̄n‖n,ρ′=ζn

‖Rn(H0n + h̄n)−H0n+1‖n+1,ρ′‖hn‖n,ρ′

ζn.

From the construction of the renormalization operator, sup‖h̄n‖n,ρ′=ζn ‖Rn(H0n + h̄n) −

H0n+1‖n+1,ρ′ can be bounded by a constant b1b2b3b4C ′, less than or equal to 1, if C ′ > 0is sufficiently small. Then, we have ‖Rn(H0n + hn) −H0n+1‖n+1,ρ′ ≤ ‖hn‖n,ρ′/ζn. Cauchy’sformula gives an estimate on the norm of the second-order remainder of the Taylor expansionof Rn about H0n,

F 2n = ‖g(1)− g(0)− g′(0)‖n+1,ρ′ ≤1

2π

∮|z|=ζn/‖hn‖n,ρ′

‖g(z)‖n+1,ρ′|z2(z − 1)|

dz

≤‖hn‖2n,ρ′

ζn(ζn − ‖hn‖n,ρ′),

(24)

for sufficiently small C ′ > 0. This provides the second desired bound. �

Remark 4.5 The renormalization operator Rn is actually analyticity improving and can bedefined from an open ball in I+nAn(ρ′′) into I

+

nAn(ρ′), with ρ′ > ρ′′, componentwise. The lossof analyticity in the transformations close to the identity can be reduced by restricting thedomain of the renormalization operator to a smaller ball.

Remark 4.6 The nth-step renormalization operator is well-defined on a space of resonantHamiltonians. For a given n ∈ N, this is not a restriction as, by construction, the renormal-ized Hamiltonians are always resonant. In order to apply the 0th-step renormalization operatorto Hamiltonians containing non-resonant modes also, one can include a pre-renormalizationstep consisting of a canonical transformation that eliminates the non-resonant modes of aHamiltonian.

5. ELIMINATION OF NON-RESONANT MODES

In this section, we construct a canonical transformation that eliminates non-resonant modesof a near-integrable Hamiltonian. The whole construction is associated to a single renormal-ization step. The index of the renormalization step will be suppressed in this section, in orderto simplify the notation. For the construction of the canonical transformation, we follow anapproach as in reference [21]. Our Hamiltonians are, however, assumed to be close to an in-tegrable Hamiltonian that contains a term quadratic in momenta. Technically, in the presentscheme, one also needs to assure that the elimination of non-resonant modes is possible ateach renormalization step. In that context, we emphasize that the constants that appear inthis section will be chosen independently of the renormalization step.

We begin by making a canonical change of coordinates (q, p) → (x, y) with x1 = ω′ · q,x2 = Ω

′· q, y1 = ω̂ · p and y2 = Ω · p. We will simplify the notation further, by writing H(x, y)instead of H(q(x, y), p(x, y)). Moreover, some of the symbols in this section will have differentmeaning than in other sections. The use of those symbols should be restricted to this section.

10 Saša Kocić

In the new coordinates, the Fourier-Taylor series and the norm of a function H ∈ A(ρ),analytic on

D(ρ) = {x ∈ C2 : |Imx1| < ρ1, |Imx2| < ρ1} × {y ∈ C2 : |y1| < ρ2, |y2| < ρ2} , (25)

where ρ = (ρ1, ρ2) > 0, componentwise, are given by

H(x, y) =∑

(v,k)∈I

Hv,kyk11 y

k22 e

iv·x , ‖H‖ρ =∑

(v,k)∈I

|Hv,k| ρk1+k22 eρ1(|v1|+|v2|) . (26)

Here I = M2 × N20, where M2 = {(ω̂ · ν,Ω · ν) ∈ R2 : ν ∈ Z2} ⊂ R2 is a set bijective to Z2that can be determined from the vectors ω and Ω.

The non-resonant index set is defined as

I−

= {(v, k) ∈ I : |v1| >σ

‖ω‖|v2| , |v1| >

κ

‖ω‖‖k‖} . (27)

The resonant index set is its complement I+

= I\I− .We state without proof the following technical Proposition. In what follows, the norm of

the functions X = (X1, X2) ∈ A2(ρ) is defined as ‖X‖ρ = max{‖X1‖ρ, ‖X2‖ρ}. We denoteby ∂iH, for i = 1, 2, the partial derivatives of H(x, y) with respect to x1 and x2, and fori = 3, 4, the partial derivatives of the same function with respect to y1 and y2, respectively.

Proposition 5.1 Let ρ = (ρ1, ρ2) and δ = (δ1, δ2) be given pairs of positive numbers andlet 0 < δ < ρ, componentwise. If f, g, h ∈ A(ρ), and X,Y ∈ A2(ρ) satisfy ‖X‖ρ ≤ δ1 and‖Y ‖ρ ≤ δ2, and U : (x, y) 7→ (x+X, y + Y ) is a given change of variables, then

(i) |f(x, y)| ≤ ‖f‖ρ , ∀(x, y) ∈ D(ρ) ,

(ii) fg ∈ A(ρ) and ‖fg‖ρ ≤ ‖f‖ρ‖g‖ρ ,

(iii) ‖∂ih‖ρ−(δ1,0) ≤ δ−11 ‖h‖ρ for i = 1, 2 ,

(iv) ‖∂jh‖ρ−(0,δ2) ≤ δ−12 ‖h‖ρ for j = 3, 4 ,

(v) ‖h ◦ U‖ρ ≤ ‖h‖ρ+δ .

Given a Hamiltonian H, close to H0, our goal is to construct a canonical transformation

UH that satisfies the equation I−H ◦ UH = 0. Such a transformation is close to the identity

as the Hamiltonian H is close to integrable. We would like to perform first a canonicaltransformation U : (x, y) 7→ (x′, y′), generated by a function φ as

x′ = x+∇y′φ(x, y′), y′ = y −∇xφ(x, y′) , (28)

where ∇x = (∂1, ∂2) and ∇y = (∂3, ∂4), which satisfies the linearized version of the aboveequation, i.e. I−(H + {H,φ}) = 0. Here, {H,φ} denotes the Poisson bracket of the functionsH and φ, defined by {H,φ} = ∇xH · ∇yφ−∇xφ · ∇yH.

We introduce ψ = ∂1φ and define the operators Di = ∂i∂−11 , for i = 1, 2, 3, 4, on I−A(ρ),

ρ > 0, componentwise.

Proposition 5.2 D2, D3 and D4 are bounded linear operators on I−A(ρ), with operator

norms satisfying

‖D2‖ ≤‖ω‖σ, ‖D3‖ ≤

‖ω‖κρ2

, ‖D4‖ ≤‖ω‖κρ2

. (29)

Proof: Consider first the action of the operators D2, D3 and D4 on Hv,k(x, y) = yk11 yk22 e

iv·x

with (v, k) belonging to I−

. We find that ‖D2Hv,k‖ρ ≤ ‖ω‖σ ‖Hv,k‖ρ, ‖(D3 + D4)Hv,k‖ρ ≤‖ω‖κρ2‖Hv,k‖ρ, ‖D3Hv,k‖ρ ≤ ‖ω‖κρ2 ‖Hv,k‖ρ, and ‖D4Hv,k‖ρ ≤

‖ω‖κρ2‖Hv,k‖ρ. These bounds extend

by linearity to the whole I−A(ρ). �


Let H = H0 + h, where H0 = ‖ω‖y1 + y22/2 and h ∈ A(%), with % > η > 0, compo-nentwise. On I−A(%− η), define the operator L(h) by the following action on an arbitraryψ ∈ I−A(%− η),

L(h)ψ =1

‖ω‖(−y2D2ψ + ∂1hD3ψ + ∂2hD4ψ − ∂3hD1ψ − ∂4hD2ψ) . (30)

Using the bounds obtained in Proposition 5.1 and Proposition 5.2, we find that

‖L(h)ψ‖%−η ≤[%2 − η2σ

+

(2

κη1(%2 − η2)+

1

ση2+

1

‖ω‖η2

)‖h‖%

]‖ψ‖%−η . (31)

As ‖ω‖ ≥ 2`, if %2 − η2 < σ and ‖h‖% is sufficiently small, then ‖L(h)ψ‖%−η ≤ A‖ψ‖%−ηfor every ψ ∈ I−A(%− η) and some positive constant A < 1. Therefore, the operator norm‖L(h)‖ ≤ A < 1.

This enables us to solve the equation I−(H + {H,φ}) = 0 for the generating function ofthe canonical transformation U .

Proposition 5.3 Let H belong to A(%), where % > η > 0, componentwise, and %2−η2 < σ. If‖h‖% is sufficiently small such that, by the inequality (31), L(h) is an operator on I

−A(%− η)bounded in norm by a positive constant A < 1, then the equation

I−

(H + {H,φ}) = 0 , I+

φ = 0 , (32)

has a unique solution φ, such that ψ = ∂1φ belongs to I−A(%− η) , and satisfies

‖ψ‖%−η ≤ (1−A)−1‖I−H‖%−η‖ω‖

, ‖{H,φ}‖%−η ≤ (1−A)−1‖I−H‖%−η . (33)

Proof: Let L±

= I±L(h)I− . Equation (32) can now be written in the form (I − L−)ψ =I−h/‖ω‖. The assumptions guarantee that the operator norms of L± satisfy ‖L±‖ ≤ A < 1.Equation (32) can then be solved by inverting the operator I− L− by means of a Neumannseries. The solution ψ satisfies the first of the bounds (33). As {H,φ} = (L(h) − I)ψ‖ω‖ =L

+

ψ‖ω‖ − I−h, using this bound one obtains the second one. �The next Proposition shows that the function φ generates an explicit canonical transfor-

mation.

Proposition 5.4 Let 0 < δ′′2 = 2σ−1�3 < �r2 = δ

′2 < %2−η2 and σ < 2`. Let also δ′ = (0, δ′2).

Define B to be the closed ball of radius δ′′2 /2 in A(%−η−δ′) centered at zero. If ψ ∈ I−A(%−η)

satisfies ‖ψ‖%−η ≤ �3/‖ω‖, the equation

K(g) = g , K(g) = ∇xφ(x, y − g) , g = g(x, y) = y − y′ , (34)

has a unique solution g ∈ B2 = B ×B and ‖g‖%−η−δ′ ≤ σ−1‖ω‖‖ψ‖%−η .

Proof: By assumption, for every g ∈ B2, ‖g‖%−η−δ′ ≤ δ′′2 /2. Using the bounds obtained inProposition 5.1 and Proposition 5.2, we find that for every g, g′ ∈ B2, there exists g∗ ∈ B2,such that

‖K(g)‖%−η−δ′ ≤ ‖∇xφ‖%−η ≤ max{1, σ−1‖ω‖}‖ψ‖%−η ≤ σ−1‖ω‖‖ψ‖%−η ≤ σ−1�3 (35)

and

‖K(g′)−K(g)‖%−η−δ′ ≤ ‖∇gK(g∗)‖%−η−δ′‖g′ − g‖%−η−δ′≤ ‖∇y′∇xφ(x, y′)‖%−η−δ′‖g′ − g‖%−η−δ′≤ ‖∇y∇xφ‖%−η−δ′/2‖g′ − g‖%−η−δ′≤ 2δ′−12 ‖∇xφ‖%−η‖g′ − g‖%−η−δ′≤ 2δ′−12 max{1, σ−1‖ω‖}‖ψ‖%−η‖g′ − g‖%−η−δ′≤ 2δ′−12 σ−1‖ω‖‖ψ‖%−η‖g′ − g‖%−η−δ′≤ 2δ′−12 σ−1�3‖g′ − g‖%−η−δ′ .

12 Saša Kocić

As, by assumption, 2δ′−12 σ−1�3 < 1, these inequalities show that K is a contraction on B2,

and thus has a unique fixed point g ∈ B2. The inequality (35) provides the desired bound onthe norm of g. �

Proposition 5.5 Let %− η − 2δ′ > 0, η > 0 and δ′ = (δ′1, δ′2) = �r > 0, componentwise, withδ′1 = σδ

′2/[κ(%2 − η2)]. Also let 0 < δ′′2 = 2σ−1�3 < δ′2 = �r2 < %2 − η2 < σ < 2`. Assume

further that H ∈ A(% − η) satisfies ‖H − H0‖%−η < b and that b > 0 is small enough suchthat the linear operator L(H−H0) is bounded on I−A(%− η) by ‖L(H−H0)‖ ≤ A < A′ < 1.If ‖I−H‖%−η ≤ (1 − A′)�3, the canonical transformation U exists and maps D(% − η − 2δ′)into D(%− η). The function H ◦U belongs to A(%− η− 2δ′) and L(H ◦U −H0) is a boundedoperator on I−A(%− η − 2δ′). They satisfy the bounds

‖I−

(H ◦ U)‖%−η−2δ′ ≤ C2(�)�4 ,‖H ◦ U −H‖%−η−2δ′ ≤ (1 + �C2(�))�3 = ∆b(�) ,‖L(H ◦ U −H0)‖ ≤ ‖L(H −H0)‖+ C3(�)(1 + �C1(�))�2 = A+ ∆A(�) ,

(36)

where

Cn(�) =σn�r2 + %2n�r2 +

12 (n�r2)

2 + ‖h‖%−ησnr2(σnr2 − �2)

,

for n = 1, 2, and

C3(�) =2(%2 − η2)

σr2(%2 − η2 − 2r2�)+

1

σr2+

1

`r2.

Proof: Our assumptions guarantee that there exists a canonical transformation U with agenerating function φ that solves the linear equation (32). More specifically, if b > 0 hasbeen chosen sufficiently small, then there exists ψ ∈ A(% − η) satisfying ‖ψ‖%−η < �3/‖ω‖and g ∈ B2 of norm ‖g‖%−η−δ′ < σ−1‖ω‖‖ψ‖%−η that solves the equation (34).

Define the following one parameter (s ∈ C) family

F (s) = −s(‖ω‖ψ(x, y − sg) + y2D2ψ(x, y − sg)) +1

2s2(D2ψ(x, y − sg))2

+ h(x+ s∇y−sgφ(x, y − sg), y − s∇xφ(x, y − sg)) ,

passing through F (0) = H −H0 and F (1) = H ◦ U −H0, with F ′(0) = {H,φ}. Assumingthat |s| ≤ s0 = σ�−3nδ′2, where n ∈ {1, 2}, and using the Proposition 5.1 and Proposition 5.2,we obtain the following bounds,

‖sg‖%−η−nδ′ ≤ s0σ−1‖ω‖‖ψ‖%−η < nδ′2 ,‖s∇xφ(x, y − sg)‖%−η−nδ′ ≤ s0σ−1‖ω‖‖ψ‖%−η < nδ′2 ,‖s∇y−sgφ(x, y − sg)‖%−η−nδ′ ≤ s0[κ(%2 − η2)]−1‖ω‖‖ψ‖%−η < nδ′1 .

(37)

These bounds, together with Proposition 5.1, imply that F (s) belongs to A(% − η − nδ′),whenever |s| ≤ s0. In fact, this is true on an open neighborhood of the disc |s| ≤ s0, as theinequalities (37) are strict due to A < A′. Now, we have

‖H ◦ U −H − {H,φ}‖%−η−nδ′ = ‖F (1)− F (0)− F ′(0)‖%−η−nδ′

=

∥∥∥∥∥ 12πi∮|s|=s0

ds

s2(s− 1)F (s)

∥∥∥∥∥%−η−nδ′

≤ 1s0(s0 − 1)

(s0‖ω‖‖ψ‖%−η + s0%2σ−1‖ω‖‖ψ‖%−η

+1

2s20σ−2‖ω‖2‖ψ‖2%−η + ‖h‖%−η)

≤σnδ′2 + %2nδ

′2 +

12 (nδ

′2)

2 + ‖h‖%−ησ�−3nδ′2(σ�

−3nδ′2 − 1)= Cn(�)�

4 .


As I−(H ◦ U) = I−(H ◦ U − H − {H,φ}), the first of the inequalities (36) immediatelyfollows from this estimate for n = 2. From Proposition 5.3 and the inequality

‖H ◦ U −H‖%−η−2δ′ ≤ ‖H ◦ U −H − {H,φ}‖%−η−2δ′ + ‖{H,φ}‖%−η−2δ′ ,

we find that ‖H ◦ U −H‖%−η−2δ′ ≤ (1− A)−1‖I−H‖%−η + C2(�)�4. This implies the second

inequality in (36). The fact that the map U takes D(%− η − 2δ′) into D(%− η) follows fromthe bounds (37) and Proposition 5.1.

From the definition of the operator L(h), one can find that

‖L(H ◦U−H0)‖ ≤ ‖L(H−H0)‖+(

2

κδ′1(%2 − η2 − 2δ′2)+

1

σδ′2+

1

‖ω‖δ′2

)‖H ◦U−H‖%−η−δ′ .

Using the inequality ‖H ◦U −H‖%−η−δ′ ≤ (1 + �C1(�))�3, which can be obtained analogouslyto the second bound in (36), one can obtain the last desired inequality. �

Theorem 5.6 Let % > %′ > 0, componentwise, %2 < σ < 2` and 0 < A′ < 1. Let B be an

open set of Hamiltonians H ∈ A(%), for which ‖H − H0‖% < b and ‖I−H‖% < (1 − A′)�3.

If b > 0 and � > 0 are sufficiently small, then for every Hamiltonian H ∈ B there exists ananalytic canonical transformation UH : D(%′)→ D(%) that solves the equation I

−H ◦UH = 0.

The map H 7→ H ◦ UH is analytic from B to I+A(%′), and

‖H ◦ UH −H‖%′ ≤ �3 +O(�4). (38)

Proof: Let % > % − η > %′ > 0 and r > 0, componentwise, with r1 = σr2/[κ(%2 − η2)]. Forsufficiently small b > 0, the norm of the operator L(H − H0) : I−A(% − η) → A(% − η) isbounded by a positive constant A < A′ < 1.

By Proposition 5.5, there exists a canonical transformation U : D(%− η− 2�r)→ D(%− η)such that ‖I−(H ◦ U)‖%−η−2�r ≤ C2(�)�4. We would like to iterate the map H 7→ H ◦ U ,indefinitely. Introducing f(�) = [�C2(�)/(1 − A′)]1/3�, we obtain ‖I

−(H ◦ U)‖%−η−2�r ≤

(1−A′)f(�)3. Let �i = f i(�), for i ∈ N0. For sufficiently small �, the sum∑∞i=0 2�ir converges

to a limit δ = O(�). The sums∑∞i=0 ∆b(�i) and

∑∞i=0 ∆A(�i) converge to ∆b = O(�3) and

∆A = O(�2), respectively. Thus, for sufficiently small � the map (H, �, %−η) 7→ (H◦U, f(�), %−η−2�r) can be iterated indefinitely and the iterations converge to a limit (H ◦UH , 0, %−η−δ).For sufficiently small �, %− η − δ > %′, componentwise.

As �i are summable, the sequence of canonical transformations U generates a uniformlyconvergent sequence on D(%− η − δ). The analyticity of the map H 7→ H ◦ UH follows fromuniform convergence of our iteration scheme. The desired bound can be obtained from thesecond inequality in (36) and its iterations. �

Let Ĥn be the Hamiltonian vector field generated by Hn, i.e. Ĥn = (J∇Hn) · ∇, whereJ(q, p) = (p,−q) and ∇ = (∇q,∇p).

The derivative of the map NHn : Hn 7→ Hn ◦ UHn at a resonant Hamiltonian H+

n is given

by DNn(H+

n ) = I+

n − I+

nĤ+

n (I−

n Ĥ+

n I−

n )−1

I−n .Let K0n = ωn · p and let EHn = K0n + fn, where fn = 12 (Ωn · p)

2 + Ehn. The derivative ofthe map NHn at a q-independent Hamiltonian EHn is

DNn(EHn) = I+

n − I+

n f̂nK̂0n

−1I−

n

(I + I

−

n f̂nK̂0n

−1I−

n

)−1I−

n . (39)

The norm of the operator f̂nK̂0n−1

I−n : An(%)→ An(%′) satisfies the bound

‖f̂nK̂0n−1

I−

n‖ ≤%′2σ

+1

‖ωn‖∑k∈N20

|(hn)0,k|(k1 + k2

‖ωn‖σ

)%′‖k‖−12 , (40)

and thus

‖f̂nK̂0n−1

I−

n‖ ≤%2σ

+

(1

‖ωn‖+

1

σ

)‖Ehn‖n,%

%2(1− %′2/%2)2. (41)

14 Saša Kocić

As %2 < σ, if ‖Ehn‖n,% is sufficiently small, then ‖f̂nK̂0n−1

I−n‖ < 1 and thus the operator(I + I−n f̂nK̂0n

−1I−n ) in (39) can be inverted by means of a Neumann series. If %2 < σ/2 and

‖Ehn‖n,% is sufficiently small, then the operator norm ‖DNn(EHn)‖ can be bounded by 1.

6. DIOPHANTINE FREQUENCY VECTORS AND SOME BOUNDS

Recall that the nth-step renormalization operator is associated to the pair of vectors(ωn,Ωn) generated from a pair (ω,Ω). We assume that ω ∈ R2 is of the form ω = `(1, α)∗,where α > 1 and ` ∈ R+. For the vectors ω = `(1, α)∗ with α < 1, a canonical transformationof the phase space can be performed, generated by a matrix from GL(2,Z), such that the“new” vector ω is of the desired form. Thus, without loss of generality we can assume thatα > 1 and, consequently, for any n ∈ N0, ωn = `(1, αn)∗ with αn > 1.

The vectors (1, αn)∗ = (−1)n(pn−1 − αqn−1)−1P ∗

−1

n−1(1, α)∗ can be obtained using the con-

vergent matrices (12). Therefore, we have

α =αnpn−1 + pn−2αnqn−1 + qn−2

. (42)

The convergents pn/qn satisfy (Theorem 171 in [15])∣∣∣∣α− pnqn∣∣∣∣ < 1qnqn+1 . (43)

They are the best rational approximates of α, in the sense that for any p ∈ Z and q ∈ N, suchthat 0 < q ≤ qn and p/q 6= pn/qn, n ∈ N, one has |pn−αqn| < |p−αq| (Theorem 182 in [15]).

From the equality (42), for n ≥ 0, we find

xn = −pn − αqn

pn−1 − αqn−1. (44)

Denoting the product of the first (n+1) numbers xi by βn =∏ni=0 xi =

∏ni=0 α

−1i+1, we obtain

that βn = (−1)n+1(pn−αqn) . Notice that det(Pn) = (−1)n+1. Using the equality (42) again,one easily finds that β−1n = qn+1 + qnxn+1 , and thus, qn+1 < β

−1n < 2qn+1 .

Define Ãn =∏ni=0 αi. As βn = α0Ã

−1n+1, the previous bound implies

α0qn < Ãn < 2α0qn , (45)

assuming that α0 > 0.Emphasizing again that the values of qn and pn are associated to α by writing explicitly

qn(α) and pn(α), notice that qn+1(α0) = pn(α1). The double inequality (45) for (n + 1)instead of n, can be written as

pn(α1) <Ãn+1α0

< 2pn(α1) , (46)

implying the following bounds on pn,

pn < Ãn < 2pn . (47)

It is easy to show that Ãn grows at least exponentially with n. If 1 < αi ≤ γ, for somei ∈ {0, . . . , n− 1}, then αiαi+1 = αi/(αi− ai) ≥ γ/(γ− 1) = γ2, where γ = (1 +

√5)/2 is the

golden mean, the limit of the sequence of ratios Fk+1/Fk of successive Fibonacci numbers Fk,

defined by Fk+2 = Fk+1+Fk, for k ∈ N, and F1 = F2 = 1. This implies that Ãn/Ãj−1 ≥ γn−j ,for 0 < j ≤ n. If α0 > 1, then Ãn ≥ γn and the previous inequality is also valid for j = 0,with Ã−1 = 1.

This growth can be controlled if α is a Diophantine number.


Definition 6.1 An irrational number α will be called Diophantine of order β ≥ 0 if thereexists a constant C > 0, such that for all p ∈ Z and q ∈ N,∣∣∣∣α− pq

∣∣∣∣ > Cq2+β . (48)The set of all Diophantine numbers of order β will be denoted by D(β). Sometimes, lessprecisely, we will call Diophantine a vector ω ∈ R2 with a Diophantine winding ratio. TheDiophantine condition on ω states that there exists C > 0, such that for all ν ∈ Z2\{0},|ω · ν| > C‖ν‖−(1+β).

The Diophantine condition on α, together with the inequality (43), imposes an upperbound on the growth rate of the denominators of its convergents. For α ∈ D(β), thereexists a constant K > 0 such that qn+1 < Kq

1+βn , for all n ∈ N0. Equivalently, using the

inequalities (45), the Diophantine condition can be written as

Ãn+1 < K̃Ã1+βn , (49)

where K̃ > 0 is a constant.In particular, constant-type numbers, which have a bounded sequence of partial quotients,

are Diophantine of order zero. Among them are quadratic irrationals, i.e. the roots ofquadratic equations with integer coefficients, whose continued fraction expansions are even-tually periodic. Constant-type numbers have zero Lebesgue measure in the real numbers.Diophantine numbers of any order β > 0 are of measure one.

Though the construction of a one-step renormalization transformation is more general, thebound (49) plays an essential role in the proof of Theorem 7.2 concerning the existence of atrivial attracting orbit of the sequence of renormalization operators which is associated to aDiophantine vector ω ∈ R2. To prove the Theorem, we will also need the bounds obtained inthe following Proposition.

Proposition 6.2 If Ω0 ∈ R2 is suitable, so is every Ωn = T−1n−1Ωn−1/‖T−1n−1Ωn−1‖, for all

n ∈ N. For a suitable choice of Ω0, we have the following bounds,

1 + α04α0

(Ãn + Ãn−1) <

n∏i=0

‖T−1i Ωi‖ < (Ãn + Ãn−1) . (50)

Proof: The first part of the claim can be proved by induction. Notice that ‖P ∗−1n Ω0‖ =‖T−1n . . . T−10 Ω0‖ =

∏ni=0 ‖T

−1i Ωi‖, as the matrices Ti are symmetric. One easily finds that

for a suitable Ω0 ∈ R2, 1/2(pn+pn−1 +qn+qn−1) ≤ ‖P ∗−1

n Ω0‖ ≤ pn+pn−1. The bounds (50)follow from the inequalities (45) and (47). �

7. CONVERGENCE OF THE RENORMALIZATION SCHEME

Recall that the orbit of an integrable Hamiltonian H00 = ω · p + 1/2(Ω · p)2 under therenormalization consists of Hamiltonians H0n = ωn ·p+1/2(Ωn ·p)2, where n ∈ N0. The mapsωn 7→ ωn+1 and Ωn 7→ Ωn+1 are induced by the Gauss map of the inverse winding ratio of ω.

The vectors ω with the winding ratio α = [a, a, . . . ], where a ∈ N, are the fixed points ofthe first of these maps. A suitable vector Ω is not necessarily a fixed point of the dynamics.However, in this case, it is possible to make a particular choice of Ω, such that it is a fixedpoint of the map. The corresponding integrable Hamiltonian H00 is then a fixed point of therenormalization.

Similarly, if the winding ratio of a frequency vector ω has a periodic continued fractionexpansion, one can make a particular choice of a suitable vector Ω, such that the HamiltonianH00 generates a periodic orbit of the renormalization. More generally, if the winding ratio of ωis a quadratic irrational and a particular choice of a suitable vector Ω is made, the dynamicsof H00 eventually settles on a periodic orbit.

16 Saša Kocić

In the following, we show that if ω0 is a Diophantine vector and Ω0 is an arbitrary suitablevector, then the orbit of a resonant Hamiltonian H0, sufficiently close to H

00 , approaches

the orbit of H00 exponentially fast, under the action of the sequence of the renormalizationoperators Rn, n ∈ N0.

We first present a description of the eigenspaces of the derivative of the nth-step renormal-

ization operator at H0n, which is given by its action on an arbitrary function fn ∈ I+

nAn(ρ′),where ρ′ > 0, componentwise, as

DRn(H0n)fn =θnµn

(I− P(0,0)n+1 − P(1,0)n+1 − P

(0,1)n+1 − P

(0,2)n+1 )DNn+1(H0n+1)fn ◦ Tn. (51)

The space of q-independent Hamiltonians is an invariant subspace of the derivative operator.A constant Hamiltonian, and the Hamiltonians (ω̂n ·p), (Ωn ·p) and (ω̂n ·p)2, are eigenvectorswith eigenvalue zero. For k ∈ N20 different from (0, 0), (1, 0), (0, 1) and (0, 2), the derivativeoperator maps the Hamiltonian (ω̂n · p)k1(Ωn · p)k2 into

DRn(H0n)(ω̂n · p)k1(Ωn · p)k2 =‖ωn+1‖k1

(αn+1‖T−1n Ωn‖)2(k1−1)+k2‖ωn‖k1(ω̂n+1 · p)k1(Ωn+1 · p)k2 .

Thus, the operator norm of DRn(H0n) acting on q-independent Hamiltonians can be boundedby

‖DRn(H0n)E‖ ≤max{1, ‖ωn+1‖‖ωn‖ }αn+1‖T−1n Ωn‖

≤ 2/3 . (52)

The derivative of the nth-step renormalization operator Rn at a q-independent HamiltonianEHn is the linear operator Ln = DRn(EHn) : I

+

nAn(ρ′)→ I+

n+1An+1(ρ′), given by its actionon an arbitrary fn ∈ I

+

nAn(ρ′),

Lnfn = θn/µnDSn+1(EH̃n+1)DNn+1(EH ′′n+1)DVn+1(EH ′n+1)fn ◦ Tn . (53)

The operators Sn+1 and Vn+1 have been introduced in Proposition 4.3 and Proposition 4.2,respectively. The next Proposition shows that there is a super-exponential shrinking of theq-dependent modes (which may also depend on the p-variables). In the case of vector fieldson a torus, a similar result has been obtained in [28].

Proposition 7.1 Let ω0 ∈ D(β), for some β ≥ 0. There exist c1, c2 > 0, such that

‖Ln ◦ · · · ◦ Lj(I− E)‖ ≤ cn−j+11Ã2n+1Ã

2n

Ã2j Ã2j−1

e−c2Λj,n , (54)

for n ≥ 0 and j = 0, . . . , n, assuming that ‖hi‖i,ρ′ ≤ ζi/2, for i = j, . . . , n, where, as before,ζi = C

′/(αiαi+1)2, C ′ > 0. Here,

Λ2+βj,n =Ãn+1

max{σ, κ}Ã1+βj−1. (55)

Proof: The following properties of the linear operator Ln will be useful to prove this Propo-sition. First, Ln = I

+

n+1 Ln. Second, when acting on a Fourier mode, this operator changesits index ν into T ∗nν, as the derivatives of the elimination, translation and scaling maps donot change the value of ν. We are interested in the action of the operator Ln ◦ · · · ◦ Lj(I−E)on the modes indexed by ν 6= 0. A mode with a particular value of k ∈ N20 can in generalproduce modes with different k′-values in N20. We will denote by ‖k′‖min the minimum of thenorms of the vectors k′ generated from a mode index k.

For every n ∈ N0, let

I1+n = {(ν, k) ∈ I+n : |ωn · ν| ≤ σ|Ωn · ν|} ,I2+n = {(ν, k) ∈ I+n : |ωn · ν| ≤ κ‖k‖} ,


be the two subsets of I+

n . We also define the following subset of I1+j , for j = 0, . . . , n,

V +j,n = {(ν, k) ∈ I1+j : (T

∗n . . . T

∗j ν, k

′) ∈ I1+n+1} .

For every (ν, k) ∈ V +j,n, ν must satisfy the resonant condition

|ωn+1 · T ∗n . . . T ∗j ν| ≤ σ|Ωn+1 · T ∗n . . . T ∗j ν| . (56)

Notice that ωn+1 · T ∗n . . . T ∗j ν = Tj . . . Tnωn+1 · ν. As T−1n ωn = α−1n+1ωn+1, we obtain

Tj . . . Tnωn+1 = ωj

n+1∏i=j+1

αi . (57)

Similarly, Ωn+1 · T ∗n . . . T ∗j ν = Tj . . . TnΩn+1 · ν. From T−1n Ωn = Ωn+1‖T−1n Ωn‖, we find

Tj . . . TnΩn+1 = Ωj

n∏i=j

1

‖T−1i Ωi‖.

The condition (56) can then be written in the form 1σ

n+1∏i=j+1

αi‖T−1i−1Ωi−1‖+1

‖ωj‖

|ωj · ν| ≤ |ω̂j · ν|+ |Ωj · ν| . (58)A lower bound on |ωj · ν| can be obtained by using the Diophantine property of ω0, i.e.

that there exists C0 > 0, such that |ω0 · ν| ≥ C0‖ν‖−(1+β), for any ν ∈ Z2. This propertyimplies that

|ωj · ν| = |ω0 · T ∗−1

0 . . . T∗−1j−1ν|

j∏i=1

αi ≥C0∏ji=1 αi

‖T ∗−10 . . . T ∗−1

j−1ν‖1+β,

for j ≥ 1. As ‖T ∗−10 . . . T ∗−1

j−1ν‖ = ‖P−1j−1ν‖ ≤ (pj−1 + qj−1)‖ν‖ ≤ (1 + 1/α0)Ãj−1‖ν‖ and

‖ν‖ ≤ 2(|ω̂j · ν|+ |Ωj · ν|), there exists c > 1, such that

|ωj · ν| ≥C0Ãj

α0(2cÃj−1)1+β(|ω̂j · ν|+ |Ωj · ν|)1+β, (59)

for j ≥ 0. The bounds (58) and (59) imply that

|ω̂j · ν|+ |Ωj · ν| ≥

(1σ

∏n+1i=j+1 αi‖T

−1i−1Ωi−1‖+ 1‖ωj‖

)C0Ãj

α0(2cÃj−1)1+β(|ω̂j · ν|+ |Ωj · ν|)1+β,

and, thus,

(|ω̂j · ν|+ |Ωj · ν|)2+β ≥C0

α0(2c)1+βσ

Ãn+1∏n+1i=j+1 ‖T

−1i−1Ωi−1‖

Ã1+βj−1.

Using the bounds obtained in Proposition 6.2, we find that there exists c′ > 0, such that

(|ω̂j · ν|+ |Ωj · ν|) ≥ c′Λ′j,n , (60)

where

Λ′2+βj,n =Ãn+1Ãn

σÃ1+βj−1 Ãj−1. (61)

18 Saša Kocić

Now consider the modes indexed by (ν, k) ∈ Z2 × N20 that belong to I+j \V+j,n. Let

(T ∗n . . . T∗j ν, k

′) ∈ I2+n+1 be the index of a mode generated by the action of the operatorLn ◦ · · · ◦ Lj(I− E) on such a mode. The following condition must be satisfied,

|ωn+1 · T ∗n . . . T ∗j ν| ≤ κ‖k′‖min ≤ κ‖k′‖ . (62)

From the inequality (62), using the identity (57) and the Diophantine bound (59), we findthat

‖k′‖ ≥ ‖k′‖min ≥C0

κα0(2c)1+βÃn+1

Ã1+βj−1 (|ω̂j · ν|+ |Ωj · ν|)1+β.

Define W+j,n = {(ν, k) ∈ I+j : (T

∗n . . . T

∗j ν, k

′) ∈ I2+n+1, |ω̂j · ν| + |Ωj · ν| ≥ ‖k′‖min}. If (ν, k) ∈(I+j \V

+j,n) ∩W

+j,n, then

(|ω̂j · ν|+ |Ωj · ν|) ≥ c′′Λ′′j,n , (63)

where

Λ′′2+βj,n =Ãn+1

κÃ1+βj−1, (64)

and c′′ > 0. If (T ∗n . . . T∗j ν, k

′) ∈ I2+n+1 and (ν, k) ∈ (I+j \V

+j,n) ∩ (I

+j \W

+j,n), then

‖k′‖min ≥ c′′Λ′′j,n . (65)

Let V+j,n : Aj(ρ′) → Aj(ρ′) be the projection operator on Aj(ρ′) over the indices in V+j,n,

defined by the action

V+j,nfj =∑

(ν,k)∈V +j,n

(fj)ν,k(ω̂ · p)k1(Ω · p)k2eiq·ν ,

on an arbitrary fj ∈ Aj(ρ′). Let V̄+j,n : Aj(ρ′)→ Aj(ρ′′) be the same projection followed byan analytic inclusion in the q variables, obtained by restricting the domain of fj ∈ Aj(ρ′) toDj(ρ′′), where ρ′1 > ρ′′1 > 0 and ρ′2 = ρ′′2 . As

‖V̄+j,nfj‖j,ρ′′ =∑

(ν,k)∈V +j,n

|(fj)ν,k|ρ′‖k‖2 e

ρ′1(|ω̂j ·ν|+|Ωj ·ν|)e−(ρ′1−ρ

′′1 )(|ω̂j ·ν|+|Ωj ·ν|) ,

we have ‖V̄+j,nfj‖j,ρ′′ ≤ ‖V̄+j,n‖‖fj‖j,ρ′ , with ‖V̄

+j,n‖ ≤ e

−(ρ′1−ρ′′1 )c′Λ′j,n , as follows from the

bound (60).Similarly, let W+j,n : Aj(ρ′)→ Aj(ρ′), be the projection operator on Aj(ρ′) over the indexes

in W+j,n and W̄+j,n : Aj(ρ′)→ Aj(ρ′′) the same projection followed by an analytic inclusion in

the q variables. Here, as before, ρ′1 > ρ′′1 > 0 and ρ

′2 = ρ

′′2 . As

‖W̄+j,nfj‖j,ρ′′ =∑

(ν,k)∈W+j,n

|(fj)ν,k|ρ′‖k‖2 e

ρ′1(|ω̂j ·ν|+|Ωj ·ν|)e−(ρ′1−ρ

′′1 )(|ω̂j ·ν|+|Ωj ·ν|) ,

from the bound (63), we find that the operator norm of W̄+j,n satisfies ‖W̄+j,n‖ ≤

e−(ρ′1−ρ

′′1 )c′′Λ′′j,n .

Let Īn : An(%′′)→ An(ρ′), be the inclusion map obtained by restricting the domain of thefunctions fn ∈ An(%′′) to Dn(ρ′), where %′′2 > ρ′2 > 0 and %′′1 = ρ′1. As

‖Īnfn‖n,ρ′ =∑

(ν,k′)∈In

|(fn)ν,k′ |%′′‖k′‖2 e

%′′1 (|ω̂n·ν|+|Ωn·ν|)e−(ln %′′2−ln ρ

′2)‖k

′‖ ≤ ‖Īn‖‖fn‖n,%′′ ,

the norm of the inclusion map Īn acting on functions that are composed only of modes with‖k′‖ ≥ ‖k′‖min satisfying the inequality (65), can be bounded by ‖Īn‖ ≤ e−(ln %

′′2−ln ρ

′2)c′′Λ′′j,n .


To obtain the desired bound (54), we write the operator Ln ◦ · · · ◦ Lj(I− E) as

Ln ◦ · · · ◦ Lj(I− E) =Ln ◦ · · · ◦ Lj+1(I− E)L(1)

j V̄+j,n

+ Ln ◦ · · · ◦ Lj+1(I− E)L(1)

j W̄+j,n(I− V

+j,n)

+ Īn+1L(2)

n ◦ Ln−1 ◦ · · · ◦ Lj(I− E)(I−W+j,n)(I− V+j,n) .

(66)

Here, L(1)n : I+

nAn(ρ′′)→ I+

n+1An+1(ρ′) and L(2)

n : I+

nAn(ρ′)→ I+

n+1An+1(%′′), n ∈ N0, are thederivatives of the nth-step renormalization operator at EHn. With superscripts (1) and (2),we explicitly emphasize that these operators actually improve analyticity in q and p variables,respectively. For some ρ′′, %′′ > 0 satisfying 0 < ρ′′1 < ρ

′1, ρ

′′2 = ρ

′2, %

′′1 = ρ

′1 and %

′′2 > ρ

′2, the

construction of the analyticity improving operators is possible, as mentioned in Remark 4.5.The norm ‖DRi(H0i )‖, i ∈ N0, can be bounded by a constant times θi/µi. By Cauchy’s

estimate, we have

‖DRi(Hi)−DRi(H0i )‖ ≤2‖hi‖i,ρ′

(ζi − ‖hi‖i,ρ′)2, (67)

where DRi(Hi) is the derivative of the ith-step renormalization operator Ri at a HamiltonianHi = H

0i + hi. In particular, the inequality (67) is satisfied by Li, the derivative of the ith-

step renormalization operator at EHi. As, by assumption, ‖hi‖i,ρ′ ≤ ζi/2, we have the bound‖Li‖ ≤ c3θi/µi, with c3 > 0.

Using the bounds on the operator norms ‖V̄+j,n‖, ‖W̄+j,n‖ and ‖Īn‖, obtained above, we find

that

‖Ln◦· · ·◦Lj(I−E)‖ ≤ c3e−c2Λj,n(

2θjµj‖Ln ◦ · · · ◦ Lj+1(I− E)‖+

θnµn‖Ln−1 ◦ · · · ◦ Lj(I− E)‖

),

where Λj,n ≤ min{Λ′j,n,Λ′′j,n}, with Λ′j,n and Λ′′j,n given by the expressions (61) and (64),respectively, and c2 = min{(ρ′1 − ρ′′1)c′, (ρ′1 − ρ′′1)c′′, (ln %′′2 − ln ρ′2)c′′}.

Using the bounds on the norms of Li, for i = j, . . . , n, we obtain

‖Ln ◦ · · · ◦ Lj(I− E)‖ ≤ 3cn−j+13 e−c2Λj,nn∏i=j

θiµi.

The bound (54) follows from this inequality and the bounds in Proposition 6.2. �

Let h0 ∈ I+

0A0(ρ′), ρ′ > 0, componentwise. After the first (n+ 1) renormalization steps theperturbation can be separated into two parts,

hn+1 = Ehn+1 + (I− E)hn+1 . (68)

The q-independent part of the perturbation can be determined by

Ehn+1 = DRn(H0n)Ehn + EO0n(‖hn‖2n,ρ′) , (69)

or after applying this equality recursively,

Ehn+1 = DRn(H0n) . . . DR0(H00 )Eh0

+

n∑j=1

DRn(H0n) . . . DRj(H0j )EO0j−1(‖hj−1‖2j−1,ρ′) + EO0n(‖hn‖2n,ρ′) . (70)

Here O0n(‖hn‖2n,ρ′) denotes the second-order remainder of the Taylor expansion of Rn(Hn)about H0n. The norm of O0n(‖hn‖2n,ρ′) will be denoted by F 2n and can be estimated by thebound (24).

In order to estimate the q-dependent part of the perturbation, we perform the Taylorexpansion of Rn(Hn) about EHn, the q-independent part of the Hamiltonian Hn. The q-dependent part of the perturbation is

(I− E)hn+1 = Ln(I− E)hn + (I− E)On(‖(I− E)hn‖2n,ρ′) , (71)

20 Saša Kocić

where On(‖(I − E)hn‖2n,ρ′) denotes the second-order remainder of the Taylor expansion ofRn(Hn) about EHn. The norm of this remainder is of the order of ‖(I − E)hn‖2n,ρ′ and willbe denoted by G2n.

After successive applications of the previous recursion relation, we obtain

(I− E)hn+1 = Ln . . .L0(I− E)h0 +n∑j=1

Ln . . .Lj(I− E)Oj−1(‖(I− E)hj−1‖2j−1,ρ′)

+ (I− E)On(‖(I− E)hn‖2n,ρ′) . (72)

We will use this identity to estimate the decrease of the norm of the q-dependent part of theperturbation in the following Theorem.

Theorem 7.2 Let ω0 ∈ D(β), 0 ≤ β <√

2 − 1, and let ρ′1 > 0. There exist τ > 2 andρ′2, σ, κ, C > 0, such that if H0 = H

00 + h0 ∈ I

+

0A0(ρ′) and ‖h0‖0,ρ′ ≤ C2 < 1, then ‖(I −E)hn‖n,ρ′ ≤ CÃ−τn < ζn/2, n ≥ 0, where ζn = C ′/(αnαn+1)2, C ′ > 0. Furthermore, ifβ < (

√161− 11)/10, then ‖hn‖n,ρ′ ≤ CÃ−2n−1 < ζn/2.

Proof: We find first, using the Diophantine bound (49), that if β ≤√

2− 1 and τ > 2, thereexists C > 0, such that

ζn2

=C ′

2α2nα2n+1

≥ C′

2K̃4Ã2βn−1Ã2βn

≥ C′

2K̃4+2βÃ2βn−1Ã2β(1+β)n−1

>C

Ã2n−1>

C

Ãτn.

Let τ ′ = ln c1/ ln γ, where c1 > 1 is the constant from Proposition 7.1. As Ãn/Ãj−1 ≥ γn−j ,0 ≤ j ≤ n, the inequality (54) implies

‖Ln ◦ · · · ◦ Lj(I− E)‖ ≤Ã4+τ

′

n+1

Ã4+τ′

j−1e−c2Λj,n . (73)

Using the inequality e−t ≤ (s/t)s, valid for any t > 0 and s > 0, we find that, for any τ ′′ > 0,

Ã4+τ′

n+1

Ã4+τ′

j−1e−c2Λj,n ≤ (max{σ, κ})τ

′′(τ ′′(2 + β)

c2

)τ ′′(2+β) Ãτ ′′(1+β)−(4+τ ′)j−1Ãτ ′′−(4+τ ′)n+1

.

Let τ ′′ > 0 be given, such that τ = τ ′′ − τ ′ − 4 > 2. There exist σ, κ > 0, such that

Ã4+τ′

n+1

Ã4+τ′

j−1e−c2Λj,n ≤ (max{σ, κ})τ

′′(τ ′′(2 + β)

c2

)τ ′′(2+β) Ãτ(1+β)+(4+τ ′)βj−1Ãτn+1

≤Ãτ(1+β)+(4+τ ′)βj−1

6Ãτn+1,

and thus,

‖Ln ◦ · · · ◦ Lj(I− E)‖ ≤Ãτ(1+β)+(4+τ ′)βj−1

6Ãτn+1. (74)

We will prove the Theorem by induction. There exists 0 < C < 1, such that

‖(I− E)h0‖0,ρ′ ≤ ‖h0‖0,ρ′ ≤ C2 < CÃ−τ0 < C < ζ0/2 . (75)

Therefore, for n = 0, the claim is true. Assume that that the claim holds for 0 < j ≤ n. Thus,there exists C > 0, such that ‖(I − E)hj‖j,ρ′ ≤ CÃ−τj < ζj/2 and ‖hj‖j,ρ′ ≤ CÃ

−2j−1 < ζj/2.

We will show that the claim is true for j = n+ 1.Using the identity (72), we obtain

‖(I−E)hn+1‖n+1,ρ′ ≤ ‖Ln . . .L0(I−E)h0‖n+1,ρ′ +n∑j=1

‖Ln . . .Lj(I−E)‖ ·G2j−1 +G2n . (76)


We will estimate the size of the terms on the right hand side of the inequality (76).As ‖h0‖0,ρ′ ≤ C2 < 1, the inequality (74), for j = 0, implies a bound on the first term

‖Ln . . .L0(I− E)h0‖n+1,ρ′ ≤C

6Ãτn+1. (77)

Using Cauchy’s formula, one can obtain an estimate on the norm of the second-order remain-der On(‖(I− E)hn‖2n,ρ′), analogous to the bound (24),

G2n ≤‖(I− E)hn‖2n,ρ′

(ζn − ‖Ehn‖n,ρ′)(ζn − ‖Ehn‖n,ρ′ − ‖(I− E)hn‖n,ρ′).

Applying the inductive hypothesis and the Diophantine condition in the form (49), or

equivalently, αn+1 ≤ K̃Ãβn, we further obtain that, for some C > 0,

G2n ≤4C2

ζ2nÃ2τn

≤4C2α4nα

4n+1

C ′2Ã2τn≤

4C2K̃2τ/(1+β)+8Ã4βn Ã4βn−1

C ′2Ã2τ/(1+β)n+1

<C

6Ãτn+1, (78)

if 0 ≤ β < 1 and τ ≥ 8β(1 + β)/(1 − β). Using the bounds (74) and (78), we can estimatethe sum

n∑j=1

‖Ln . . .Lj(I− E)‖ ·G2j−1 ≤2C2

3Ãτn+1

n∑j=1

Ãτ(1+β)+(4+τ ′)βj−1

ζ2j−1Ã2τj−1

≤ 2C2K̃8

3C ′2Ãτn+1

n∑j=1

1

Ãτ(1−β)−(12+τ ′)βj−1

.

Since for α0 > 1, we have Ãj−1 ≥ γj−1, the previous sum can be bounded by some positiveconstant, if 0 ≤ β < 1 and τ > β(τ ′ + 12)/(1− β). Thus, there exists C > 0, such that

n∑j=1

‖Ln . . .Lj(I− E)‖ ·G2j−1 ≤C

6Ãτn+1. (79)

Finally, if 0 ≤ β < 1 is given, and constants τ > 2 and σ, κ, C > 0 are chosen according tothe various conditions stated above, using the bounds (77), (78) and (79), we obtain

‖(I− E)hn+1‖n+1,ρ′ ≤C

6Ãτn+1+

C

6Ãτn+1+

C

6Ãτn+1=

C

2Ãτn+1<

C

Ãτn+1<ζn+1

2. (80)

Concerning the q-independent part of the perturbation, from the identity (70), we find that

‖Ehn+1‖n+1,ρ′ ≤ ‖DRn(H0n) . . . DR0(H00 )Eh0‖n+1,ρ′

+

n∑j=1

‖DRn(H0n) . . . DRj(H0j )E‖ · F 2j−1 + F 2n . (81)

Using the bound (52) on the derivative of a one-step renormalization operator acting on q-independent Hamiltonians and the first of the inequalities (50), one obtains that there existsC > 0, such that

‖DRn(H0n) . . . DR0(H00 )Eh0‖n,ρ′ ≤4α20

(1 + α0)Ã2n‖h0‖0,ρ′ ≤

4α20(1 + α0)Ã2n

C2 <C

6Ã2n. (82)

From the estimate (24) obtained in Theorem 4.4, we have the following bound on the normof the second-order remainder, F 2n ≤ ζ−1n (ζn − ‖hn‖n,ρ′)−1‖hn‖2n,ρ′ . Using the inductivehypothesis and the Diophantine property of ω0, we find that there exists C > 0, such that

F 2j ≤2C2

ζ2j Ã4j−1≤

2C2K̃4/(1+β)α4jα4j+1

C ′2Ã4/(1+β)j

≤2C2K̃4/(1+β)+8Ã4βj−1Ã

4βj

C ′2Ã4/(1+β)j

<C

6Ã2j, (83)

22 Saša Kocić

for 0 ≤ β < (√

41− 5)/8. Since, from the bound (52) and the inequalities (50), one has

‖DRn(H0n) . . . DRj(H0j )E‖ ≤8α0ÃjÃj−1

(1 + α0)Ã2n,

the sum on the right hand side of the inequality (81) can be estimated by

n∑j=1

‖DRn(H0n) . . . DRj(H0j )E‖ · F 2j−1 ≤16α0C

2K̃8

(1 + α0)C ′2Ã2n

n∑j=1

ÃjÃ1+4βj−1 Ã

4βj−2

Ã4j−2

≤ 16α0C2K̃11+5β

(1 + α0)C ′2Ã2n

n∑j=1

1

Ã4−(2+5β)(1+β)−4βj−2

.

If 2− 11β − 5β2 > 0, the sum on the right side can be bounded by a positive constant, as Ãjgrows at least exponentially with j. Thus, there exists C > 0, such that

n∑j=1

‖DRn(H0n) . . . DRj(H(0)j )E‖ · F

2j−1 ≤

C

6Ã2n. (84)

Using the bounds (82), (83) and (84), the inequality (81) implies

‖Ehn+1‖n+1,ρ′ ≤C

6Ã2n+

C

6Ã2n+

C

6Ã2n=

C

2Ã2n. (85)

Finally, taking into account the estimates (80) and (85), we find that

‖hn+1‖n+1,ρ′ = ‖(I− E)hn+1‖n+1,ρ′ + ‖Ehn+1‖n+1,ρ′ ≤C

2Ãτn+1+

C

2Ã2n≤ CÃ2n

<ζn+1

2.

This completes the proof of the claim. �

Corollary 7.3 Let ω0 ∈ D(β), with 0 ≤ β < (√

161 − 11)/10. Given ρ′1 > 0, there existρ′2, σ, κ, C̃ > 0 and a non-empty open neighborhood B

+0,ρ′ ⊂ I

+

0A0(ρ′) of H00 , such that for allHamiltonians H0 ∈ B+0,ρ′ and n ∈ N0, ‖Hn −H0n‖n,ρ′ ≤ C̃γ−2n.

Proof: The proof of this Corollary follows directly from Theorem 7.2 and the fact that Ãngrows at least exponentially with n ≥ 0, i.e. Ãn ≥ γn, for α0 > 1. �

Remark 7.4 In the context of Remark 4.6, the set of Hamiltonians in I+0A0(ρ′) that satisfiesthe assumptions of Theorem 7.2 and the neighborhood B+0,ρ′ in Corollary 7.3 could be replaced

with an open ball B0,ρ′ ⊂ A0(ρ′).

8. DEFINITION OF INVARIANT TORI AND FORMAL IDENTITIES

In the remaining part of this paper, we apply the result about the convergence of theconstructed renormalization scheme for ω0 ∈ D(β), with 0 ≤ β < (

√161 − 11)/10, to prove

a KAM theorem for near-integrable Hamiltonians. We construct the invariant tori withDiophantine frequency vectors for Hamiltonians approaching the trivial limit set under therenormalization. We start by providing some definitions and some formal identities that willbe used in the construction.

Definition 8.1 Given δ1 > 0, we define

Dn,0(δ1) = {q ∈ C2 : |Imω′n · q| < δ1, |Im Ω′n · q| = 0} × {0} . (86)


Definition 8.2 Given r > 0 and δ1 > 0, let An,0(δ1) be the space of functions f , analytic onDn,0(δ1), 2π-periodic in both q-variables, with the finite norm

‖f‖n,δ1 =∑ν∈Z2

|fν |(1 + |Ωn · ν|)reδ1|ω̂n·ν| , (87)

where fν are the Fourier coefficients of f .

Definition 8.3 We say that Hn has an invariant torus with frequency vector ωn if thereexists a continuous map Γn : Dn,0(δ1)→ Dn(ρ), with ρ > 0, componentwise, and a continuousfunction t : R→ R, such that for all s ∈ R,

Φt(s)n ◦ Γn = Γn ◦Ψsn. (88)

Here Φn is the flow for the Hamiltonian Hn and Ψn is the flow for the Hamiltonian Kn = ωn·p,i.e. Ψsn(q, 0) = (q + ωns, 0).

Notice that an invariant torus of a Hamiltonian is defined as the conjugacy between theHamiltonian flow and a linear flow of an integrable Hamiltonian.

The flow Φn for the Hamiltonian Hn and the flow Φn+1 for the renormalized HamiltonianHn+1 = Rn(Hn) can be related by

Λn ◦ Φtn+1 = Φθ′ntn ◦ Λn. (89)

This identity is valid for t ∈ C on any domain where the compositions are well-defined. Theidentity formally follows from

d

dtf ◦ Φθ

′ntn ◦ Λn |t=0= θ′n{f,Hn} ◦ Λn =

θ′nµ′n{f ◦ Λn, Hn ◦ Λn} = {f ◦ Λn,Rn(Hn)},

where we have used the identity µ′n−1{f ◦ Λn, g ◦ Λn} = {f, g} ◦ Λn, for complex valued

functions f and g defined on a neighborhood of T2×R2. Here {f, g} = ∇qf ·∇pg−∇qg ·∇pfis the Poisson bracket of the functions f and g.

Now, we have the identities

Λn ◦ Γn+1 ◦ T −1n ◦Ψsn = Λn ◦ Γn+1 ◦Ψα−1n+1s

n+1 ◦ T −1n

= Λn ◦ Φtn+1(α

−1n+1s)

n+1 ◦ Γn+1 ◦ T −1n = Φθ′ntn+1(α

−1n+1s)

n ◦ Λn ◦ Γn+1 ◦ T −1n . (90)

Thus, formally, if Γn+1 is an invariant torus with frequency vector ωn+1 of Hn+1, then

Γn = Λn ◦ Γn+1 ◦ T −1n =Mn(Γn+1) (91)

is the invariant torus with frequency vector ωn of Hn. The flow parameters tn and tn+1are related by tn(s) = θ

′ntn+1(α

−1n+1s). One solution of this functional equation is tn(s) =

ξs/∏ni=1 τi, where ξ is a constant.

9. CONSTRUCTION OF INVARIANT TORI

The formal relationship (91) between invariant tori of a Hamiltonian and its renormalizationimage motivates the construction of an invariant torus. In this section we study the propertiesof the mapsMn, defined by (91), and use them to construct invariant tori for near-integrableHamiltonians in a space of functions of low regularity. In the following, assume that ω0 ∈D(β), with 0 ≤ β < (

√161− 11)/10, and that Hn ∈ I

+

nAn(ρ′), n ∈ N0, is the renormalizationorbit of a given Hamiltonian H0 ∈ B+0,ρ′(C2) in the domain of attraction of the integrablelimit set.

Introduce the coordinates x1 = ω′n · q, x2 = Ω′n · q, y1 = ω̂n · p and y2 = Ωn · p, for a given

n ∈ N0. Denote the partial derivative with respect to x1, x2, y1 and y2 as ∂i, for i = 1, 2, 3and 4, respectively. We can define the components of a vector valued function f = (fq, fp)on Dn,0(δ1), δ1 > 0, as f1 = ω′n · fq, f2 = Ω′n · fq, f3 = ω̂n · fp, f4 = Ωn · fp.

24 Saša Kocić

Definition 9.1 Given r > 0 and δ1 > 0, let Bn,0(δ1) be the Banach space of vector-valuedfunctions f = (fq, fp) on Dn,0(δ1), whose components f i, i = 1, 2, 3, 4, belong to An,0(δ1),with the norm

‖f‖n,δ1 = max{cn‖f1‖n,δ1 , ‖f2‖n,δ1 , ‖f3‖n,δ1 , ‖f4‖n,δ1}, (92)

where cn = Ãn∏n−1i=0 ‖T

−1i Ωi‖.

The maps Mn are defined formally by the action on a vector valued function F = I + f ,where I is the identity map and f ∈ Bn+1,0(δ1),

Mn(F ) = Λn ◦ F ◦ T −1n . (93)

The weight factor cn has been introduced in the norm (92) in order to make the map Mna contraction; if the norm were defined without that factor, the scaling map in Λn wouldactually expand the first component of F .

Similarly, for a vector valued function g = (gq, gp) on Dn(ρ) with components g1 = ω′n · gq,g2 = Ω′n · gq, g3 = ω̂n · gp and g4 = Ωn · gp in An(ρ), define the norms

‖g‖n,ρ = max{cn‖g1‖n,ρ, ‖g2‖n,ρ, ‖g3‖n,ρ, ‖g4‖n,ρ}, (94)

and

‖g‖′n,ρ = max{‖∂1g1‖n,ρ + cn4∑i=2

‖∂ig1‖n,ρ, c−1n ‖∂1g2‖n,ρ +4∑i=2

‖∂ig2‖n,ρ,

c−1n ‖∂1g3‖n,ρ +4∑i=2

‖∂ig3‖n,ρ, c−1n ‖∂1g4‖n,ρ +4∑i=2

‖∂ig4‖n,ρ}. (95)

Before proving that in the above norm the maps Mn are contractions, let us state thefollowing Proposition.

Proposition 9.2 Let δ1 > 0, δ<1 = δ1‖ωn+1‖/(αn+1‖ωn‖) < 2δ1/3 and ρ > 0, component-

wise, with ρ1 > δ1. Let also fn, gn ∈ An,0(δ1), hn ∈ An(ρ) and Xn, Yn ∈ A2n,0(δ1), such thatδ1 + ‖ω′n · Xn‖n,δ1 < ρ1, ‖Ω′n · Xn‖n,δ1 < ρ1, ‖ω̂n · Yn‖n,δ1 < ρ2 and ‖Ωn · Yn‖n,δ1 < ρ2. IfU(q, 0) = (q +Xn(q, 0), Yn(q, 0)) is a given change of variables, then

(i) |fn(q, 0)| ≤ ‖fn‖n,δ1 , ∀(q, 0) ∈ Dn,0(δ1) ,

(ii) fngn ∈ An,0(δ1) and ‖fngn‖n,δ1 ≤ ‖fn‖n,δ1‖gn‖n,δ1 ,

(iii) ‖fn+1 ◦ T −1n ‖n,δ1 ≤ ‖T−1n Ωn‖r‖fn+1‖n+1,δ 0 in the affine space I + Bn,0(δ1), centered at the identity.

Lemma 9.3 Let r < 1 and let 0 < δ1 < ρ′1. For sufficiently small b, C > 0 and all Hamilto-

nians Hn, n ∈ N0, of the renormalization orbit of H0 ∈ B+0,ρ′(C2) ⊂ I+

0A0(ρ′), the map Mnis a contraction with the contraction rate a < 1 (independent of H0 and n) from Bn+1(b) intoBn(b).

Proof: Let Fn+1 ∈ Bn+1(b) and let fn+1 = Fn+1 − I. Define the map Nn by

Nn(fn+1) =Mn(Fn+1)− I = Λn ◦ (I + fn+1) ◦ T −1n − I .

The fact that the maps VH′n+1 , UH′′n+1 and SH̃n+1 , included in Λn, are close to the identity

motivates us to write Λn = Tn ◦ (I + gn+1). Therefore,

Nn(fn+1) = Tn ◦ (fn+1 + gn+1 ◦ (I + fn+1)) ◦ T −1n .


As the maps VH′n+1 , UH′′n+1 and SH̃n+1 depend only on the q-dependent part of the Hamil-

tonian, ‖gin+1‖n+1,ρ′ ≤ C1‖(I − E)hn‖n,ρ′ , for i = 1, 2, 3, 4, i.e. ‖gin+1‖n+1,ρ′ ≤ CC1Ã−τn ,where C1 > 0 is an n-independent constant. Here, we assume that C > 0 is sufficiently smallsuch that the estimates of Theorem 7.2 are valid. Using the Diophantine bound (49) and theinequalities (50), we find that, if τ ≥ 2 + β, then

‖gn+1‖n+1,ρ′ ≤ Ãn+1n∏i=0

‖T−1i Ωi‖CC1

Ãτn≤ CC2Ãτ−2−βn

≤ CC2,

where C2 > 0 is a constant. These inequalities show that the growth of the weight factorwith n is slower than the decrease of the norm of the q-dependent part of Hamiltonian.

The derivative of the operatorNn at fn+1 is given by its action on an arbitrary f̃ ∈ Bn+1(b),

DNn(fn+1)f̃ = Tn ◦ (f̃ +Dgn+1 ◦ (I + fn+1)f̃) ◦ T −1n .

Using the Proposition 9.2, we find that for (δ1, 0) < ρ′′ < ρ′, componentwise,

‖DNn(fn+1)f̃‖n,δ1 ≤ ‖T−1n Ωn‖r−1‖f̃‖n+1,δ 0,tm, sm ∈ R and for all m ∈ N0, and if C > 0 is sufficiently small, then

‖(Φtmm ◦ Φ−smm,0 − I)‖m,δ1 ≤ b, (96)

where Φm,0 is the flow for the Hamiltonian H0m restricted to Dm,0(δ1).

Proof: Let s ∈ R be given and let (q, 0) ∈ Dm,0(δ1). The flow Φm for the Hamiltonian Hmsatisfies the equation,

d

dt(Φtm ◦ Φ−sm,0 − I)(q, 0) = (J · ∇Hm ◦ Φtm ◦ Φ

−sm,0)(q, 0), (97)

where J(q, p) = (p,−q) and ∇ = (∇q,∇p). Introducing the function

Υs(q, t) = (Φtm ◦ Φ−sm,0 − I)(q, 0),

2π-periodic in both q-variables, we can integrate the equation (97) to obtain the integralequation

Υs(q, t) = −(sωm, 0) +∫ t

0

dt′{J · ∇Hm ◦ [(q, 0) + Υs(q, t′)]}. (98)

26 Saša Kocić

This equation can be viewed as the fixed point equation of the functional Ξs, defined by theaction

Ξs(Υs(q, t), q) = −(sωm, 0) +∫ t

0

dt′{J · ∇Hm ◦ [(q, 0) + Υs(q, t′)]}, (99)

on a space of functions Υs : Dm,0(δ1)× J → C2 × C2, J = [0, t] ⊂ R, with the norm

|||Υs||| = supt′∈J

max{cm‖Υ1s(·, t′)‖m,δ1 , ‖Υ2s(·, t′)‖m,δ1 , cm‖Υ3s(·, t′)‖m,δ1 , cm‖Υ4s(·, t′)‖m,δ1},

where Υ1s = ω′m ·Υqs,Υ2s = Ω′m ·Υqs, Υ3s = ω̂m ·Υps ,Υ4s = Ωm ·Υps , and Υs = (Υqs,Υps).

We consider an open ball of functions Υs in that space that satisfy |||Υs||| < b, with0 < b < min{ρ′1 − δ1, ρ2}. This justifies the formal use of equations (98) and (99), as for anygiven t′ ∈ J , one has ‖Υs(·, t′)‖m,δ1 ≤ |||Υs|||.

Now, if Υ(1)s and Υ

(2)s are two functions from that ball, we have

Ξs(Υ(2)s (q, t), q)− Ξs(Υ(1)s (q, t), q) =

∫ t0

dt′∫ Υ(2)s (q,t′)

Υ(1)s (q,t′)

dΥ · ∇(J · ∇Hm) ◦ [(q, 0) + Υ].

Thus,∣∣∣∣∣∣∣∣∣Ξs(Υ(2)s , ·)− Ξs(Υ(1)s , ·)∣∣∣∣∣∣∣∣∣ ≤ |t| · ‖Υ(2)s −Υ(1)s ‖ · ‖∇(J · ∇Hm) ◦ [(q, 0) + Υ]‖, (100)where

‖∇(J · ∇Hm) ◦ [Υ + (q, 0)]‖ ≤ max{cm‖∂2∂3Hm‖m,ρ′′ +4∑

i=1,i6=2

‖∂i∂3Hm‖m,ρ′′ ,

‖∂2∂4Hm‖m,ρ′′ + c−1m4∑

i=1,i6=2

‖∂i∂4Hm‖m,ρ′′ , cm‖∂2∂1Hm‖m,ρ′′ +4∑

i=1,i6=2

‖∂i∂1Hm‖m,ρ′′ ,

cm‖∂2∂2Hm‖m,ρ′′ +4∑

i=1,i6=2

‖∂i∂2Hm‖m,ρ′′},

with b+ δ1 < ρ′′1 < ρ

′1 and b < ρ

′′2 < ρ

′2.

Using Cauchy estimates on the norms of the derivatives and the fact that ‖∂2∂iHm‖m,ρ′′ ≤C6‖(I − E)Hm‖m,ρ′ , for i = 1, 2, 3, where C6 > 0, this norm can be bounded by an m-independent constant. Here, we again use the fact that ‖(I−E)Hm‖m,ρ′ drops faster with m,than cm increases. The inequality (100) then implies that, if |t| = |tm| < C5, H0 ∈ B+0,ρ′(C2)and C > 0 is sufficiently small, Ξs is a contraction with a contraction rate ā < 1.

Notice that

Ξs(0, q) = ((t− s)ωm, 0) + J · ∇hm(q, 0)t,

and that ∇Ehm(q, 0) = 0. Therefore, since |tm−sm| ≤ C4‖(I−E)Hm‖m,ρ′ , where C4 > 0 and|tm| < C5, if t = tm and s = sm, then, for sufficiently small C > 0, we have |||Ξsm(0, q)||| <(1− ā)b. This shows that Ξsm(·, q) is a contraction on a ball of radius b > 0, and has a uniquefixed point Υsm , of norm |||Υsm ||| < b, that satisfies the equation (98). The claim follows fromthe fact that for |tm| < C5, one has ‖Υsm(·, tm)‖m,δ1 ≤ |||Υsm |||. �

Let, in the following, b > 0 and C > 0 be chosen sufficiently small such that the assumptionsof Lemma 9.3 and Proposition 9.4 are satisfied. Let Fm, m ∈ N0, be arbitrary maps in Bm(b).Define

Γn,m = (Mn ◦ · · · ◦Mm−1)(Fm), (101)

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Renormalization of Hamiltonians for Diophantine frequency ...of Diophantine frequency vectors and obtain some bounds that will be used in section 7 to prove the convergence of the