In view of the great relevance of the work of Kolmogorov and of its impact on the

PERSISTENCE OF

INVARIANT TORI

This chapter is devoted to the celebrated theorem of Kolmogorov on persistence of invariant tori in near to integrable Hamiltonian systems. The relevance of the theorem for Classical and Celestial Mechanics was emphasized by V.I. Arnold, who wrote in [2]:

“ One of the most remarkable of A.N. Kolmogorov’s mathematical achieve- ments is his work on classical mechanics of 1954. A simple and novel idea, the combination of very classical and essentially modern methods, the so- lution of a 200 year–old problem, a clear geometric picture and a breadth of outlook — these are the merits of the work. ”

Kolmogorov announced his theorem at the International Congress of Mathemati- cians held in Amsterdam in 1954. Then he published a sketch of the proof in the short note [58]. The note contains in a synthetic form the formal scheme of an iter- ative procedure and a few essential hints on the proof of convergence. Kolmogorov also professed a series of lectures in Moskow where he gave the complete proof, as witnessed by some Russian mathematicians (see [19], ch. 11), but it seems that the text of the lectures has never been published. A proof of a related theorem for maps has been published by Moser [80]. Two papers with proofs for Hamiltonian systems that generalize and extend the ideas of Kolmogorov have been published by Arnold in [2] and [3]. Proofs based on the original paper of Kolmogorov may be found, e.g., in [9], [10], [11], [51]. The work of Kolmogorov, Arnold and Moser marked the begin- ning of a wide research field nowadays known as KAM theory. A lot of papers have been published on the subject: an exhaustive list would fill several pages.

7.1 The normal form of Kolmogorov

In view of the great relevance of the work of Kolmogorov and of its impact on the

progress of our knowlenge about non integrable systems this section is devoted to a

short exposition of the main ideas. The exposition closely follows the scheme that

can be found in the short note of Kolmogorov [58]. It goes without saying that the exposition in this section reflects the personal experience of the author.

The general framework is still the general problem of dynamics as stated by Poincar´e (see chapter 4). With a negligible change of notation I shall write the Hamil- tonian as

(7.1) H(p, q) = h(p) + εf (p, q, ε) ,

were p ∈ G ⊂ R ⁿ are action variables in an open set G and q ∈ T ⁿ are angle variables. The Hamiltonian is assumed to be holomorphic in all variables and in the small parameter ε.

We know that the unperturbed system, namely for ε = 0, is characterized by quasi–periodic motions on invariant tori parameterized by the actions p (see sect. 4.1).

The classical question is whether such a behaviour persists for the perturbed system, at least for ε small enough. That this can not be true for all tori is stated by Poincare’s theorem on non integrability (see chapter 4). The theorem of Kolmogorov claims that invariant tori characterized by strongly non resonant frequencies (e.g., diophantine frequencies in the sense of sect. 4.2.3) are slightly deformed, but not destroyed by the perturbation; they continue to exist for ε small enough and still carry quasi–periodic orbits.

The first idea is that there is a very simple Hamiltonian which brings immediately to evidence the existence of an invariant torus. Precisely, let the Hamiltonian be

(7.2) H(q, p) = hω, pi + R(q, p)

where ω ∈ R ⁿ and R(q, p) is at least quadratic in the actions p, i.e., R(q, p) = O(p ² ).

Hamilton’s equations read

˙q _j = ω _j + ∂R

∂p _j , ˙p _j = − ∂R

∂q _j , j = 1, . . . , n . Let now the initial point be p = 0 with arbitrary q. Since _∂p ^∂R

j = O(p) and _∂q ^∂R

j = O(p ² ) it is immediate to conclude that the torus p = 0 is invariant for the flow, and carries a Kronecker flow with frequencies ω. We shall say that the Hamiltonian (7.2) is in Kolmogorov’s normal form.

The suggestion of Kolmogorov is to reduce a Hamiltonian to the normal form above in a neighbourhood of a strongly non resonant unperturbed torus.

7.2 The formal scheme

Referring to the Hamiltonian (7.1), assume h(p) be nondegenerate. Pick a point p ^∗ ∈ G such that the corresponding unperturbed frequency ω = ^∂h _∂p (p ^∗ ) satisfies the diophan- tine condition

(7.3) |hk, ωi| ≥ γ|k| ^τ for all k ∈ Z ⁿ , k 6= 0 ,

with some γ > 0 and τ ≥ n − 1. By expanding the Hamiltonian in Taylor’s series of the actions around the point p ^∗ and forgetting unessential constants we get

H(q, p) = hω, p − p ^∗ i + εA(q) + εhB(q), p − p ^∗ i + 1

2 hC(q)(p − p ^∗ ), p − p ^∗ i + g(q, p − p ^∗ ) , where g(q, p − p ^∗ ) = O (p − p ^∗ ) ³
. The functions A(q), B _j (q), C _j,k (q) are calculated as

A(q) = f (q, p ^∗ , ε) , B _j (q) = ∂f

∂p _j (q, p ^∗ , ε) , C _j,k (q) = ∂ ² h

∂p j ∂p k

(p ^∗ ) + ε ∂ ² f

∂p j ∂p k

(q, p ^∗ , ε) .

We may perform a translation setting the point p ^∗ as the origin of the actions, which is tantamount to putting p ^∗ = 0 in the formulæ above, so that the Hamiltonian takes the form

(7.4) H(q, p) = hω, pi + εA(q) + εhB(q), pi + 1

2 hC(q)p, pi + g(q, p) ,

where the parameter ε has been added for convenience in the next calculation. If A(q) = B(q) = 0, then the Hamiltonian is in Kolmogorov’s normal form.

Kolmogorov’s suggestion is to kill the unwanted terms A(q) and hB(q), pi) in (7.4) by applying a canonical transformation of the special form

(7.5)

q ^′ _j = q _j + εY _j (q) p j = p ^′ _j − ε

 n

X

l=1

p ^′ _l ∂Y j (q)

∂q _l + ξ l + ∂X(q)

∂q _l



where ξ ∈ R ⁿ and X(q) and Y _j (q) are still periodic in the angles q. The effect of X(q) + hY (q), pi is a deformation of the torus, while hξ, qi induces a small translation.

It is matter of a moment to check that such a canonical transformation is produced by a generating function (in mixed variables)

S(p ^′ , q) = hp ^′ , qi + εX(q) + hξ, qi + hY (q), p ^′ i .

Exploiting the formalism of Lie series we may avoid inversions and substitutions by using the generating function

χ(q, p) = X(q) + hξ, qi + hY (q), pi ,

where the function X(q), the vector function Y (q) and the real vector ξ are determined so that the transformation kills the unwanted terms A(q) + hB(q), pi.

Let calculate the transformed Hamiltonian H ^′ = exp εL _χ
H, keeping only terms

of order ε. Here the Poisson bracket with the linear term hω, pi plays a special role,

so let us introduce the notation ∂ _ω = L _hω,pi . We get (recall that the names of the

variables are irrelevant)

H ^′ =hω, pi + C(q)p, p + g(p, q) + εL _χ g(p, q) + εA(q) − ∂ _ω X + hω, ξi 

+ ε



B(q), p +



C (q)p, ξ + ∂X

∂q



− ∂ _ω hY (q), pi



+ O(ε ² )

Here the first line contains the part already in normal form (at least quadratic in p, including L _χ g). Terms of order O(ε ² ) are left unhandled, and must be removed later.

The second and third line contain the parts that should be cleared. To this end we write the equations

(7.6)

A(q) − ∂ ω X = 0 , C ξ + B + ∂X

∂q = 0 , B(q) + C(q)



ξ + ∂X

∂q



− ∂ _ω Y = 0 .

Here the overline denotes the average with respect to the angles q, i.e., the term independent of q in the Fourier expansion of a function. The first and third equation (usually called homological equations) can be solved provided the average of the known term is zero (see next section). In the first equation the average is a constant that can be neglected. The second equation aims at determining the real vector ξ precisely in order to clear the average of the known term in the third equation. It may be solved provided the constant matrix C is not degenerate, which is initially assured by non degeneracy of h(p). The translation vector ξ keeps the frequency fixed.

Having determined the generating function we may perform the transformation and then rearrange the Hamiltonian in a form similar to (7.4), namely

H ^′ = hω, pi + C ^′ (q)p, p + ε ² A ^′ (q) + ε ² B ^′ (q), p + g ^′ (p, q) , g ^′ (p, q) = O(|p| ³ ) , with a new symmetric matrix C ^′ (q) which is a small correction of the previous one.

The factor ε ² in front of A(q) and B(p, q) reminds us that these terms have been produced by the transformation, but have been left unhandled. Explicit formulæ will be given later.

Thus, the consistency of the procedure depends on the existence of the solution of the equations (7.6). If so, then the procedure may be iterated in order to hopefully reduce the size of the unwanted terms to zero, thus giving the Hamiltonian the normal form of Kolmogorov.

7.2.1 Small divisors and the problem of convergence

The problem of solving the homological equation has been already discussed in

sect. 4.3.1, in connection with nonexistence of first integrals. However, here there

is a nice difference: the frequencies are constant and non resonant. Let us state the

problem as follows. Given a known function ψ(p, q) with zero average, namely ψ = 0,

find χ such that ∂ ω χ = ψ. The actions p here are just parameters. The procedure is quite standard. Expand in Fourier series

ψ(p, q) = X

06=k∈Z ⁿ

ψ _k (p) exp ihk, qi
, χ(p, q) = X

k∈Z ⁿ

c _k (p) exp ihk, qi
, with coefficients ψ k (p) known and c k (p) to be found. Calculate

∂ ω χ = i X

k

hk, ωic k (p) exp ihk, qi
.

Therefore, assuming that the frequencies ω are non resonant, we get the formal solution with coefficients

c _k (p) = −i ψ _k (p) hk, ωi .

The solution can be proved to be holomorphic on the basis of the following con- siderations, already made by Poincar´e. If ψ(p, q) is holomorphic, then the coeffi- cients ψ k (p) decay exponentially, i.e., 

 ψ k (p) 

 ∼ e ^−|k|σ for some σ. Therefore we get 

 c k (p) 

 ∼ |k| ^τ e ^−|k|σ ∼ e ^{−|k|σ ′} with some σ ^′ < σ. This shows that χ(p, q) is still holo- morphic, thus making every single step of Kolmogorov to be formally consistent.

The problem now is that iterating the procedure we produce an accumulation of small divisors: at every step the coefficients gain a new small divisor, which makes convergence doubtful. Here comes the second, great idea of Kolmogorov: apply the so called generalized Newton method (in his own terms), based on the work of Kan- torovich [57]. Do not use expansions in a parameter. At every step collect all contri- butions independent of and linear in p in a single pair of functions A(q) and B(p, q).

In very rough heuristic terms this is what happens. Starting with functions of size ε and forgetting for a moment the contribution of small divisors the procedure reduces step by step the size of the unwanted terms to ε ² , ε ⁴ , ε ⁸ , . . . that decrease quadrati- cally, as it happens in Newton’s method (as remarked by Kolmogorov himself). Such a strong decrease compensates the dramatically growing factors due to small divisors, eventually assuring convergence of the procedure. The latter heuristic argument was commonly used in the past, and it has been often synthetized in the words “quadratic method”, “quadratic convergence”, “superconvergence” and so on.

The heuristic considerations above on the fast convergence are too optimistic, because the contribution of the small divisors is ignored. As a matter of fact, in the proofs available in literature it is shown that after r iteration steps the size of the functions A(q) and B(q) decreases as ε ^{r 3 /2} or some similar power less than two, or geometrically as c ^r with some c < 1, or even as an inverse power r ^−k with some k > 1.

Anyway, the procedure can be proven to be actually convergent, and this is indeed the wonderful result of Kolmogorov.

The matter concerning fast convergence methods deserves a short extra discussion which is relevant here because the proof given in this chapter uses classical expansions.

Let us quote a sentence from a paper of Helmut R¨ ussman [91]: “It has often been said

that the rapid convergence of the Newton iteration is necessary for compensating the

influence of small divisors. But a deeper analysis shows that this is not true. (. . .)

Historically, the Newton method was surely necessary to establish the main theorems of the KAM–theory. But for clarifying the structure of the small divisor problems the Newton method is not useful because it compensates not only the influence of small divisors, but also many bad estimates veiling the structure of the problems.”

Up to the author’s knowledge the first proof of existence of invariant tori that does not make use of the fast convergence assured by the quadratic method has been published by R¨ ussman ^[92] . Other proofs have been published by Ugo Locatelli and the author [36] [37] [34].

The main reason for avoiding Newton’s method is the better understanding of the mechanism of accumulation of small divisors. A second relevant reason is that we may produce a constructive algorithm, that can be implemented with algebraic manipulation.

7.2.2 Statement and proof of the theorem

In order to reduce the technical troubles to a minimum, let us restrict our attention to a rather simple model. Consider a system of coupled rotators as described by the Hamiltonian H(p, q) = H ₀ (p) + εH ₁ (p, q), where

(7.7) H ₀ (p) = 1 2

n

X

j=1

p ² _j , H ₁ (p, q) = X

|k|≤K

c _k (p)e ^ihk,qi , p ∈ R ⁿ , q ∈ T ⁿ ,

with a fixed K > 0 and coefficients c k (p) that are polynomials of degree at most 2. The choice is made in order to reduce technicalities to a minimum (though there remain enough), but all the crucial difficulties of the problem are accounted for. The extension to the general case is matter of not being scared by long and boring calculations. Some hints on how to deal with the general case are provided later, in sect. 7.4.

The following statement is adapted to the particular case we are considering.

Theorem 7.1: On the phase space T ⁿ × R ⁿ consider the Hamiltonian (7.7) where H ₁ (q, p, ε) is a polynomial of degree at most 2 in the action variables p and is a trigonometric polynomial. For ε = 0, let p ^∗ be an unperturbed torus with frequencies ω = p ^∗ satisfying a diophantine condition (7.3). Then there exists a positive ε _∗ such that for every |ε| < ε _∗ the Hamiltonian (7.7) possesses an invariant torus ε ^a –close to p ^∗ for some positive a < 1. The flow on the torus is quasi periodic with frequencies ω.

7.2.3 The formal constructive algorithm

According to the procedure outlined in sect. 7.1, and considering the Hamilto- nian (7.7), we select an unperturbed torus p ^∗ such that the corresponding unperturbed frequency ω = p ^∗ satisfies the diophantine condition (7.3). By translating the origin of the action variables to p ^∗ , we write the Hamiltonian as

(7.8) H(q, p) = hω, pi + 1

2 hp, pi + εA 1 (q) + B ₁ (p, q) + C ₁ (p, q) 

This is rather obvious, because H 1 is supposed to be a polynomial of degree 2.

The aim is to construct an infinite sequence H ⁽⁰⁾ (p, q), H ⁽¹⁾ (p, q), H ⁽²⁾ (p, q), . . . of Hamiltonians, with H ⁽⁰⁾ coinciding with H in (7.8), which after r steps of normal- ization turn out to be written in the general form

(7.9) H ^(r) = ω · p +

r

X

s=0

ε ^s h _s (p, q) + X

s>r

ε ^s h

A ^(r) _s (q) + B _s ^(r) (p, q) + C _s ^(r) (p, q) i ,

where h 0 (p) = ¹ ₂ hp, pi. The Hamiltonian H ^(r) (p, q) is in Kolmogorov’s normal form up to order r, in formal sense. The following properties will be kept along the whole procedure:

(i) h 1 (p, q), . . . , h r (p, q) are quadratic in p, so that they are in normal form, and do not change after step r;

(ii) A ^(r) s (q) is independent of p;

(iii) B _s ^(r) (q) is linear in p.

(iv) C s ^(r) (q) is a quadratic polynomial in p;

(iv) A ^(r) s (q), B s ^(r) (q) and C s ^(r) (q) are trigonometric polynomials of degree sK, where K is the degree of H ₁ in the original Hamiltonian.

The properties appear to be quite strange, are precisely the characteristics that simpli- fies the proof, allowing us to concentrate on the crucial problems — e.g., the impact of small divisors on convergence. The original Hamiltonian (7.8) obviously satifies these properties.

The normalization process is worked out with a minor recasting of the method of Kolmogorov. Assuming that r − 1 steps have been performed, so that the Hamiltonian H ^(r−1) (p, q) has the wanted form (7.9) with r−1 in place of r, we construct in sequence the new Hamiltonians

H ˆ ^(r) = exp ε ^r L _{χ r,1}
H ^(r−1) , H ^(r) = exp ε ^r L _{χ r,2}
H ˆ ^(r) ,

the first one being an intermediate Hamiltonian, and the second one being in normal form up to order r. At every step r we apply a first canonical transformation with generating function χ _r,1 (q) = X _r (q)+hξ _r , qi, followed by a second transformation with generating function χ _r,2 (p, q) = hY ^(r) (q), pi. The explicit constructive algorithm for a single step is summarized in table 7.2. A reader who has devolped some expertise in perturbation methods will probably be able to check the table by himself. However, some help on how to construct the table may be welcome; hence, a detailed (very close to pedantic) explanation is included here.

We calculate the Hamiltonian ˆ H ^(r) by applying exp(L _{χ r,1} ) to every term of H ^(r−1) .

It may be useful to represent the Lie triangle, splitting it in two parts. The part that

involves terms of order ε ⁰ and ε ^r is the following:

ε ⁰ : hω, pi h ₀

ε ^r : L _{χ r,1} hω, pi L _{χ r,1} h ₀ A ^(r−1) _r B _r ^(r−1) C _r ^(r−1)

ε ^2r : 0 ¹ ₂ L ² _{χ r,1} h ₀ 0 L _{χ r,1} B _r ^(r−1) L _{χ r,1} C _r ^(r−1)

ε ^3r : 0 0 0 0 ¹ ₂ L ² _{χ r,1} C _r ^(r−1)

Since the generating function is of order ε ^r the rows of the triangle proceed by powers of ε ^r . The second line contains all terms that are involved in the process of removing the unwanted functions.

Extracting from the second line the two contributions independent of p and clear- ing them we get

1

2 L χ r,1 hω, pi + A ^(r−1) _r = −∂ ω X r + A ^(r−1) _r − ∂ ω hξ, qi = 0 .

Ignoring the last term ∂ ω hξ, qi, which is a constant that can be ignored, we get the first homological equation in table 7.2 that determines the function X _r (q). Forgetting also the average of A ^(r−1) r (q), which is a constant, the equation is solved as explained in sect. 7.2.1.

Extracting from the second line the contributions linear in p we get

1

2 L χ r,1 hp, pi + B _r ^(r−1) = − D ∂X r

∂q , p E

− hξ r , pi + B _r ^(r−1) .

Here we remark that the average of B r ^(r−1) (q) might be non zero, so that it would introduce an unwanted correction of the frequencies ω in ˆ H ^(r) . We avoid it by de- termining ξ r from the second equation in table 7.2, which is a trivial one in view of h 0 = ¹ ₂ hp, pi. Having removed the average, what is left is the function ˆ B r ^(r) in the second line of table 7.2.

The remaining term C r ^(r−1) is quadratic in p, and is left unchanged. Similarly the term ¹ ₂ L ² _{χ r,1} h 0 , of order ε ^2r and linear in p, is included in ˆ B r ^(r) ; the terms L χ r,1 B r ^(r−1)

of order ε ^2r , and ¹ ₂ L ² _{χ r,1} C r ^(r−1) , of order ε ^3r , are independent of p, and are included in A ˆ ^(r) _2r and ˆ A ^(r) _3r , respectively. The triangle does not include anything else, due to χ r,1

being independent of p. No term of degree higher than 2 in p is generated.

For h 1 , . . . , h r−1 , which are all quadratic functions, we have ε ^s exp L ε ^r χ r,1 h s = ε ^s h s + ε ^r+s L χ r,1 h s + ^{ε 2r+s} ₂ L ² _{χ r,1} h s ,

all the rest of the series being zero. The new term L χ r,1 h s is linear in p, and is included

in ˆ B _r+s ^(r) in table 7.2. Similarly, ^{ε 2r+s} ₂ L ² _{χ r,1} h s is independent of p, and is included in

A ˆ ^(r) _r+s . For the functions A ^(r−1) _s , B _s ^(r−1) and C _s ^(r−1) of higher order s > r we get a small triangle, namely

ε ^s : A ^(r−1) _s B _s ^(r−1) C _s ^(r−1) ε ^r+s : 0 L _{χ r,1} B _s ^(r−1) L _{χ r,1} C _s ^(r−1) ε ^2r+s : 0 0 ¹ ₂ L ² _{χ r,1} C _s ^(r−1)

The triangle contains all terms that are generated, which are at most quadratic in p, as wanted. Here, L _{χ r,1} B s ^(r−1) and ¹ ₂ L ² _{χ r,1} C s ^(r−1) are independent of p, and are included in ˆ A ^(r) _r+s and ˆ A ^(r) _2r+s , respectively. Finally, L χ r,1 C s ^(r−1) is linear in p and is included in B ˆ _r+s ^(r) . All unchanged functions are also included, as appropriate. The functions C s ^(r−1)

for s > r remain unchanged in this step.

Having performed the transformation with χ _r,1 step we have an intermediate Hamiltonian

H ˆ ^(r) = ω ·p+

r−1

X

s=0

ε ^s h _s (p, q)+ ˆ B _r ^(r) +C _r ^(r−1) + X

s>r

ε ^s h ˆ A ^(r) _s (q) + ˆ B ^(r) _s (p, q) + C _s ^(r) (p, q) i ,

where all functions are defined in table 7.2. Remark that there is no term ˆ A ^(r) r , which has been cleared.

We come now to calculating the Hamiltonian H ^(r) = exp L χ r,2 H ˆ ^(r) . We split again the triangle, first considering terms of order ε ⁰ and ε ^r . We get

ε ⁰ : hω, pi h 0

ε ^r : L _{χ r,2} hω, pi L _{χ r,2} h ₀ B ˆ ^(r) _r C _r ^(r−1)

ε ^2r : ¹ ₂ L ² _{χ r,2} hω, pi ¹ ₂ L ² _{χ r,2} h ₀ L _{χ r,2} B ˆ _r ^(r) L _{χ r,2} C _r ^(r−1)

ε ^3r : _3! ¹ L ³ _{χ r,2} hω, pi _3! ¹ L ³ _{χ r,2} h ₀ ¹ ₂ L ² _{χ r,2} B ˆ _r ^{(r) 1} ₂ L ² _{χ r,2} C _r ^(r−1)

ε ^4r : _4! ¹ L ⁴ _{χ r,2} hω, pi _4! ¹ L ⁴ _{χ r,2} h _{0 3!} ¹ L ³ _{χ r,2} B ˆ _r ^(r) _3! ¹ L ³ _{χ r,2} C _r ^(r−1)

.. . .. . .. . .. .

The triangle here is infinite, because χ r,2 is linear in p and so it does not change the degree of a function.

Extracting the linear terms in p from the second line and forcing it to zero we get L χ r,2 hω, pi + ˆ B _r ^(r) = −∂ ω χ r,2 + ˆ B _r ^(r) = 0 ,

which is the third equation for the generating functions in table 7.2. In order to show

that it may be solved we need a few more considerations. Recall that χ _r,2 = hY _r (q), pi.

On the other hand, since ˆ B _r ^(r) is linear in p, we may write it as, e.g., ˆ B _r ^(r) = hΨ(q), pi.

Then the equation to be solved splits into the n equations ∂ _ω Y _r,j (q) = Ψ _j (q), which can all be solved because the right members have null average (it has been removed by appropriately finding ξ _r ). Having determined χ _r,2 we can construct all the rest of the triangle, and move every element to the appropriate function in 7.2. Only the functions B _kr ^(r) deserve some more explanation. For k > 1 we observe that in view of the homological equation we have

1

k! L ^k _{χ r,2} hω, pi = − _k! ¹ L ^k−1 _{χ r,2} ∂ _ω χ _r,2 = − _k! ¹ L ^k−1 _{χ r,2} B ˆ _r ^(r) . Thus we calculate

1

k! L ^k _{χ r,2} hω, pi + _(k−1)! ¹ L ^k−1 _{χ r,2} B ˆ _r ^(r) = 

1

(k−1)! − _k! ¹ 

L ^k−1 _{χ r,2} B ˆ _r ^(r) = ^k−1 _k! L ^k−1 _{χ r,2} B ˆ _r ^(r) . The result is included in B _kr ^(r) in 7.2.

For the functions h ₁ , . . . , h _r we have ε ^m exp ε ^r L _χ 2 h _m
= X

k≥0

ε ^kr+m

k! L ^k _{χ r,2} h _m , m = 0, . . . , r − 1 .

For k = 0 we get h _m , which is left unchanged. For k > 0 we include every term in the sum in the corresponding function C _kr+m ^(r) . Finally, for s > r the functions ˆ A ^(r) s , ˆ B s ^(r)

and C s ^(r−1) are transformed as

ε ^s exp ε ^r L _χ 2 A ˆ ^(r) _s = ε ^s X

k≥0

ε ^r

k! L ^k _{χ r,2} A ˆ ^(r) _s , ε ^s exp ε ^r L _χ 2 B ˆ _s ^(r) = ε ^s X

k≥0

ε ^r

k! L ^k _{χ r,2} B ˆ _s ^(r) , ε ^s exp ε ^r L _χ 2 C ˆ _s ^(r−1) = ε ^s X

k≥0

ε ^r

k! L ^k _{χ r,2} C ˆ _s ^(r−1) .

Every term in the sums should be moved to the appropriate function in table 7.2.

It remains to show that also the property (v) is satisfied, namely that every func- tion of order ε ^s is a trigonometric polynomial of degree sK. The generating functions X r and χ r,2 have degree rK because they are solutions of homological equations with known members of degree rK. On the other hand, if a function f _l has degree lK then L χ r,j f l clearly has degree (r + l)K. Just apply this rule to every Lie derivative in the algorithm. Thus, the justification of the formal algorithm in table 7.2 is complete.

7.3 Quantitative estimates

We come now to the crucial problem that has challenged mathematicians for a couple

of centuries: the accumulation of small divisors, which is indeed the most challenging

part of the proof that the sequence of formal transformations to normal form actually

converges to a Hamiltonian possessing the normal form of Kolmogorov. In this section

we construct the scheme of analytic estimates, paying particular attention to small divisors.

In view of the form of the Hamiltonian, which is a polynomial in the actions p and a trigonometric polynomial in the angles, we shall consider all functions as analytic in a domain D (̺,σ) = ∆ ̺ (0) × T ⁿ _σ , where the choice of ̺ and σ is rather arbitrary.

Thus we pick some values for them, and keep them constant in the whole proof. Our choice is intended to make easier to adapt the proof to more generale situation, as will sketched later in sect. 7.4.

In view of our purposes it is convenient to use the weighted Fourier norm, for which the estimates of Poisson brackets are provided in sect. 6.7.2, forgetting the obvious fact that the estimate the derivatives of a polynomial of degree two does not require the sophisticated tool of Cauchy estimates. An exception is represented by the function hξ, qi with ξ ∈ R ⁿ , which is part of the generating functions χ 1 , but is not trigonometric. However, we shall see that this causes a very little trouble. We simplify a little the notation by writing the norm as, e.g., k · k _1−d in place of k · k _{(1−d)(̺,σ)} . In particular we write k · k ₁ in place of k · k _̺,σ .

The strategy of the proof follows the following scheme:

(i) Translate the formal algoritm into a sequence of estimates for the norms of all functions involved.

(ii) Isolate the problem of accumulation of small divisors, and show that they obey some strict, perhaps surprising rules that make their contribution to grow not faster than geometrically with the order (no factorials).

(iii) Prove that the norms of the generating functions satisfy the convergence con- dition of proposition 6.11.

Thus the proof of the theorem of Kolmogorov will be complete.

7.3.1 Estimates for the generating functions

The formal algorithm is based on the operations of solving the homological equations and of calculating Lie derivatives. In order to use the generalized Cauchy estimates of sect. 6.7 we should make a choice for the restrictions of domains. We need an infinite sequence of restrictions d ₁ < d ₂ < d ₃ < . . . tending to a limit d < 1. We shall actually impose the stronger condition d ≤ 1/6. To this end let us introduce the arbitrary sequence for r > 0

(7.10) δ r = 1

2π ² · 1

r ² , X

r>0

δ r = 1 12 . Then set

(7.11) d ₀ = 0 , d _r = 2(δ ₁ + . . . + δ _r ) , r > 0 ,

which satisfies our request. Our aim is to find recurrent estimates for ˆ H ^(r) in the domain D (1−d r−1 −δ r )(̺,σ) and for H ^(r) in the domain D (1−dr)(̺,σ) .

Let us begin with an estimate of the solution of a homological equation. Having

given a non resonant frequency vector ω ∈ R ⁿ we introduce the real, non increasing

Table 7.2. The formal constructive algorithm for Kolmogorov’s normal form.

• Equations for the generating functions χ r,1 = X r + hξ r , qi and χ r,2 = hY r (q), pi:

∂ _ω X _r = A ^(r−1) _r , hξ _r , pi = B r ^(r−1) ,

∂ _ω χ _r,2 = ˆ B _r ^(r) ,

B ˆ _r ^(r) = B _r ^(r−1) − B r ^(r−1) −  ∂X _r

∂q , p

 .

• Intermediate Hamiltonian ˆ H ^(r) = exp L χ r,1
H ^(r−1) : A ˆ ^(r) _r = 0

A ˆ ^(r) _s =



 

 

 

 

 

 

A ^(r−1) _s , r < s < 2r ;

1

2 L ² _{χ r,1} h s−2r + L χ r,1 B _s−r ^(r−1) + A ^(r−1) _s , 2r ≤ s < 3r ; 1

2 L ² _{χ r,1} C _s−2r ^(r−1) + L _{χ r,1} B _s−r ^(r−1) + A ^(r−1) _s , s ≥ 3r . B ˆ _s ^(r) =







L _{χ r,1} h _s−r + B _s ^(r−1) , r < s < 2r ; L _{χ r,1} C _s−r ^(r−1) + B _s ^(r−1) , s ≥ 2r .

• Transformed Hamiltonian H ^(r) = exp L _{χ r,2} H ˆ ^(r)
(set k = ⌊s/r⌋, m = s (mod r), s = kr + m):

h r = L χ r,2 h 0 + C _r ^(r−1) . A ^(r) _s =

k−1

X

j=0

1

j! L ^j _{χ r,2} A ˆ ^(r) _s−jr , s > r .

B _s ^(r) =



 

 

 



 

 

 

 k − 1

k! L ^k−1 _{χ r,2} B ˆ _r ^(r) +

k−2

X

j=0

1

j! L ^j _{χ r,2} B ˆ _s−jr ^(r) , k ≥ 2 , m = 0 ;

k−1

X

j=0

1

j! L ^j _{χ r,2} B ˆ _s−jr ^(r) , k ≥ 1 , m 6= 0 .

C _s ^(r) = 1

k! L ^k _{χ r,2} h _m +

k−1

X

j=0

1

j! L ^j _{χ r,2} C _s−jr ^(r) , s > r .

Table 7.3. Quantitative estimates for the normalization scheme.

• Generating functions χ _r,1 = X _r + hξ _r , qi and χ _r,2 = hY _r (q), pi:

kX _r k _1−d

r−1 ≤ 1

α _r kA ^(r−1) _r k _1−d

r−1 ,   ξ _r,j 

 ≤ 2

̺ B

(r−1) r

1−d r−1

kχ _r,2 k _1−d

r−1 −δ r ≤ 1

α _r k ˆ B _r ^(r) k _1−d

r−1 −δ r

• Intermediate Hamiltonian ˆ H ^(r) = exp L χ r,1
H ^(r−1) . Set G _r,1 = e

̺σ

 kA ^(r−1) _r k _1−d

r−1 + 2α _r δ _r σ

̺ B

(r−1) r

1−d r−1

 . For r < s < 2r , 2r ≤ s < 3r and s ≥ 3r , respectively, get

k ˆ A ^(r) _s k _1−d

r−1 −δ r ≤



 

 

 

 

 

 

 

 

kA ^(r−1) _s k _1−d

r−1 ;

 G _r,1 δ _r ² α _r

 2

kh _s−2r k _1−d

s−2r + G r,1

δ _r ² α _r kB _s−r ^(r−1) k _1−d

r−1 + kA ^(r−1) _s k _1−d

r−1 ;

 G _r,1 δ _r ² α _r

 2

kC _s−2r ^(r−1) k _1−d

s−2r + G _r,1

δ _r ² α _r kB _s−r ^(r−1) k _1−d

r−1 + kA ^(r−1) _s k _1−d

r−1 ; For r < s < 2r and s ≥ 2r , respectively, get

k ˆ B _s ^(r) k _1−d

r−1 −δ r ≤



 



 

 G _r,1

δ ² _r α _r kh _s−r k _1−d

s−r + kB _s ^(r−1) k _1−d

r−1 ; G _r,1

δ ² _r α r

kC _s−r ^(r−1) k _1−d

s−r + kB _s ^(r−1) k _1−d

r−1 .

• New Hamiltonian H ^(r) = exp L χ r,2 H ˆ ^(r)
. Set G r,2 = _̺σ ^e k ˆ B ^(r) r k _(1−d

r−1 −δ r ) . For s ≥ r get

kh r k _{1−d r} ≤ G r,2

δ _r ² α _r kh 0 k ₁ + kC _r ^(r) k _1−d

r−1 . kA ^(r) _s k _{1−d r} ≤

k−1

X

j=0

 G _r,2 δ _r ² α r

 j

k ˆ A ^(r) _s−jr k _1−d

r−1 −δ r ; kB _s ^(r) k _{1−d r} ≤

k−1

X

j=0

 G _r,2 δ _r ² α _r

 j

k ˆ B _s−jr ^(r) k _1−d

r−1 −δ r ; kC _s ^(r) k _{1−d r} ≤  G _r,2

δ ² _r α _r

 k

kh _s−kr k _1−d

s−kr +

k−1

X

j=0

 G _r,2 δ _r ² α _r

 j

k ˆ C _s−jr ^(r) k

1−d r−1 −δ r .

sequence {α r } r≥0 defined as

(7.12) α ₀ = 1 , α _r = min 

1, min

0<|k|≤rK

 hk, ωi  

 .

That is, α _r is the smallest divisor that may appear in the solution of the homological equation for the generating functions χ r,1 and χ r,2 at step r of the normalization process. If the frequencies are non resonant then the sequence has zero limit for r → ∞.

A further conditions that characterizes strong non resonance will be found later.

Lemma 7.2: Let ψ s (p, q) be a trigonometric polynomial of degree s, analytic and bounded in a domain D ̺,σ . Let also ω ∈ R ⁿ be a non resonant vector. Then the homological equation ∂ ω χ = ψ s possesses a solution with zero average satisfying

(7.13) kχk _̺,σ ≤ kψk _̺,σ

α s

with α _s defined by (7.12) .

Proof. The solution is constructed as explained in sect. 7.2.1: the coefficient c _k (p) of the Fourier expansion of χ are

c _k (p) = −i ψ _s,k (p) hk, ωi ,

where ψ _s,k (p) are the coefficients of ψ _s (p, q). Hence we estimate |c _k | _̺ ≤ ^{|ψ s,k} _α ^{| ̺}

s . Then we apply the definition (6.52) of the weighted Fourier norm. Q.E.D.

The lemma is used in order to produce the estimates of the generating functions χ _r,1 and χ _r,2 in table 7.3. The estimates of X _r and χ _r,2 are a direct application of lemma 7.2. The estimate of |ξ _r | requires a couple of elementary considerations.

The function B r ^(r−1) is linear in p. On the other hand, in view of the special form of h 0 (p) = ¹ ₂ hp, pi the equation for ξ r in table 7.2 has the very simple form hξ, pi = B r ^(r−1) . Therefore, by Cauchy’s estimate on the center of the disk ∆ _(1−d)̺ (0) and in view of the inequality d < 1/2 that we shall use we get

(7.14) |ξ _r,j | ≤ 1

(1 − d)̺

B r ^(r−1)

1−d ≤ 2

̺

B r ^(r−1)

1−d .

Lemma 7.3: Let ξ ∈ R ⁿ . If f (p, q) is analytic and bounded in D 1 then

L _hξ,qi f

(1−d) ≤ 1

d̺ |ξ| kf k ₁ , |ξ| = |ξ 1 | + . . . + |ξ n | .

Proof. An almost verbatim repetition of the proof of lemma 6.14. The angles q play the role of parameters. For any (p, q) ∈ D 1 the function F (τ ) = f (p + τ ξ, q) is analytic and bounded by kf k ₁ in the disk |τ | < _|ξ| ^d̺ . Use L _hξ,qi f = ^{dF (τ )} _dτ 



 τ =0 , and apply the

Cauchy estimate. Q.E.D.

7.3.2 The scheme of estimates

We are now ready to construct a recurrent scheme of estimates, which is collected in table 7.3. Constructing the table is a tedious but straightforward operation. Here are some hints that the reader may find useful.

The estimates of the generating functions X _r and χ _r,2 follow from lemma 7.2. The estimate for the components ξ r,j of ξ r are a copy of (7.14). However, we shall actually use lemma 7.3 for the Poisson bracket. If we know the norm kf k _1−d

r−1 of a function f (p, q) (which is precisely what we need) then a straightforward calculation gives

(7.15)

L _{χ r,1} f

1−d _r−1 −δ r ≤ kX _r k _1−d

r−1

δ _r ² kf k _1−d

r−1 + 2

B r ^(r−1)

1−d r−1

δ _r ̺ ² kf k _1−d

r−1

≤



kA ^(r−1) _r k _1−d

r−1 + 2α _r δ _r σ

̺

B r ^(r−1) 1−d _r−1

 kf k _1−d

r−1

δ _r ² α _r ̺σ . That is: the quantity between parentheses plays the role of the norm kχ _r,1 k _1−d

r−1 . Remark the extra factor α r δ r which will be used in order to control the accumulation of small divisors. It is matter of a moment to realize that the estimates (6.56) of lemma 6.15 for powers of L χ r,1 remain true, because they are based on the estimate of a single Lie derivative.

All estimates of the functions ˆ A ^(r) s and ˆ B s ^(r) that enter the expansion of ˆ H ^(r) are nothing but an application of the estimates of multiple Lie derivatives of lemma 6.15 using (7.15) for the estimate of χ _r,1 . All functions C ^(r−1) remain unchanged, so that there is no need of estimating them: just keep their norms. The constant G r,1 is defined so that in all estimates we collect all constants together and bring into evidence the divisor δ _r ² α _r which represents the real trouble in the proof of convergence. ¹

The estimates of h _r and of all functions A ^(r) s , B ^(r and C s ^(r) that enter the expansion of H ^(r) do not require any further comment, since it is only matter of applying again lemma 6.15 with the norm of χ ₂ estimated at the beginning of the table. The functions h ₁ , . . . , h _r−1 remain unchanged, and so are their norms.

The fact to be remarked and exploited is that all estimates exhibit a common structure: all estimates are sums of different contributions obtained by multiplying a factor _δ ^G 2 ^r,1

r α r or _δ ^G 2 ^r,2

r α r (or a power of it) by the known norm of some function.

In view of the latter remark we are led to consider the quantities β _r = δ _r ² α _r as the small divisors to be put under observation. It will be convenient also to define β 0 = 1.

1 The reader will remark that the role of the small divisors is played not only by the

quantities α r that depend on the frequencies, but also by the restrictions δ r of the

domain that are requested by Cauchy estimates of Lie derivatives. By comparison, the

reader my observe that in the case of formal expansion of first integrals of chapter 5 the

sources of divergence are both the divisors α r and the factors coming from the exponents

of polynomials, due to derivatives.

7.3.3 The game of small divisors

The accumulation of small divisors can be analyzed paying particular attention to the remark that we have just made. First, note that the mechanism of accumulation has nothing to do with the actual value of divisors: it is rather matter of indices. Consider any function h s , A ^(r−1) s , B s ^(r−1) and C s ^(r−1) in the expansion of H ^(r−1) . In view of the recurrent scheme of estimates of table 7.3 the norm of every function is estimated by a sum of terms with general form c(β j 1 · · · β j m ) ⁻¹ , where c is a positive number and m is the number of divisors that have been generated through the scheme of estimates.

E.g., we construct the collection of terms to be added up in the estimate for k ˆ A ^(r) s k (the domain is not relevant here) by multiplying each term in kh _s k by ^G _β ^r,1

r

2

, then multiplying every term in kB _s−r ^(r−1) k by ^G _β ^r,1 _r , then taking the collection of terms in kA ^(r−1) s k as is, and making the union of the three collections so constructed without applying any algebraic simplification. This applies to every estimate in the table, and it reflects precisely the scheme of estimates.

Let us now say that a function f owns a list of indices I = {j ₁ , . . . , j _k } if its estimate contains a term with the divisor β _j 1 · · · β _{j k} . The following considerations are immediate.

(i) If ψ _r (trigonometric polynomial of degree r) owns a list I, then the solution χ _r of the homological equation ∂ _ω χ _r = ψ _r owns the list {r} ∪ I, the union meaning concatenation of lists (repeated indices are kept in the list).

(ii) If f _s (of order s) owns a list I ^′ and χ _r is as at point (i) then L _{χ r} f _s owns the list {r} ∪ I ∪ I ^′ .

(iii) For the multiple Lie derivative the following scheme applies:

f _s owns I ^′ ;

L _{χ r} f owns I ₁ = {r} ∪ I ∪ I ^′ L ² _{χ r} f owns I ₂ = {r} ∪ I ∪ I ₁ L ³ _{χ r} f owns I 3 = {r} ∪ I ∪ I 2

and so on, so that we may proceed by recurrence.

Hence the points (i) and (ii) contain all the relevant information about the mechanism of accumulation of divisors.

7.3.4 The kindness of small divisors

Let us open a long parenthesis. Forget for a moment the technicalities of the theorem of Kolmogorov, and focus attention on the propagation of lists of indices in the simplest case.

We call list of indices a collection ² {j ₁ , . . . , j _s } of non negative integers, with length s ≥ 0. The empty list {} of length 0 is allowed, as well as repeated indices. The index 0 is allowed, too, and will be used in order to pad a short list to the wanted length, when needed. The lists of indices provide a full characterization of the products of

2 The name list is used in order to emphasize that repeated elements are allowed.

Table 7.4. The special lists I _s ^∗ for 0 ≤ s ≤ 16.

s I _s ^∗

1 {}

2 {1}

3 {1 , 1}

4 {1 , 1 , 2}

5 {1 , 1 , 1 , 2}

6 {1 , 1 , 1 , 2 , 3}

7 {1 , 1 , 1 , 1 , 2 , 3}

8 {1 , 1 , 1 , 1 , 2 , 2 , 4}

9 {1 , 1 , 1 , 1 , 1 , 2 , 3 , 4}

10 {1 , 1 , 1 , 1 , 1 , 2 , 2 , 3 , 5}

11 {1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 3 , 5}

12 {1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 3 , 4 , 6}

13 {1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 3 , 4 , 6}

14 {1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 2 , 3 , 4 , 7}

15 {1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 3 , 3 , 5 , 7}

16 {1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 2 , 3 , 4 , 5 , 8}

small divisors: to the list {j ₁ , . . . , j _s } we associate the product {β _j 1 , . . . , β _{j s} }. Adding any number of zeros to a list of indices is harmless, for we have set β ₀ = 1.

On the set of lists of indices we introduce a partial ordering as follows. Let I, I ^′ be lists with the same length s. We say that I ⊳ I ^′ in case there is a permutation of the indices such that the relations j 1 ≤ j ₁ ^′ , . . . , j s ≤ j _s ^′ hold true. If the lists have different lenghts then we pad the shorter one with zeros, and apply the criterion. The comparison is made easy by just ordering all elements of every list and comparing element by element. The order is clearly partial: e.g., the criterion does not apply to the lists {1, 3} and {1, 1}. This will not harm us, however.

A central role will be played by special lists of indices that we denote by I _s ^∗ , with s ≥ 0. We define

(7.16) I _s ^∗ =  s

s

 ,

 s s − 1



, . . . ,  s 2



. In table 7.4 we give examples of the special lists just defined.

Lemma 7.4: For the sets of indices I _s ^∗ = {j 1 , . . . , j s } the following statements hold true:

(i) the maximal index is j _max =  _s

2  ;

(ii) for every k ∈ {1, . . . , j _max } the index k appears exactly  _s

k  −  _k+1 ^s  times;

(iii) for 0 < r ≤ s one has

{r} ∪ I _r ^∗ ∪ I _s ^∗
⊳ I _r+s ^∗ .

Proof. The claim (i) is a trivial consequence of the definition.

(ii) For each fixed value of s > 0 and 1 ≤ k ≤ ⌊s/2⌋ , we should determine the cardinality of the set M _k,s = {m ∈ N : 2 ≤ m ≤ s , ⌊s/m⌋ = k}. For this purpose, we use the trivial inequalities

 s

⌊s/k⌋



≥ k and

 s

⌊s/k⌋ + 1



< k .

After having rewritten the same relations with k + 1 in place of k , one immediately realizes that a index m ∈ M _k,s if and only if m ≥ ⌊s/(k + 1)⌋ + 1 and m ≤ ⌊s/k⌋.

Therefore #M _k,s =  _s

k  −  _k+1 ^s  , as claimed.

(iii) Since r ≤ s , the definition in (7.16) implies that neither {r} ∪ I _r ^∗ ∪ I _s ^∗ nor I _r+s ^∗ can include any index exceeding (r + s)/2 . Let us define some finite sequences of non-negative integers as follows:

R _k = #j ∈ I _r ^∗ : j ≤ k , S _k = #j ∈ I _s ^∗ : j ≤ k , M k = #j ∈ {r} ∪ I _r ^∗ ∪ I _s ^∗ : j ≤ k , N k = #j ∈ I _r+s ^∗ : j ≤ k ,

where 1 ≤ k ≤ ⌊(r + s)/2⌋ . When k < r , the property (ii) of the present lemma allows us to write

R _k = r − j r k + 1

k , S _k = s − j s k + 1

k , N _k = r + s − j r + s k + 1

k ;

using the elementary estimate ⌊x⌋ + ⌊y⌋ ≤ ⌊x + y⌋ , from the equations above it follows that M k ≥ N k for 1 ≤ k < r . In the remaining cases, i.e., when r ≤ k ≤ ⌊(r + s)/2⌋ , we have that

R k = r − 1 , S k = s − j s k + 1

k

, N k = r + s − j r + s k + 1 k

;

therefore, M k = 1 + R k + S k ≥ N k . Since we have just shown that M k ≥ N k ∀ 1 ≤ k ≤ ⌊(r + s)/2⌋ , it is now an easy matter to complete the proof. Let us first reorder both the set of indices {r}∪I _r ^∗ ∪I _s ^∗ and I _r+s ^∗ in increasing order; moreover, let us recall that # {r} ∪ I _r ^∗ ∪ I _s ^∗
= #I _r+s ^∗ = r + s − 1 , because of the definition in (7.16). Thus, since M 1 ≥ N 1 , every element equal to 1 in {r} ∪ I _r ^∗ ∪ I _s ^∗ has a corresponding index in I _r+s ^∗ the value of which is at least 1 . Analogously, since M ₂ ≥ N ₂ , every index 2 in {r} ∪ I _r ^∗ ∪ I _s ^∗ has a corresponding index in I _r+s ^∗ which is at least 2 , and so on up to k = ⌊(r + s)/2⌋ . We conclude that {r} ∪ I _r ^∗ ∪ I _s ^∗ ⊳ I _r+s ^∗ . Q.E.D.

We shall also need the following

Lemma 7.5: Let I r and I s be lists of length r − 1 and s − 1, respectively, with 1 ≤ r ≤ s (pad them with zeros, if needed). If I _r ⊳ I _r ^∗ and I _s ⊳ I _s ^∗ then we have

r ∪ I _r ∪ I _s ⊳ I _r+s ^∗ . The proof is elementary: just reorder the indices.

We come now to exploit the relation between lists of indices and products of small

divisors. Let 1 = α 0 ≥ α 2 ≥ α 3 be a sequence of positive numbers; so is the sequence

of small divisors, with the additional property that the sequence tends to zero. To a list I associate the quantity Q(I) = Q

j∈I 1

α j . The following property is obvious:

if I ⊳ I ^′ then Q(I) > Q(I ^′ ) . Let us consider the special sequence

(7.17) Q ^∗ _s = Y

j∈I _s ^∗

1 α j

;

We look for a sufficient condition assuring that the sequence has a finite limit. In view of lemma 7.4 we may evaluate

(7.18) ln Q ^∗ _s = ln Y

j∈I _s ^∗

1 α _j ≤ −

s

X

k=1

j s k

k − j s k + 1

k α _k ≤ −s X

k≥1

ln α k

k(k + 1) . We are thus led to introduce

Condition τ: The sequence {α _r } _r≥0 satisfies

(7.19) − X

r≥1

ln α _r

r(r + 1) = Γ < ∞ .

If so, then we also have the estimate Q ^∗ _s < e ^sΓ , that is, it grows not faster than geometrically.

Taking the sequence of small divisors associated to the frequencies ω ∈ R ⁿ as in (7.12) condition τ will provide the strong non resonance condition for the validity of the theorem of Kolmogorov.

In his original proof Kolmogorov used the diophantine condition. More recently other conditions have been introduced with the aim of finding the optimal one. The most known is the so called Bruno condition, introduced by Alexander Bruno.For the problem of Schr¨ oder–Siegel (a map of the complex plane with the origin a fixed point) the optimality of the condition of Bruno has been proved by Jean Christophe Yoccoz.

For the problem of Kolmogorov the question is still open.

Example 7.1: Comparison with other conditions Here are a few notes that allow the reader to compare condition τ with other commonly used conditions that are found in literature.

(i) The diophantine condition introduced by Siegel says α _r = r ^−k with k > 1 (an innocuous multiplicative constant is omitted). This gives

− X

r≥1

ln α r

r(r + 1) = k X

r≥1

ln r

r(r + 1) < ∞ .

This shows that diophantine frequencies satisfy also contidion τ. By the way, this

also shows that if the sequence α _r satisfies condition τ then so does the sequence

α r = δ _r ² α r with δ r ∼ r ⁻² that appears in our estimates for the case of Kolmogorov,

for it just adds to the estimate (7.18) a convergent series.

(ii) Condition τ is weaker than the diophantine one. E.g., if α r = e ^{−r/ ln 2 r} then

− X

r≥1

ln α _r

r(r + 1) = X

r≥1

1

(r + 1) ln ² r < ∞ .

(iii) There are ω’s that violate condition τ. For instance, if α r = e ^−r then

− X

r≥1

ln α _r

r(r + 1) = X

r≥1

1

(r + 1) = ∞.

(iv) The condition of Bruno writes

− X

r≥1

ln α ₂ r−1

2 ^r = B < ∞ .

It is equivalent to condition τ, for one gets Γ < B < 2Γ. The proof is left to the reader.

7.3.5 Small divisors in the algorithm of Kolmogorov

We can now go back to the estimates for the normal form of Kolmogorov, and focus on the accumulation of small divisors. The problem is to identify the the worst possible product of divisors in every coefficient of every function. To this end, in view of the discussion of the previous section, we concentrate on the indices, and look for two informations, namely

(i) the number of divisors β j ;

(ii) a selection rule that specifies which lists of coefficiens may really show up.

Definition 7.6: For all integers r ≥ 0 and s > 0 , we introduce the collection of lists (7.20) J r,s = I = {j 1 , . . . , j s−1 } : 0 ≤ j m ≤ min{r, ⌊s/2⌋} , I ⊳ I _s ^∗ ,

Lemma 7.7: The following properties hold true: for r ^′ < r we have (7.21) J _r ^′ _,s ⊂ J r,s for r ^′ < r ;

for 0 ≤ r ≤ s we have

(7.22) {r} ∪ J r−1,r ∪ J r,s ⊂ J r,r+s .

Proof. The first property is obvious. For the second one check that: the number of indices in the two members is the same; the maximum value of the indices is respected;

the selection rule is respected in view of property (iii) of lemma 7.4. Q.E.D.

Our goal now is to identify the list of divisors owned by every function, with particular care for the generating functions. This will be made later with lemma 7.8.

However, before stating the lemma we need some considerations concerning the norm

of the vector ξ r , for a straightforward estimate will add a divisor that apparently

breaks the toy constructed in sect. 7.3.4. This may be seen by looking at the expression of G _r,1 in table 7.3, namely

G r,1 = e

̺σ



kA ^(r−1) _r k _{1−d r−1} + 2α _r δ _r σ

̺ B

(r−1) r

1−d r−1

 . The immediate inequality (the domains are not relevant here)

B r ^(r−1) ≤

B r ^(r−1) suggests that the second term should contain a divisor δ ² _r α r in addition to those in the first term A ^(r−1) r . That divisor is only partially compensated by the factor δ r α r . The point is that a careful estimate of

B s ^(r)

contains an extra factor δ _r which removes the extra divisors. Regrettably, the proof requires some tedious calculations.

We proceed by induction, as usual. The problem does not show up for G _r,1 , be- cause there are no small divisors in B s ⁽⁰⁾ . For r > 1 we should take into consideration the expressions of ˆ B _s ^(r) and B _s ^(r) in table 7.2. Let us recall them:

(7.23) B ˆ _s ^(r) = (L χ r,1 h _s−r + B _s ^(r−1) , r < s < 2r ; L _{χ r,1} C _s−r ^(r−1) + B _s ^(r−1) , s ≥ 2r . and, recalling that k = ⌊s/r⌋,

(7.24) B _s ^(r) =



 

 

 

 

 

  k − 1

k! L ^k−1 _{χ r,2} B ˆ _r ^(r) +

k−2

X

j=0

1

j! L ^j _{χ r,2} B ˆ _s−jr ^(r) , k ≥ 2 , m = 0 ;

k−1

X

j=0

1

j! L ^j _{χ r,2} B ˆ _s−jr ^(r) , k ≥ 1 , m 6= 0 . In the expressions for ˆ B in (7.23) the estimate of B ^(r−1) s is assumed to contain the wanted extra factor δ _r by the inductive hypothesis. On the other hand, the two ex- pressions are much the same, the irrelevant difference being in the functions h _s−r and C _s−r ^(r−1) . Hence let us work out the estimate only for the second one. Denoting by x k ∈ C ⁿ and by c k (p) the coefficients of the Fourier expansions of X r and C s ^(r−1) , respectively, we obviously have

L X r C _s−r ^(r−1) = −i X

k∈Z ⁿ n

X

l=1

k l x k ∂c −k

∂p _l , L _hξ,qi C _s−r ^(r−1) =

n

X

l=1

ξ r,l ∂c 0

∂p _l .

Hence, taking into account that c _k (p) is a homogeneous polynomial of degree 2, we estimate the first expression as

L ^{X r} C _s−r ^(r−1) _1−d

r−1

≤ 2 X

k

|k| |x _k |   c _−k 

 (1−d _r−1 )̺

≤ 2 kC _r−1 ^(r−1) k _1−d

r−1

X

k

|x _k | e ^{(1−d r−1 )|k|σ} |k| e ^{−δ r |k|σ}

≤ 2

eδ _r σ kX r k _{1−d r−1}

C _s−r ^(r−1)

1−d r−1 ̺