Whiskered tori

Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions.

Abstract.

We present theorems which provide the existence of invariant whiskered tori in finite-dimensional exact symplectic maps and flows. The method is based on the study of a functional equation expressing that there is an invariant torus.

We show that, given an approximate solution of the invariance equation which satisfies some non-degeneracy conditions, there is a true solution nearby. We call this an a posteriori approach.

The proof of the main theorems is based on an iterative method to solve the functional equation.

The theorems do not assume that the system is close to integrable nor that it is written in action-angle variables (hence we can deal in a unified way with primary and secondary tori). It also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant.

The a posteriori formulation allows us to justify approximate solutions produced by many non-rigorous methods (e.g. formal series expansions, numerical methods). The iterative method is not based on transformation theory, but rather on succesive corrections. This makes it possible to adapt the method almost verbatim to several infinite-dimensional situations, which we will discuss in a forthcoming paper. We also note that the method leads to fast and efficient algorithms. We plan to develop these improvements in forthcoming papers.

Key words and phrases:
Keywords: whiskered tori, hamiltonian systems, small divisors, KAM theory
2000 Mathematics Subject Classification:
MSC 37J40
Contents

1. Introduction

The goal of this paper is to prove some results on persistence of invariant tori for symplectic and exact symplectic maps and flows.

We will assume that the motion on the torus is a Diophantine rotation and that the remaining directions are as hyperbolic as allowed by the symplectic structure (if the remaining directions are not void such tori are commonly called whiskered tori).

More precisely, as it is well-known, the preservation of the symplectic structure, together with the fact that the motion on the torus is a rotation, implies that the symplectic conjugate direction to the tangent of the torus is not hyperbolic. We will assume that the remaining directions in the tangent bundle of the phase space at the torus are spanned by a basis of vectors which contract exponentially in the future or in the past.

To make the previous statements more precise, we discuss first the case of maps. As we will show, results for flows can be readily deduced from the ones for maps. Given an exact symplectic map from an exact symplectic manifold into itself (for the purposes of this preliminary exposition, we will take to be an Euclidean manifold, even if we will indicate how to eliminate this restriction later), and a frequency vector , we seek an embedding satisfying

(1)

Equation (1) implies that the range of is invariant under . If is an embedding, we obtain that is a torus contained in , invariant by and that the dynamics on it is, up to a change of coordinates, just a rotation of rotation vector .

The main result of this paper will show that if we can find a function which satisfies some non-degeneracy assumptions and which satisfies (1) up to a sufficiently small error, then there is a true solution nearby.

Differentiating the functional equation (1) with respect to one gets

Geometrically, this shows that the tangent vector-field is invariant and does not grow or contract under iteration of the action by the map.

As we will see in more detail in Section 4.2.1, if the map preserves the symplectic form and is a solution of the invariance equation, there exists an analytic matrix valued function , such that

(2)

where is the matrix representation of the symplectic form and

As a consequence, cannot grow more that polynomially. Hence we obtain that the center subspace of is at least a -dimensional space spanned by and (we will show that range range because the image of the torus is a co-isotropic manifold.)

For approximately invariant systems, the previous identities are just approximate and this implies that the center direction is at least . We will assume that indeed the dimension of the center subspace is exactly . That is, we will assume that the tori we consider are as hyperbolic as allowed by the fact that the motion on them are rotations and that the system preserves the symplectic structure.

The main non-degeneracy assumptions on the approximate solution are a) that the other directions in are hyperbolic. That is, they are spanned by vectors which contract exponentially fast in the future or in the past. b) that there is some twist condition, that is, that the matrix in (2) is invertible.

We will use a KAM iterative method to show that, if we are given a function which solves (1) up to an error which is sufficiently small with respect to the properties of the non-degeneracy conditions a), b) above, then there is a true solution close to this approximate solution.

These results based on validating an approximate solution — which we call a posteriori — imply the usual persistence results (one can take as approximate solution of the modified system the exact solution for the original one). Nevertheless, the a posteriori results can be used for other purposes. For example a posteriori results can be used to validate solutions obtained through any method such as numerical approximations or asymptotic methods. The validation of Lindstedt series leads to estimates on their domain of analyticity. The paper [Mas05] for instance considers Lindstedt series of whiskered tori.

A posteriori results also lead automatically to Lipschitz dependence on paramaters and, with a bit more of work, to differentiable dependence on parameters. The a posteriori approach to KAM theorem was emphasized in [Mos66b, Mos66a, Zeh75, Zeh76]. There, it was pointed out that this a posteriori approach automatically allows to deduce results for finitely differentiable systems. We refer the reader to [dlL01c] for a comparison of different KAM methods.

In the present paper we deal with finite-dimensional maps and flows. In the forthcoming second part of it we consider coupled map lattices [FdlLS08]. The case of Partial Differential Equations, which can be treated in a similar way but involves technical difficulties, is postponed to a forthcoming paper [dlLS07].

Results on whiskered tori similar to the finite-dimensional ones of this paper have been considered several times in the literature. The first ones are [Gra74, Zeh76].

The approach in [Zeh76] — which also takes the a posteriori format — is based on [Zeh75] which consists of finding a change of variables which reduces the system to a normal form which obviously possesses an invariant torus. This change of variables is accomplished by applying a sequence of canonical transformations. The method of proof introduced here is not based on successive transformations but rather on successive corrections introduced additively. This makes the estimates easier to establish and it leads to efficient numerical implementations. In order to be able to solve the equations, we take advantage of some cancellations due to the preservation of the symplectic structure that were also pointed out in [JdlLZ99, dlLGJV05, dlL01c].

The method of [Zeh76] proves the result for periodic Hamiltonian flows. The result for diffeomorphisms is proved in [Zeh76] by interpolating diffeomorphisms by periodic flows and then applying the results for periodic flows. The proof we present here proceeds along the opposite route. We prove first the result for diffeomorphisms and, then, deduce the result for flows taking time-one maps. Giving a direct proof of the result for persistence of whiskered tori for maps has been suggested as somewhat desirable in J. Moser’s Mathematical review for [Zeh76]. We also provide such a direct proof. Of course, if one uses normal forms — as in [Zeh76] — it is natural to consider flows since the normal forms require only the study of the Hamiltonian function, which transforms very well. In the method presented here, the geometric cancellations are much more transparent in the case of diffeomorphisms.

Among other results for finite-dimensional systems, we call attention to [Val00], which uses a method similar to that of [Arn63]. The paper [Val00] has the advantage that it is a first order method (i.e. that each step of the Newton iteration requires to solve only one small divisors equation). As a consequence, the size of the gaps among tori in near integrable systems, the loss of regularity as a function of the Diophantine exponent and the required minimum regularity are smaller than these of the second order methods. A comparison between first and second order methods to prove KAM results can be found in [dlL01c]. The paper [Sor02] (see also the sketch in [Moe96]) uses a reduction to a normally hyperbolic manifold and then applies the standard KAM theorem for Lagrangian tori. Of course, since normally hyperbolic manifolds are in general only , the above method cannot produce or analytic tori. On the other hand, we note that the method of [Sor02, Moe96] leads to very good regularity conclusions for finite differentiable systems and also to good estimates on the measure occupied by the tori. We also call attention to [ZLL08, HLY06, LY05b, LY05a, Sev06, Sev99, Eli01, Eli89] which consider also tori with hyperbolic and elliptic directions and relax the twist conditions and the differentiability requirements. The paper [GG02] considers analytic perturbations which depend only on the angles of reducible tori satisfying a twist condition and uses a direct resummation method.

As compared with previous finite-dimensional results, the method presented here has the advantage that one does not need to assume that the hyperbolic bundles are trivial (and much less that the motion in the hyperbolic directions is reducible to a constant linear map). Tori with non-trivial invariant bundles appear naturally in one parameter families after crossing a resonance, see [HdlL07].

Also, we do not need to assume that the system is given in action-angle coordinates, something which is convenient if we are working in situations when the action-angle coordinates are singular. For instance, in the study of diffusion one is lead naturally to the study of whiskered tori near resonances (see [DdlLS03, DdlLS06]). In this case, the action-angle variables are singular and avoiding its use leads to better estimates.

For symplectic ODE’s we will also prove a translated tori theorem. From this general version we will deduce the results for exact symplectic ODE’s using a vanishing lemma. We note that the approach of proving a translated torus theorem was introduced in [Rüs76a] in the one degree of freedom case.

The method presented here lends itself to a very efficient numerical implementation (see [HdlLS08]). The only functions to be considered are functions with a number of variables equal to the dimension of the torus itself (independently of the number of variables of the ambient space). Of course, when studying infinite-dimensional systems — PDE’s or coupled map lattices or chains of oscillators — studying functions with the number of variables of the phase space is prohibitive. When implementing our method, if we discretize the tori by Fourier coefficients, the algorithm presented here only requires storage of order of and a Newton step takes only order of operations using the Fast Fourier Transform. This seems to be significantly faster than other algorithms. Actual implementations are now being pursued and will be the subject of a forthcoming paper (see [HdlLS08]). We refer the reader to [HdlL06b, HdlL06a, HdlL07] for analysis and implementation of related algorithms.

2. Definitions and notations

Before presenting the basic ideas and the results of our method, we introduce some notations and definitions which are useful for our purposes. All definitions are rather standard and we collect them here mainly to set the notation.

2.1. Diophantine vectors

In the study of invariant tori one needs an arithmetic condition over the frequency vector. In the case of maps the notion of Diophantine vector is the following.

Definition 2.1.

Given and , we define as the set of frequency vectors satisfying the Diophantine condition:

where means scalar product, and are the coordinates of . We will say that is Diophantine.

For vector-fields the corresponding notion is the following.

Definition 2.2.

Given and , we define as the set of frequency vectors satisfying the Diophantine condition:

with the same notation as in Definition 2.1.

The two conditions are closely related since with if and only if for some . The geometric and measure properties of the sets of Diophantine vectors have been extensively studied. These results translate immediately into statements about the abundance of KAM tori.

Definition 2.3.

Let and . We denote its average on the -dimensional torus, i.e.

Definition 2.4.

Given we introduce the rotation over of rotation vector :

2.2. Functional spaces, functions and operators

We will denote the complex extension of the torus of width , i.e.

(3)

We denote by the supremum norm on or . The sup norm makes several estimates independent of the dimension of the manifold, which are useful when considering infinite-dimensional problems. However on we will use the norm Furthermore, for finite differentiability purposes, we consider the following norms: given analytic, with bounded derivatives in a complex domain , and we introduce the following -norm for

Let be the set of continuous functions on , analytic in the interior of with values on a manifold , which is assumed to be Euclidean. We endow the space with the usual supremum norm

We have that is a Banach space. In particular, .

We also recall the following convexity property (see [Rud74, Lemma 12.8]).

Proposition 2.5.

Let and assume that . Then, for every we have:

(4)

In particular, taking and ,

(5)

We will also consider spaces of continuous functions on analytic in its interior and taking values on finite-dimensional vector spaces, for instance in spaces of matrices. When endowed with the supremum norm, these function spaces are also Banach spaces.

In particular, we will also need some norms of linear maps on the tangent space with , where is an embedding. More concretely, let be a continuous linear operator from into itself depending on the variable . Then we define by

3. Setting of the problem and results

3.1. Geometric setup

We will consider the Euclidean manifolds and . In the second case we can consider the universal covering of and lift the maps defined on to maps , defined on , such that , where is the canonical projection. Even if we pass to the covering we will use the symbol to refer to the manifold. These manifolds obviously admit complex extensions by considering and . As we will see, these different possibilities are convenient when we consider tori whose embeddings are topologically different. For example, tori which are contractible to tori with different dimensions. We will use the same symbol for the complex extension of the manifold or its covering.

For convenience of notation, we will endow these manifolds with the standard Riemannian metric, even if this may not be natural for the problem at hand. For us, the metric will only play the role to measure sizes and therefore any equivalent metric will give a similar result. The standard metric will have the advantage that it will allow us to use matrix notation for adjoints. In matrix notation, thinking of vectors as column vectors, we can write . On the other hand, we note that the length of vectors will always be the supremum norm and the norm of matrices will be the operator norm associated to the supremum norm on vectors. Of course, for finite-dimensional problems the supremum norm is equivalent to the Euclidean norm.

We will assume that the Euclidean manifold has an analytic exact symplectic form with primitive , i.e. . For each , let be the isomorphism such that

where is the Euclidean product on .

We will not assume that has the standard form. We do not assume either that induces an almost-complex structure on . This generality is useful in some applications, (celestial mechanics, numerics, …) when we use some system of coordinates — e.g. polar coordinates — which lead to non-standard symplectic matrices.

Remark 3.1.

As we will see in the proof, we are not using much the Euclidean structure of the manifolds. In Section 7.6, we will present the modifications needed to work on other manifolds.

More precisely, we will show that it is possible to work out the proof in a neighborhood of the zero section of a bundle . The bundle will be shown to be trivial (as a consequence of the fact that the motion on the torus is a rotation, the preservation of the symplectic structure and the fact that the dimension of the center space is , see Section 4.2.2), but the others bundles — which correspond to the hyperbolic directions need not be trivial.

We note that equation (1) is geometrically natural since it can be formulated in any manifold.

In the following write up the Euclidean structure enters in two ways: one is a purely notational one and can be eliminated at the price of a typographical nightmare. When we only have approximate solutions, we will denote the error just as rather than . We will also compare vectors in with vectors in . This can be done by introducing connectors as in [HPPS70], so that what we denote is really . See Definition 3.9 and the discussion after (particularly equation (17)).

A second, and more serious way in that the Euclidean space enters is that, to implement the iterative step in KAM theory, we will use Fourier series. This certainly requires that the functions take values in a vector space. Fortunately, this happens only in the center directions. In the hyperbolic directions there are geometrically natural ways to solve the iterative equation. This is why we are requiring that the center bundle is trivial, but we do not need the triviality of the hyperbolic bundles.

Of course, the fact that we work in a set as above is no loss of generality because, if there is a whiskered torus, by the tubular neighborhood theorem, we can identify a neighborhood of the torus with a neighborhood of the zero section of the normal bundle.

3.2. Setting of the problem and results for maps

The main purpose of the theory we are going to develop is to construct invariant tori for exact symplectic maps. We recall the following

Definition 3.2.

Let be an exact symplectic manifold. A map from into itself is exact symplectic if there exists a smooth function on such that

In particular, every exact symplectic map is symplectic, i.e. .

Heuristically, our problem is the following: let be an exact symplectic map and . We want to construct an invariant torus for such that the dynamics of on it is conjugated to . To this end, we search for an embedding in such that for all , satisfies the functional equation

(6)

Notice that if (6) is satisfied, the image under of a point in the range of will be also in the same range. Hence, since is an embedding, the range of will be an invariant torus.

The assumptions of our results will be that we are given a mapping that satisfies (6) up to a very small error and which satisfies some non-degeneracy and hyperbolicity assumptions. We will prove that then, there is a true solution of (6) close to . We will also prove that the solution of (6) is unique up to composition on the right with translations.

The exactness of the map is important for the existence of a solution to (6). It is easy to construct examples of symplectic non-exact symplectic maps without invariant tori. For instance, consider with the standard symplectic structure. The translation in the -direction is a symplectic non-exact symplectic map without any invariant torus.

To construct the desired invariant torus, we consider a parameter and introduce a translation term in equation (6) depending on .

We then consider the following functional equation, where is a suitably chosen function of taking values in matrices and whose unknowns are both and

(7)

The introduction of this parameter will allow us to sidestep several technical complications and then we will show that, since is exact symplectic, the geometry implies that . The fact that the dimension of the parameter is is important for our purpose. We also mention that it is possible to use the parameter to weaken non-degeneracy conditions by taking instead of . In such a case, is a matrix.

Remark 3.3.

The introduction of the parameter is also motivated by numerical calculations (see [HdlLS08]). It leads to more stable computations. More importantly, it is useful in the numerical computation of secondary tori (i.e. tori generated by resonances, which have some contractible directions).

We go through a KAM technique to prove the existence of such a pair . To this end, we introduce the operator

(8)

where

(9)

is a function defined on and where stands for an approximate whiskered torus. We will write instead of its explicit form in many of the following results. As we will see later, the important property of is that translations along the direction of can change the cohomology of the pushforward in the center directions.

The method is based on a careful study of the linearization (around a given pair ) of the operator . We will show that this linear operator is approximately invertible in a suitable sense.

For that, we have to introduce several non-degeneracy conditions.

Definition 3.4.

Given and an embedding we say that the pair is non-degenerate for the functional equation (7) (and we denote ) if it satisfies the following conditions:

  • Spectral condition: the tangent space has an invariant splitting for all ,

    (10)

    where , and are the stable, center and unstable invariant spaces respectively, i.e.

    This splitting is analytic in . To this splitting we associate the projections , and respectively, which are analytic with respect to .

    Moreover, the splitting (10) is characterized by asymptotic growth conditions (co-cycles over ): there exist , such that , and such that for all and

    (11)
    (12)

    and

    (13)
  • Furthermore, we assume that the dimension of the center subspace is .

    That is, the torus is as hyperbolic as allowed by the symplectic structure and there are no elliptic directions in the normal direction.

  • Twist condition: We introduce the notation

    (14)

    Assume that the averages on of the matrices

    (15)

    and

    (16)

    are non-singular.

Remark 3.5.

With a view to applications, we note that in Proposition 5.2, we will show that we can deduce the existence of an invariant splitting from the existence of an approximately invariant one which satisfies the hyperbolicity Conditions (11)–(13). Consequently, Definition 3.4 can be verified with a finite precision calculation on a given numerical approximation. We anticipate that the basic idea is that, if we can verify that for some operator we have for some , it follows that for all . This gives a way to obtain all inequalities from finite computations.

Remark 3.6.

Note that since is an embedding — hence is one to one for all — and we have that is invertible for all .

Remark 3.7.

If we take , becomes and hence one of the twist conditions becomes automatic because, under the smallness assumptions, and is invertible. Indeed, assume that . Then . This last expression is approximately

which implies for all . Since is one to one for all we obtain . Hence the condition on the invertibility of is just a quantitative statement of the fact that is indeed an embedding. The condition on is a twist condition.

Remark 3.8.

Note that if the torus was exactly invariant (i.e. ) then

so that Conditions (11)–(13) are the usual growth conditions in the theory of normally hyperbolic manifolds (see [Fen74, HPS77, Pes04]). Of course, for our applications, we only assume that the tori are approximately invariant.

When the manifolds are Euclidean, the conditions (11)–(13) make perfect sense. Nevertheless, if the phase space is a general manifold , we have . If , then, we should write the conditions (11)-(13) using connectors (see [HPPS70]).

We recall that

Definition 3.9.

A connector is an isomorphism from to , defined when is small enough, such that and , when both make sense.

A concrete way of implementing the connectors is to take parallel transport along the shortest geodesic joining (equivalently, the differential of the exponential map).

In the case that we formulate the result in a general manifold, (11) should be written

(17)

and analogously the others.

Remark 3.10.

The technical reason why we introduced the extra parameter in (7) is the following: in the iteration of the KAM scheme, one has to prove that some equations are approximately solved up to a quadratic error. To this end, we have to show that some averages are quadratic in the error. To avoid these technicalities, we introduce this parameter which allows us to cancel some terms in the equation so that we can reach the suitable approximate solution (See Propositions 4.18 and 4.19). Then we use the exact symplecticness of the map to keep the parameter under control.

We can now state our main theorem, which provides the existence of a solution to the functional equation (6) with exact symplectic, provided we are given a sufficiently approximate one.

Theorem 3.11.

Let for some . Assume that

  1. is an exact symplectic map and is an open connected set, which we will assume without loss of generality has a smooth boundary.

  2. (the embedding is non-degenerate) in the sense that it satisfies the spectral condition in Definition 3.4 and the average on of the matrices and are non-singular, where and are as and in Definition 3.4 with .

  3. The map is real analytic and it can be extended holomorphically to some complex neighborhood of the image under of :

    for some and such that is finite.

Denote the initial error. Then there exists a constant depending on , , , , , , , , and the norms of the projections such that, if satisfies the estimates

(18)

and

where is fixed, then there exists an embedding such that

Furthermore, we have the following estimate

(19)
Remark 3.12.

The previous theorem provides a construction of whiskered tori without assuming the existence of action-angle variables for the original system. Moreover, the method of proof does not involve the sequence of transformations by symplectomorphisms, which is often used to prove this kind of results, but hard to implement numerically.

Remark 3.13.

It is important to remark that the non-degeneracy conditions we use in Theorem 3.11 depend only on the approximate solution under consideration. As one can see, Definition 3.4 only depends on averages of the approximately computed solutions. This latter fact is useful in the validation of numerical computations. Indeed, numerical computations provide an approximate solution and this is the only information that is available. The non-degeneracy conditions needed to apply Theorem 3.11 can be verified by straightforward computations on the numerical approximation.

This leads directly to the so-called small twist theorems. See Section 7.3 and in particular Proposition 7.1 and the subsequent comments for more details on the dependence of the constants on the non-degeneracy assumptions.

After introducing an additional term in the functional equation (6), namely

and performing a KAM iteration on , the final task consists of proving that using the geometry. This is done by using the exact symplecticness of and a suitable representation of the center subspace. Indeed, the center subspace in , which will be shown to be non-trivial, will be very close to the vector space spanned by and its symplectically conjugate .

3.3. Uniqueness

A natural question to ask is whether the embedding provided by Theorem 3.11 is unique. Notice that if is a solution of (6), for any , is also a solution, hence one can only hope for uniqueness up to a composition with a translation on the right.

The following theorem provides a local uniqueness result. We will see in the next section that there is a simple general argument that shows that uniqueness results allow us to deduce results for flows from results for diffeomorphisms.

Theorem 3.14.

Let be exact symplectic and analytic in . Let for some . Assume with are two solutions of equation (6) such that . Then there exists a constant depending on , , , , , , , , such that if for some the norm satisfies

(20)

with , there exists a phase such that in . Moreover .

The proof of this theorem is postponed to Section 6.

3.4. Result for flows

As a by-product of the previous uniqueness theorem, we get a result on the existence of invariant whiskered tori for flows. This follows from a time-one map argument (see [Dou82]). The argument we present here comes from [dlLGJV05, CFdlL03].

Theorem 3.15.

Let for some . Let be the flow generated by a finite-dimensional analytic exact symplectic vector-field

where . Assume that there exists a time and an embedding for some such that for all . Then for all time , we have

Proof.

If we have , then for all this yields

By Theorem 3.14, if is sufficiently small, which is achieved if is sufficiently small, this implies that there exists a phase such that . From the flow property and the fact that is one to one, we have . We now prove that the function is continuous. The map from into its image is one to one and continuous over a compact (for the topology of ). Then its inverse is continuous. This leads to the continuity of the function . Using this fact and the additivity condition we deduce, that for small enough, for some . Then in this case we have

(21)

Since both sides of (21) are analytic with respect to we obtain the result for all . Putting we get . Expression (21) shows that the torus is invariant by the flow. Since the torus is compact, the flow on it is defined for all and hence (21) holds for all . This ends the proof. ∎

In Section 9, we will give a more precise version of this result and a direct proof (i.e. a proof which does not pass through a reduction to a time- map). This is useful since the method of proof leads to numerical algorithms for differential equations. The direct proof can also be used as a model for results for some ill-posed partial differential equations (see [dlLS07]).

4. The linearized operator

In this section, we describe the inductive step of the procedure. As most of the KAM proofs, it will be a modification of the classical Newton method.

Using the Taylor theorem, given an approximate solution, we write

and, following the idea of Newton’s method, we look for such that is quadratically small so we are lead to consider the following equation

(22)

where is a pair satisfying approximately equation (7) with an error with . Using the definition of the operator in (8), we see that the derivative of the operator can be written more explicitly as:

The study of the Newton equation (22) is mainly done in three steps:

  • One projects equation (22) on the hyperbolic space and the center space, by using the invariant splitting (see Definition 3.4).

  • One reduces the equation of the projection on the center subspace to two classical small divisors equations. Thanks to a suitable change of coordinates on the tangent space (which does not use action-angle variables) these equations are then solved approximately (i.e. up to quadratic error) by using the extra variable .

  • One solves (with “tame” estimates) the equations corresponding to the projections onto the stable and unstable invariant subspaces, by using the conditions on the co-cycles over .

Remark 4.1.

We note that the equation on the center subspace will not be solved exactly. We will just solve it up to quadratic errors. The reason is that the change of variables mentioned in the above discussion will be constructed taking advantage of approximate identities obtained by differentiating with respect to the equation for the initial error and applying geometric identities. The procedure of comparing the linearized Newton equation with the equations that appear taking derivatives is very common in KAM theory. It is certainly used systematically in [Mos66b, Mos66a, Zeh76]. See [Zeh75, Section 5] for some remarks on the relation of these identities with a group structure of conjugacy problems. We note that some of these remarks in the above references work also for some semi-conjugacy problems.

Of course, the above-mentioned strategy uses the non-degeneracy assumptions. In subsequent sections, we will show that these assumptions are changed only by a small amount, so that the procedure can be iterated.

The main goal of this section is to prove the following result.

Lemma 4.2.

Consider the linearized equation

(23)

Then there exists a constant that depends on , , , , , , , and the hyperbolicity constants such that assuming that satisfies

(24)

we have

  1. There exists an approximate solution of (23), in the following sense: there exits a function such that solves exactly

    with the following estimates: for all

    (25)
    (26)
    (27)
    (28)
  2. If and solve the linearized equation in the previous approximate sense, then there exists such that for all

    (29)
Remark 4.3.

The form of the previous inductive lemma corresponds very closely to Zehnder’s implicit function theorem in [Zeh75]. Once Lemma 4.2 is proved, we then follow the strategy in [Zeh75]. The most crucial step is the verification of how the hypothesis of hyperbolicity are changed when the embedding changes in the iterative step.

More precise information on the dependence of the constants on the non-degeneracy conditions will be provided in Proposition 7.1. We anticipate that, roughly speaking , the constants can be bounded by universal powers of the non-degeneracy constants. We postpone the precise formulation since it will involve some notations that will be developed along the proof. This power dependence on the constants has some applications to the study of tori close to resonance and to small twist theorems.

We will need the following classical proposition (see [Rüs76a], [Rüs76b], [Rüs75], [dlL01c]) which provides existence of a solution together with estimates for small divisors equations.

Proposition 4.4.

Let and assume the mapping is analytic on and has zero average. Then for any the difference equation

has a unique zero average solution , real analytic on for any . Moreover, we have the estimate

(30)

where only depends on and the dimension of the torus .

Remark 4.5.

It is important for our purposes to have estimates independent of the dimension of the manifold since in a followup paper [FdlLS08] we apply the procedure of this paper in an infinite-dimensional context.

The independence of the estimates on the number of dimensions comes from the fact that we consider the supremum norm and the equation is solved component-wise.

4.1. Geometric considerations

Isotropic character of the torus

We start by recalling the definition of isotropy.

Definition 4.6.

Let be a symplectic manifold. A submanifold of is isotropic if , where is the orthogonal space of with respect to the -form .

We formulate in our framework the well-known fact that a torus supporting an irrational rotation is isotropic. The manifold is isotropic if the pull-back vanishes for all . In other words, noting

for all , the isotropic character is equivalent to

for all . We first deal with the case of an exact solution of (6) (see Lemma 4.7). The approximate case is the purpose of Lemma 4.8. We note that the fact that exactly invariant tori are isotropic manifolds remains true for all irrational rotations and is well known [Zeh76]. The fact that approximately invariant tori carrying an irrational rotation are approximately isotropic seems to require that the rotation is Diophantine, see [dlLGJV05]. For the sake of completeness, we present the simple proofs of both results.

Lemma 4.7.

Assume that is exact symplectic, satisfies (6) and is rationally independent. Then is identically zero.

Proof.

Since is symplectic we have

Consequently, this yields

Since is rationally independent, is ergodic and this implies that is constant and so is . Using that is exact symplectic, we have that and, the only constant form which is exact is zero.

Similarly, a computation shows that has the form for some matrix . Since the average on of is zero, we get the result. ∎

Lemma 4.8.

Assume that is an exact symplectic manifold, is analytic and symplectic. Let be real analytic on the complex strip for some and such that . Assume also that and denote

Then there exists a constant depending on , , , , such that for all we have

(31)
Proof.

We want to estimate the norm of the matrix . Recalling , one gets

Performing the same computations as in [dlLGJV05], this leads to the following equation

where is a function on such that (here we just use Cauchy estimates)

We now make use of Proposition 4.4 to complete the proof. ∎

Recall that we are assuming that is an embedding. Hence the range of is -dimensional.

Vanishing lemma

This section is devoted to an estimate which allows to control the extra parameter through the iterative step. We consider the functional equation

where is exact symplectic (see Definition 3.2) and .

Recall that and . Note that the term is very close to