For perturbations of integrable Hamiltonians systems, the Nekhoroshev theorem shows that all solutions are stable for an exponentially long interval of time, provided the integrable part satisfies a steepness condition and the system is analytic. This fundamental result has been extended in two distinct directions. The first one is due to Niederman, who showed that under the analyticity assumption, the result holds true for a prevalent class of integrable systems which is much wider than the steep systems. The second one is due to Marco-Sauzin but it is limited to quasi-convex integrable systems, for which they showed exponential stability if the system is assumed to be only Gevrey regular. If the system is finitely differentiable, the author showed polynomial stability, still in the quasi-convex case. The goal of this work is to generalize all these results in a unified way, by proving exponential or polynomial stability for Gevrey or finitely differentiable perturbations of prevalent integrable Hamiltonian systems.
1. Consider a near-integrable Hamiltonian system, that is a perturbation of an integrable Hamiltonian system, which is of the form
Here are angle-action coordinates, and is a small perturbation, of size , in some suitable topology defined by a norm . In the absence of perturbation, that is when is zero, the action variables are integrals of motions and all solutions are quasi-periodic. Therefore it is a natural question, which is in fact motivated by concrete problems of stability in celestial mechanics, to study the evolution of the action variables after perturbation, that is for but arbitrarily small.
2. If the system is analytic and if satisfies a steepness condition, which is a quantitative transversality condition, it is a remarkable result due to Nekhoroshev ([Nek77], [Nek79]) that the action variables are stable for an exponentially long interval of time with respect to the inverse of the size of the perturbation: one has
for some positive constants and provided that the size of the perturbation is smaller than a threshold . This is a major result in the theory of perturbations of Hamiltonian systems, and it gives complement to two previous and equally important results. The first one is that “most” solutions (in a measure-theoretical sense) are quasi-periodic and hence stable for all time, in the sense that
for some positive constant , provided that the system is sufficiently regular (analytic, Gevrey or even for ), satisfies a mild non-degeneracy assumption and is smaller than a threshold . This is the content of KAM theory, see [Kol54] for the original statement and, for instance [Rüs01], [Sal04] and [Pop04] among the enormous literature, for various improvements regarding the hypotheses of non-degeneracy and regularity. Hence Nekhoroshev estimates give new information for solutions living on the phase space not covered by KAM theory, and even though the latter has a small measure, it is usually topologically large (at least for ). The second important previous result is that there exist “unstable” solutions, satisfying
This was discovered by Arnold in his famous paper [Arn64], and it has become widely known as “Arnold diffusion”. Even though this phenomenon has been intensively studied, very little is known. Here Nekhoroshev estimates give an exponentially large lower bound on the time of instability , explaining (in part) why such instability properties are so hard to detect.
3. Now returning to Nekhoroshev estimates, the original proof is rather long and complicated. It is naturally divided into two parts. The first part, which is analytic, is the construction of general resonant normal forms, up to an exponentially small remainder (this is where the analyticity of the system is used), on local domains of the action space where one has a suitable control on the so-called small divisors. Then, the second part, which is geometric, consists in the construction of a partition of the action space where one can use such normal forms. This is where the steepness of the integrable system enters, basically it rules out the existence of solutions which cannot be controlled by these normal forms, so eventually the exponentially small remainders easily translate into an exponentially long time of stability for all solutions.
4. It has been noticed by the Italian school ([BGG85], [BG86]) that using preservation of energy, the geometric part of the proof can be simplified for the simplest steep Hamiltonians, namely strictly convex or strictly quasi-convex Hamiltonians (recall that quasi-convexity means that the energy sub-levels are convex subsets). Then, much work have been devoted to this special case. In particular, Lochak introduced in [Loc92] a new method leading in particular to an extremely simple and elegant proof of these estimates under the quasi-convexity assumption. His approach only relies on averaging along periodic frequencies, which enables him to construct special resonant normal forms (periodic frequencies are just resonant frequencies of codimension-one multiplicities), and the use of the most basic result in simultaneous Diophantine approximation, namely Dirichlet’s theorem, to cover the whole action space by domains where such special normal forms can be used. This also brought to light the surprising phenomenon of “stabilization by resonances”, which implies that the more resonant the initial condition is, the more stable (in a finite time-scale) the solution will be. This method had several applications, and an important one, that we shall be concerned here, was the extension of these stability estimates for non-analytic systems. Indeed, using Lochak’s strategy, it was shown in [MS02] that the exponential estimates are also satisfied for Gevrey regular systems, and in [Bou10b] it was proved that polynomial estimates hold true if the system is only of finite differentiability (this is obviously the best one can expect under such a weak regularity assumption). Let us point out that for non-analytic systems, the construction of these normal forms, which is the only new ingredient one has to add since the geometric part of the proof is insensible to the regularity of the system, is more difficult than for an analytic system (this is especially true for Gevrey systems). Indeed, one cannot simply work with -norms (on some complex strip, then by the usual Cauchy estimates one has a control on all derivatives on smaller complex strips), so one has to work directly with all the derivatives (and moreover keep a control on the growth of these derivatives in the Gevrey case). In [MS02] and [Bou10b], such normal forms were constructed but only because it was enough to consider periodic frequencies, which boils down to what is classically known as a one-phase averaging.
5. However, all these results were restricted by the quasi-convexity hypothesis. A study of Nekhoroshev estimates under more general assumptions on the integrable part has been initiated by Niederman. First, in [Nie04], he introduced new geometric arguments based on simultaneous Diophantine approximation, leading to a great simplification in the geometric part of the proof of Nekhoroshev’s result under the steepness condition. Then, in [Nie07], he realized that this method allows in fact to obtain the result for a much wider class of unperturbed Hamiltonians, which he called “Diophantine steep”, and which are prevalent (in the sense of Hunt, Sauer and York, recall that prevalence is a possible generalization of full Lebesgue measure for infinite dimensional linear spaces). However, the analytic part of Niederman’s proof was still based on averaging along general frequencies and hence required the construction of general resonant normal forms (which was taken from [Pös93]). In fact, in the non-convex case, a simple averaging along a periodic frequency, which corresponds to studying the dynamics in the neighbourhood of a resonance of codimension-one multiplicity, cannot be enough since solutions will necessarily explore resonances associated to different, and possibly all, multiplicities. But then in [BN10] we were able to construct normal norms associated to any multiplicities by making suitable composition of periodic averagings, with periodic vectors which are independent and sufficiently close to each other. This was an extension of Lochak’s method, in the sense that no small divisors were involved, only (a composition of) periodic averagings and simultaneous Diophantine approximation were used. This proof was not only simpler than the previous one, but also opened the way to several applications. For instance, in [Bou10a], it was shown how one can easily obtain more general results of stability in the vicinity of linearly stable quasi-periodic invariant tori.
6. The aim of this paper is to extend the above results by proving stability estimates for Hamiltonian systems with a prevalent integrable part, but which are not necessarily analytic. In the Gevrey case, this will lead to exponential estimates of stability for perturbation of a generic integrable Hamiltonian, as stated below.
For , consider an arbitrary -Gevrey integrable Hamiltonian defined on an open ball in . Then for almost any , the integrable Hamiltonian is exponentially stable.
This will be a direct consequence of Theorems 2.2 and Theorem 2.4 below. This result generalizes the main results of [Nie07] and [BN10] which were restricted by the analyticity assumption (), and the main stability result of [MS02] which was restricted by the quasi-convexity assumption (our condition on the integrable part is much more general than quasi-convexity). In the finitely differentiable case, we will obtain polynomial estimates of stability for perturbation of a generic integrable Hamiltonian.
For , consider an arbitrary integrable Hamiltonian defined on an open ball in . Then for almost any , the integrable Hamiltonian is polynomially stable.
2 Main results
In order to state our results, we now describe our setting more precisely. We let be the open ball of , centered at the origin, of radius with respect to the supremum norm . Our phase space will be the domain .
1. Let us first explain our prevalent condition on the unperturbed Hamiltonian , which comes from [BN10]. Let be the Grassmannian of all vector subspaces of of dimension . We equip with the Euclidean scalar product, stands for the Euclidean norm, and given an integer , we define as the subset of consisting of those subspaces whose orthogonal complement can be spanned by vectors with , where is the -norm.
A function is said to be Diophantine Morse if there exist and such that for any , any and any , there exists (resp. ), an orthonormal basis of (resp. of ), such that the function defined on by
satisfies the following: for any ,
for any .
In other words, for any , we have the following alternative: either or for any . This technical definition is basically a quantitative transversality condition which is stated in adapted coordinates. It is inspired on the one hand by the steepness condition introduced by Nekhoroshev ([Nek77]) where one has to look at the projection of the gradient map onto affine subspaces, and on the other hand by the quantitative Morse-Sard theory of Yomdin ([Yom83], [YC04]) where critical or “nearly-critical” points of have to be quantitatively non degenerate. It is in fact equivalent to the condition introduced by Niederman in [Nie07]: there the author considered the subset of consisting of those subspaces which can be spanned by vectors with , but one can check that is included in (similarly, is included in ). Hence we will stick with the terminology “Diophantine Morse” introduced in [Nie07], and the equivalent term “Simultaneous Diophantine Morse” introduced in [BN10] will not be used any more.
The set of Diophantine Morse functions on with respect to and will be denoted by , and we will also use the notations
Let and . Then for Lebesgue almost all , the function belongs to .
We already mentioned that there is a good notion of “full measure” in an infinite dimensional vector space, which is called prevalence (see [OY05] and [HK10] for nice surveys), and the previous theorem has the following immediate corollary.
For , is prevalent in .
2. Now let us introduce our regularity assumption, starting with the Gevrey case. Given and , a real-valued function is -Gevrey if, using the standard multi-index notation, we have
where is the usual supremum norm for functions on . The space of such functions, with the above norm, is a Banach algebra that we denote by , and in the sequel we shall simply write . Analytic functions are a particular case of Gevrey functions, as one can check that is exactly the space of bounded real-analytic functions on which extend as bounded holomorphic functions on the complex domain
where is the imaginary part of , the supremum norm on and the associated distance on .
3. Therefore we shall consider a Hamiltonian
Our main result in the Gevrey case is the following.
Let be as in ( ‣ 2), and assume that the integrable part belongs to , with and . Let us define
Then there exists a constant , depending on and , such that if , for every initial action the following estimates
For , we exactly recover the main theorem of [BN10] including the value of the exponents, therefore the latter result is generalized to the Gevrey classes. Moreover, quasi-convex Hamiltonians are a very particular case of our class of Morse Diophantine Hamiltonians, hence the stability result of [MS02] is also generalized, but not with the same exponents (we did not try to improve our exponents). Let us now explain several consequences of our result.
First, note that the only property used on the integrable part to derive these estimates is a specific steepness property, therefore the proof is also valid assuming a Diophantine steepness condition as in [Nie07], which is much more general than the original steepness condition of Nekhoroshev. Indeed, the class of Diophantine Morse functions (and a fortiori the class of Diophantine steep functions) contains fairly degenerate Hamiltonians, as for instance linear Hamiltonians with a Diophantine frequency, which of course are far from being steep. As a direct consequence, our main theorem also gives an alternative proof of exponential stability in the neighbourhood of a Gevrey Lagrangian quasi-periodic invariant torus, a fact which was only recently proved by Mitev and Popov in [MP10] by the construction a Gevrey Birkhoff normal form.
Then, as it was proved in [Bou10a], the method we are using is relatively intrinsic and does not depend much on the choice of coordinates. This remark is particularly useful when studying the stability in the neighbourhood of an elliptic fixed point, and more generally in the neighbourhood of a linearly stable lower-dimensional torus, under the common assumptions of isotropicity and reducibility (which are automatic for a fixed point or a Lagrangian torus). As in [Bou10a], one can easily prove results of exponential stability in the Gevrey case under an appropriate Diophantine condition, therefore extending the results of exponential stability obtained in [Bou10a] which were valid in the analytic case. This also gives an extension of the stability result of [MP10] which is only available for a Gevrey Lagrangian torus.
In [Bou10a], using the idea introduced by Morbidelli and Giorgilli ([MG95]) to combine Birkhoff normal forms and Nekhoroshev estimates, we also had results of super-exponential stability under a Diophantine condition on the frequency and a prevalent condition on the formal series of Birkhoff invariants. Using the Gevrey Birkhoff normal form of [MP10], we can also extend this super-exponential stability result to Gevrey classes, but only for a Lagrangian torus. For a more general linearly stable torus (isotropic, reducible), the existence of a Gevrey Birkhoff normal form undoubtedly holds true but it is still missing.
As a last remark, we would like to point out that one can also extend a fairly different result of stability, which is due to Berti, Bolle and Biasco ([BBB03]). This concerns perturbations of a priori unstable Hamiltonians systems, which have intensely studied since instability properties in this context are much more simple to exhibit. In the analytic case, if the size of the perturbation is , it was proved in [BBB03] that the optimal time of instability is . The upper bound follows from a specific construction of an unstable solution, while the lower bound was a consequence of a stability result, where the analyticity of the system was only necessary to apply Nekhoroshev estimates both in the quasi-convex and steep case on certain regions of the phase space (but because of the presence of “hyperbolicity”, the global stability time is far from being exponentially large). In [BP10], we introduced yet another technique (which pertains more to dynamical systems, as opposed to the variational arguments of [BBB03]) to construct a solution for which , but only the Gevrey case. Now having at our disposal Nekhoroshev estimates in the Gevrey case for both quasi-convex and steep integrable systems, this implies that the lower bound can also be obtained and that the time of instability is also optimal in the Gevrey case (in fact, using Theorem 2.5 below, this is also true if the system is for large enough). This justify the optimality we claimed in [BP10].
4. Let us now explain our result in the finitely differentiable case. Here we assume that is of class , i.e. it is -times differentiable and all its derivatives up to order extend continuously to the closure . In order to have non-trivial results, we shall assume a minimal amount of regularity, that is and it will convenient to introduce another parameter of regularity satisfying . We denote by the space of functions of class on , which is a Banach algebra with the norm
where we have used the standard multi-index notation and where still denotes the usual supremum norm for functions on . Once again, for simplicity, we shall only write .
5. So now we consider a Hamiltonian of the form
Our main result in the finitely differentiable case is the following.
Let be as in ( ‣ 2), assume that the integrable part belongs to , with and , and that for some . Let us define
Then there exists a constant , depending on and , such that if , for every initial action the following estimates
The above theorem extends the main result of [Bou10b], which was only valid for quasi-convex integrable Hamiltonians. Let us point out that in this result (as in the one contained in [Bou10b]) we have decided to consider only the case of integer values of , but the results can also be extended to real values (that is, to Hölder spaces) and this would have given a more precise exponent of stability in terms of the regularity of the system, but we decided not to pursue this further.
6. Let us now conclude with some notations that we shall use throughout the text.
First, we have define norms for Gevrey and functions, but we shall need corresponding norms for vector-valued functions (in particular for diffeomorphisms). Hence given a vector-valued function , and , we say that is -Gevrey if , for , and we will write . Similarly, is of class if , for , and we will write .
Then, to avoid cumbersome expressions, we will replace constants depending only on and (resp. on and ) in the Gevrey case (resp. in the case) with a dot. More precisely, an assertion of the form “there exists a constant depending on the above parameters such that ” will be simply replaced with “”, when the context is clear.
3 Analytical part
In this part, we shall describe and prove some normal forms that we will need for the proofs of Theorem 2.4 and Theorem 2.5. More precisely, Gevrey Hamiltonians will be considered in section 3.1 and finitely differentiable Hamiltonians in section 3.2, and eventually in section 3.3 we will explain the dynamical consequences of these normal forms.
But first we need to recall the following basic definition, which will be crucial to us.
A vector is said to be periodic if there exists a real number such that . In this case, the number
is called the period of .
The easiest example is given by a vector with rational components, the period of which is just the least common multiple of the denominators of its components. Geometrically, if is -periodic, an invariant torus with a linear flow with vector is filled with -periodic orbits.
3.1 The Gevrey case
As in [MS02], we shall start with perturbations of linear integrable Hamiltonians, for which we will obtain a global normal form (Lemma 3.3 below), and then the latter will be used to obtain local normal forms for perturbations of general Hamiltonians (Proposition 3.4 below).
1. Let be a -periodic vector, and let be the linear integrable Hamiltonian with frequency . In the following, we shall consider a “large” positive integer and a “small” parameter , which will eventually depend on . We shall also use a real number independent of , to be fixed below. The following result is due to Marco and Sauzin ([MS02]).
Lemma 3.2 (Marco-Sauzin).
Consider the Hamiltonian defined on , with and . Assume that
Then there exist , for some constant , and an -Gevrey symplectic transformation
with such that
with and the estimates
One can choose the constant . The statement above is exactly Proposition 3.2 in [MS02], where the authors state their result for , but it also holds trivially for smaller , that is . The use of this artificial parameter will make subsequent arguments easier.
It is perhaps useful to understand this lemma in the very special case where is the first vector of the canonical basis of : the equality simply means that is independent of the first angle , and therefore the evolution of the first action component is only governed by the remainder .
2. Now we are going to make suitable “compositions” of the above lemma, but first we shall explain heuristically what we are planning to do formally in the sequel.
So we consider another periodic vector , with period , which is independent of , and we let . If the suitable hypotheses are met, by Lemma 3.2 we can transform , where the size of is of order , into
with of order and of order . Now if is close enough to , that is , and if , we can write
where satisfies and its size is of order . For a moment, let us forget about , which is already exponentially small with respect to , and consider as a perturbation of . Under the suitable assumptions, we can apply once again Lemma 3.2 and find a transformation that sends into , where is exponentially small with respect to and .
Now the key point is the following: as satisfies , and also satisfy , hence . Indeed, it is enough to show that, denoting the Hamiltonian flow of , if then
also satisfy and . This can be easily proved by direct computations, but this is a general fact in normal form theory (sometimes known as a “normal form with symmetry”) and a nicer way to see this goes as follows. Since , the linear operators and commutes, so that the kernel of is invariant by , and as is semi-simple, the kernel of is also invariant under the projection onto the kernel of which is given by the map . This explains why . Now is in the kernel of , and its unique pre-image by is given by , hence . Put it differently, if a Hamiltonian (above ) has an integral (in our case, ), then the integral is invariant under the normalizing transformation (in our case, ) and remains an integral of the normalized Hamiltonian (that is ).
To conclude, taking into account , the map sends to which remains exponentially small with respect to , and so is . Therefore
then setting , we can write again
Finally, we have found such that
with and exponentially small.
3. Now let us make our previous discussion rigorous.
For , let be independent -periodic vectors, and we denote by the linear integrable Hamiltonian of frequency . We consider a positive integer and a sequence of small parameters , for . As before, and , for , will eventually depend on . Now to fix the ideas, we define the increasing sequence
We shall need some assumptions on these parameters, so we define the condition for by
and for ,
Let , and consider the Hamiltonian defined on , with and . Assume that is satisfied for . Then there exists a symplectic transformation
where is -Gevrey and for , such that
with , for , and the estimates
The proof goes by induction. For , this is nothing but Lemma 3.2 with and . Now assume that the result holds true for some , and let us show that it remains true for .
By assumption, is satisfied, in particular
with such that
with and the estimates
Now let us introduce . Obviously, we have . Moreover, by assumption we have and so that
Then we can write
Furthermore, as , the Hamiltonian is well-defined on , we have with .
Now recall that holds true for , hence we can eventually apply our hypothesis of induction to the Hamiltonian : there exists a symplectic transformation
where is -Gevrey and for , such that
with , for , and the estimates
hold true. Moreover, as , we also have and therefore , for .
Then we set so that
We will prove below that , and since we know that the function , this will easily implies that . Now let us define and so that we can eventually write
Since then obviously , for . Therefore it remains to prove the estimates. First, we have
Then, we know that , so we only need to estimate . For that, recall that
and for , we have the estimates
Therefore a repeated use of lemma A.1 yields
which is the required estimate. ∎
Here also, it is perhaps useful to understand this lemma in the special case where is the canonical basis of : the equality for means that is independent of the first angles , and therefore the evolution of the first action components is only governed by the remainder . In any cases, since we are assuming that are linearly independent, then for , is integrable and the action variables can only evolve according to .
4. Now we shall come back to our original setting ( ‣ 2), that is
For , we still consider a sequence of -periodic vectors , a sequence of small parameters and an integer .
Let us fix . If we were able to find a -periodic action linked to , that is satisfying , then on a small ball of radius around , we could perform some standard scalings to reduce the study of perturbations of to the study of perturbations of the linear Hamiltonian , and so we could use the results of the previous section. However, in the sequel we will construct , but since we are not assuming that the gradient map of is invertible, we cannot construct a corresponding action. In fact, this is not a serious problem, but this is just meant to explain why we need to use some slightly twisted arguments below. In [BN10], we used the idea, coming from [Nie07], to define domains directly in the space of frequencies, however this lead to a rather cumbersome definition of domains. Here we shall use a simpler approach that will enable us to work in the space of actions.
For , we consider a sequence of actions which is -linked to the sequence of independent periodic vectors , in the sense that
By the construction of our periodic vectors, such actions will indeed exist.
Taking into account this sequence of actions