Patrick Bernard
email: patrick.bernard@ujf-grenoble.fr

Une propri ét é de transfert en approximation diophantienne

Étant donné un vecteur ω ∈ R n , on définit la suite T i des périodes de ω comme la suite des temps de meilleur retour près de l'origine de la translation x -→ x + ω sur le tore T n . On étudie comment les propriétés diophantiennes du vecteur ω peuvent être exprimées à l'aide de la suite des périodes. Plus précisément, on montre que si le vecteur ω est non résonant, et si ses périodes vérifient l'inégalité T i+1 CT 1+τ i avec τ < (n -1) -1 , alors le vecteur ω est diophantien.

Introduction

0.1

Soit ω un vecteur de R n . On note T n le tore (R/Z) n , et t ω : T n -→ T n le difféomorphisme x -→ x + ω. On s'intéresse au système dynamique discret engendré par t ω . On dit que ω est résonant si il existe k ∈ Z n tel que le produit scalaire ω, k est entier (∈ Z). Dans ce cas, le tore T n est fibré en tores de plus petite dimension invariants par t ω . Sinon, on dit que ω est non résonant, et t ω est uniquement ergodique et minimal, c'est à dire que T n est le seul compact non vide invariant par t ω , et que la mesure de Haar est la seule mesure de probabilité invariante.

Il est très utile en théorie de la moyennisation (théorie KAM par exemple) de quantifier le caractère non résonant de ω. Le plus simple consiste à minorer les "petits diviseurs" d( k, ω , Z), k ∈ Z n . On parle alors de propriétés diophantiennes linéaires. Cette approche est assez naturelle, car les petits diviseurs apparaissent directement au dénominateur des coefficients des séries de perturbation dans de nombreux problèmes. Intuitivement, ceci revient a quantifier l'ergodicité de t ω , mais le sens dynamique des petits diviseurs n'est pas très clair.

On peut aussi mesurer la qualité de l'approximation de ω par des vecteurs rationnels, c'est à dire estimer d(T ω, Z n ), T ∈ Z. On parle alors de propriétés diophantiennes simultanées. Cette approche n'a été utilisée que beaucoup plus récemment dans les problèmes de moyennisation, voir [START_REF] Lochak | Canonical perturbation theory via simultaneous approximation[END_REF], où elle permet une simplification remarquable du théorème de Nekhoroshev. On peut enfin se restreindre à la suite des meilleures approximations de ω, comme suggéré dans [START_REF] Lochak | Canonical perturbation theory via simultaneous approximation[END_REF]. On définit pour ceci les périodes T i (ω) de ω ∈ R n en posant T 0 (ω) = 1 et

T i+1 (ω) = min T ∈ N tel que T ω Z < T i (ω)ω Z . (1)
Voir 0.2 ci-dessous pour les notations. La croissance des périodes détermine de manière très condensée les propriétés diophantiennes de ω.

Un intérêt majeur de l'approximation simultanée est son caractère profondément dynamique. En effet, si l'on considère un homéomorphisme φ d'un espace métrique et une orbite récurrente φ n (x) de cet homéomorphisme, on peut définir les écarts d(φ n (x), x) et les périodes T i de cette orbite par T 0 = 1 et

T i+1 = min T ∈ N tel que d(φ T (x), x) < d(φ T i (x), x) .
Ces définition coïncident avec les notions vues au dessus lorsque φ est la translation sur le tore de vecteur ω.

La correspondance précise entre les différents types de propriétés diophantiennes n'est pas immédiate, bien que ces propriétés soient visiblement de même nature. Nous rappelons ici le très classique théorème de transfert de Khintchine, et l'étendons aux propriétés diophantiennes définies en terme de croissance des périodes, qui n'avaient pas été étudiées jusqu'ici. Cet article doit beaucoup à mes conversations avec Pierre Lochak sur le lien entre les méthodes de perturbation par approximation simultanées développées dans [START_REF] Lochak | Canonical perturbation theory via simultaneous approximation[END_REF] et la théorie KAM. Je tiens aussi a remercier Xavier Buff, qui m'a suggéré une amélioration de la Propriété 1.3, ainsi que le referee pour sa lecture attentive qui a permis de nombreuses améliorations du texte.

0.2

Notations : On utilise pour l'essentiel les notations de l'appendice 1 de [START_REF] Lochak | Canonical perturbation theory via simultaneous approximation[END_REF]. Soit

ω = (ω 1 , . . . , ω n ) ∈ R n , on note |ω| = sup i |ω i |. Si ω ′ = (ω ′ 1 , . . . , ω ′ n ) ∈ R n , on note ω, ω ′ = ω 1 ω ′ 1 + . . . + ω n ω ′ n le produit scalaire standard, et ω = ω, ω .
Pour ω ∈ R n , on note ω Z = min k∈Z n |ω -k|, et de la même façon, x Z = min k∈Z |x -k| pour x réel.

Théorèmes de Dirichlet

Il est utile pour fixer les idées de rappeler les deux résultats les plus simples de l'approximation diophantienne. On pourra consulter [START_REF] Schmidt | Diophantine Approximation[END_REF] pour les preuves.

1.1

Théorème de Dirichlet : Considérons un vecteur ω ∈ R n , pour tout réel Q > 0, il existe un entier positif T < Q tel que

T ω Z Q -1 n .
En conséquence, il existe une infinité d'entiers T tels que

T ω Z < T -1 n . 1.2 Théorème : Considérons un vecteur ω ∈ R n , pour tout réel Q > 0 il existe un vecteur k ∈ Z n vérifiant |k| < Q et tel que k, ω Z Q -n .
En conséquence, il existe une infinité de vecteurs entiers k tels que k, ω Z < |k| -n .

1.3

Propriété : La suite T i (ω) des périodes de ω satisfait :

1 T i (ω) + T i+1 (ω) T i (ω)ω Z 1 T i+1 (ω) 1/n .
L'inégalité de droite est une conséquence directe du théorème de Dirichlet avec Q = T i+1 (ω). En effet, on a alors

T i (ω)ω Z = min 1 T <T i+1 (ω) T ω Z 1 T i+1 (ω) 1/n .
Pour montrer l'inégalité de gauche, considérons des vecteurs entiers w i tels que

T i (ω)ω Z = |T i (ω)ω -w i |. Comme T i (ω)w i+1 -T i+1 (ω)w i est un vecteur entier non nul, on a 1 T i (ω)T i+1 (ω) T i (ω)w i+1 -T i+1 (ω)w i T i (ω)T i+1 (ω) = w i+1 T i+1 (ω) - w i T i (ω) w i+1 T i+1 (ω) -ω + ω - w i T i (ω) T i+1 (ω)ω Z T i+1 (ω) + T i (ω)ω Z T i (ω) < T i (ω) + T i+1 (ω) T i (ω)T i+1 (ω) T i (ω)ω Z ,
la Propriété en découle.

2 Propriétés de transfert

2.1

Rappelons que la suite T i (ω) est la suite des périodes introduite dans l'introduction. On définit les ensembles

Ω n (τ ) = ω ∈ R n / ∃C > 0, ∀T ∈ N, T ω Z CT -(1+τ )/n , Ω n (τ ) = ω ∈ R n / ∃C > 0, ∀k ∈ Z n -{0}, k, ω Z C|k| -(1+τ )n , Ω(τ ) = ω ∈ R n / ∃C > 0, ∀i ∈ N, T i+1 (ω) CT i (ω) 1+τ , Ω(τ ) = {ω ∈ Ω(τ )/ ∀k ∈ Z n -{0}, k, ω ∈ Z} .
Au vu des théorèmes de Dirichlet de la section 1, les ensembles Ω n (τ ) et Ω n (τ ) sont vides pour τ < 0, il en va évidemment de même des ensembles Ω(τ ) et Ω(τ ). On dit que les éléments de Ω n (τ ) pour τ 0 satisfont une propriété diophantienne linéaire, et que les éléments de Ω n (τ ) satisfont une propriété diophantienne simultanée. Dans cette note, on montre le

2.2

Théorème de transfert : Pour tout τ 0,

Ω n τ (n -1)τ + n ⊂ Ω τ (n -1)τ + n ⊂ Ω n (τ ) ⊂ Ω n (nτ ).
On en déduit par exemple que toutes les notions de vecteurs mal approchables coïncident :

Ω n (0) = Ω n (0) = Ω(0).
Lorsque n = 1, ces nombres sont dits de type constant. On déduit aussi qu'une propriété diophantienne linéaire implique toujours une propriété diophantienne simultanée. L'appartenance à Ω n (τ ) n'implique cependant une propriété diophantienne linéaire que si τ < (n -1) -1 . L'existence de ce seuil sera expliquée dans la section 3.

2.3

Mentionnons que les inclusions suivantes, dues à Khintchine, sont bien connues depuis longtemps (voir [START_REF] Lochak | Canonical perturbation theory via simultaneous approximation[END_REF], [START_REF] Schmidt | Diophantine Approximation[END_REF]) : Pour tout τ 0,

Ω n τ (n -1)τ + n ⊂ Ω n (τ ) ⊂ Ω n (nτ ).

2.4

La section suivante est consacrée à quelques discussions sur le théorème de transfert. On y réduit la preuve des deux nouvelles inclusions à celle la Proposition 3.10.

3 Vecteurs rationnels et résonances

3.1

Le module de résonance de ω, R(ω), est l'ensemble des vecteurs k ∈ Z n pour lesquels k, ω ∈ Z. On appelle ordre de résonance de ω le rang de ce sous groupe de Z n , c'est à dire le cardinal de ses bases. Le vecteur ω est dit résonant si son module de résonance n'est pas trivial, c'est à dire si son ordre de résonance est non nul.

3.2

Le vecteur ω est dit rationnel si ses composantes sont rationnelles. Le vecteur ω est rationnel si et seulement si toutes les orbites des t ω sont périodiques. Elles ont alors toutes la même période, qui est le plus petit dénominateur commun des composantes de ω. Les vecteurs rationnels sont les vecteurs maximalement résonants, c'est à dire ceux dont l'ordre de résonance est n. En effet, tout module R de rang n contient le module T Z n pour une certain T . Les vecteurs ayant R pour module de résonance sont donc nécessairement T -périodiques. avec des composantes a i /d i irréductibles.

3.4

Démonstration : Soit R le module de résonance de ω. Considérons une base k 1 , . . . , k r de R, et associons lui la matrice B à coefficients entiers de l'application linéaire R n ∋ x -→ ( k i , x ) ∈ R r . Par un résultat classique, voir [START_REF] Jacobson | Basic Algebra I[END_REF], théorème 3.8, il existe une matrice A ∈ Gl n (Z), une matrice C ∈ Gl r (Z) et une matrice diagonale

Λ =    d 1 0 0 • • • . . . 0 d r 0 . . .   
telle que B = CΛA, où les coefficients diagonaux d i sont des entiers tels que d i divide d i+1 .

La propostion en découle puisque

k, Aω ∈ Z ⇐⇒ A t k, ω ∈ Z ⇐⇒ A t k ∈ Im(B t ) = Im(A t Λ t ) ⇐⇒ k ∈ Im(Λ t ).

3.5

Remarque : Tout sous groupe de Z n n'est pas un module de résonance, et les invariants d'un module de résonance satisfont certaines contraintes. En particulier, il est facile de vérifier que les invariants qui ne sont pas égaux à 1 sont tous distincts :

d i = d i+1 ⇒ d i = 1.

3.6

Lorsque n = 1, un rationnel ω = w/T (sous forme irréductible) de grand dénominateur T a un comportement très proche d'un irrationnel en ce sens que les points kω, k ∈ Z sont bien répartis sur le cercle (ce sont précisément les points de la forme l/T , l ∈ Z). En dimension supérieure, certains vecteurs périodiques de grande période se comportent essentiellement comme des vecteurs non-résonants, et d'autres se comportent plutôt comme des vecteurs résonants (par exemple le vecteur (0, w/T ) ∈ R 2 ). En raison de la non compacité du groupe Gl n (Z) pour n 2, il n'est pas possible de quantifier la présence effective ou non de résonances pour un vecteur périodique ω de grande période par des quantités invariantes par l'action de Gl n (Z). Par exemple, étant donné un vecteur ω ∈ R 2 , l'algorithme des fractions continues permet de trouver une suite (non bornée) A n de matrices de Gl 2 (Z) et une suite de rationnels x n ∈ Q tels que A n (0, x n ) -→ ω lorsque n tend vers l'infini.

Il est donc utile d'introduire la quantité

e(ω) = min k∈R(ω)-{0}
|k|.

Pour illustrer son rôle, considérons un vecteur réel ω et une suite d'approximations rationnelles w n /T n de ω, où la période T n tend vers l'infini. Il n'est pas difficile de voir que la limite ω est résonante si et seulement si la suite e(w n /T n ) est bornée. Notons que, pour un vecteur rationnel w/T de période T , on a l'estimation 1 e(w/T ) T 1/n .

En effet, le théorème 1.2 donne l'existence d'un vecteur entier k tel que |k| T 1/n et k, w/T Z < 1/T, mais alors k ∈ R(w/T ). Réciproquement, pour la plupart des vecteurs rationnels w/T de dénominateur T , e(w/T ) est au moins de l'ordre de T 1/(n+1) . Ceci est rendu plus précis par la Propriété 3.7 ci-dessous. Les vecteurs périodiques pour lesquels la valeur e est grande sont ceux dont l'orbite est constituée de points bien répartis sur le tore.

3.7

Propriété : Pour tout réel strictement positif τ , la proportion de vecteurs entiers w tels que e(w/T ) T points entiers w dans [0, T -1] n tels que e(w/T ) A. On termine alors la démonstration en remarquant que

n 2T 1-τ n+1 + 1 n+1 T n-1 = o(T n ).

3.8

En ne retenant d'un vecteur rationnel que sa période T , on perd donc une information essentielle. Par exemple, la suite des distances T (ω, 0) Z , T ∈ Z, associée au vecteur (ω, 0) ∈ R n+1 est la même que la suite T ω Z associée à ω, alors que le vecteur (ω, 0) est fortement résonant et engendre des orbites très mal réparties dans T n+1 . De la même façon, la suite des périodes T i (ω, 0) est la même que celle de ω. Il est remarquable qu'une simple connaissance de la suite T ω Z puisse suffire à déterminer le caractère non résonant de ω, voir 3.9. Il n'est pas possible de tirer ce caractère non résonant d'une connaissance de la suite des périodes. Pourtant, si l'on suppose à priori le vecteur ω non résonant, on montre que certaines estimées sur la croissance des périodes permettent de déduire des propriétés diophantiennes linéaires vérifiées par ω. Cette remarque, à ma connaissance nouvelle, fait l'objet de la proposition 3.10, qui sera démontrée dans la section suivante.

3.9

Propriété : Pour tout τ < (n -1) -1 , les vecteurs de Ω n (τ ) sont non résonants. Ce résultat est bien sur impliqué par les inclusions de Khintchine 2.3. Nous allons cependant en donner une preuve élémentaire. Considérons un vecteur résonant ω. D'après 3.3, il existe une matrice A ∈ SL n (Z) telle que Aω = (ω ′ , w/p) ∈ R n-1 × Q, (w, p) ∈ Z 2 . Appliquons le théorème de Dirichlet à pω ′ . Il existe une infinité d'entiers T tels que T pω ′ Z < T -1/n-1 , et donc tels que

T (pω ′ , w) Z < 1 T 1/(n-1) ⇐⇒ T pAω Z < p 1/(n-1) (pT ) 1/(n-1) .
Il y a donc une infinité d'entiers T ′ = pT et une constante C tels que 1) .

T ′ ω Z C (T ′ ) 1/(n-
Ceci exclut que ω puisse appartenir à l'ensemble Ω n (τ ) lorsque

1 + τ n < 1 n ⇐⇒ τ < 1 n -1 .

3.10

Proposition : Si ω est non résonant et si il existe τ ∈ 0, 1 n-1 et C > 0 tels que T i+1 (ω) CT i (ω) 1+τ alors il y a une constante D > 0 telle que, pour tout k ∈ Z n , on a k, ω Z D|k| -1+µ n , où µ = nτ n -(n -1)(1 + τ ) .

3.11

Démonstration du théorème de transfert : Nous ne prouverons que les deux première inclusions. Nous renvoyons à [START_REF] Schmidt | Diophantine Approximation[END_REF] pour la preuve de la troisième inclusion. La deuxième inclusion découle directement de la propriété 3.10. La première inclusion découle de la propriété 3.9 et de la remarque suivante : Si ω ∈ Ω n (τ ′ ), on obtient en appliquant théorème de Dirichlet avec Q = T i+1 (ω) que

CT i (ω) -(1+τ ′ )/n T i (ω)ω Z T i+1 (ω) -1/n , et donc que ω ∈ Ω(τ ′ ) (voir [2]).
4 Démonstration de la proposition 3.10

Le vecteur ω est fixé une fois pour toutes, on notera donc T i les périodes T i (ω). On considère une suite de meilleures approximations ω i , c'est à dire une suite de vecteurs rationnels telle que T

i (ω i -ω) = T i ω Z et T i ω i = w i ∈ Z n . Soit k ∈ Z n ,
comme ω est non résonant, les produits scalaires k, ω i ne peuvent être entiers que pour un nombre fini de valeurs de i, et on peut définir

i(k) = 1 + max i tel que k, ω i ∈ Z .

4.1

Établissons pour commencer les inégalités

k T i(k)-1 2 √ n T 1-1/n i(k) 1 2 √ n C 1 1+τ T n+(1-n)(1+τ ) n(1+τ ) i(k) . On a k, ω i(k)-1 ∈ Z, et k, ω i(k) Z 1/T i(k) donc | k, ω i(k) -ω i(k)-1 | 1/T i(k) .
On en déduit que

k 1 2T i(k) ω i(k)-1 -ω .
En appliquant le Théorème de Dirichlet avec Q = T i+1 , on obtient

ω i(k)-1 -ω √ n|ω i(k)-1 -ω| √ n T i(k)-1 T 1/n i(k)
, ce qui donne la première égalité. La seconde inéqualité est alors une conséquence directe de l'hypothèse.

4.2

Un calcul simple donne alors que implique, puisque T j(k)+1 CT 1+τ j(k) , que

T i(k) 2 √ n C
T j(k) 2 √ n C 1 1+τ ) n 1+µ 1+τ C 1 1+τ k n 1+µ 1+τ .
D'un autre coté, on a la majoration

T j(k) 2 √ n C 1 1+τ k n(1+µ) .
On a donc, comme j(k) i(k),

k, ω j(k) Z 1 T j(k) 2 √ n C 1 1+τ k -n(1+µ) . (2)
En utilisant encore le théorème de Dirichlet, on obtient

|ω j(k) -ω| 1
T 1+1/n j(k) et donc | k, ω j(k) -ω | k T 1+1/n j(k) 1 T j(k) C k τ -µ 1+τ , (3)
où C ne dépend pas de k. Il est facile de voir que τ < µ, et donc, par (2) et (3) que

k, ω Z 1 T j(k) (1 -ǫ( k )) 2 √ n C 1 1+τ k -n(1+µ) (1 -ǫ( k )),
où ǫ(x) est une fonction qui tend vers 0 lorsque x tend vers l'infini. La proposition annoncée en découle.

3. 3

 3 Proposition : Soit ω un vecteur résonant d'ordre r. Il existe une matrice A ∈ Gl n (Z) et une famille d i , 1 i r, d'entiers tels que d i divise d i+1 et tels que (d 1 e 1 , . . . , d r e r ) forme une base du module de résonance de Aω, où (e 1 , . . . , e n ) est la base standard de R n . En particulier, Aω = (w/T, ω ′ ) ∈ Q r × R n-r , où ω ′ est un vecteur non résonant et w/T est un vecteur périodique (rationnel) de période T = d r . Lorsque ω est périodique, on lui associe donc n invariants d i , où d n = T est la période. Il existe alors une matrice A ∈ Gl n (Z) telle que Aω = (a 1 /d 1 , a 2 /d 2 , . . . , a n /d n )

  lorsque T tend vers l'infini. Démonstration : Nous allons majorer le nombre de points entiers w de [0, T -1] n tels qu'il existe un vecteur entier k vérifiant |k| A et k, w = T l, l ∈ Z. Pour chaque couple (k, l) ∈ Z n × Z, l'hyperplan k, w = T l contient au plus T n-1 points entiers dans [0, T -1] n . Par ailleurs, pour un vecteur entier k donné, le nombre d'entiers l tels que l'hyperplan k, w = T l intersecte [0, T -1] n est au plus n(2|k|+1), puisque | k, w | n|k||w|. Le nombre de couples (k, l) ∈ Z n × Z vérifiant |k| A et tels que l'hyperplan d'équation k, w = T l intersecte [0, T -1] n est donc inférieur à n(2A + 1) n+1 . Il y a donc au plus n(2A + 1) n+1 T n-1

  où µ est donné dans l'énoncé. Définissons l'indice j(k) = min j i(k) tels que T j+1 2


Homoclinic orbits to invariant sets of quasi-integrable exact maps

Patrick Bernard october 98

In Hamiltonian mechanics, various questions and results concern the becoming of the invariant tori of an integrable system after perturbation. It is well known, by KAM theory, that many of these invariant tori are deformed but not destroyed. But it is known as well that many of them are destroyed. They give rise to more or less complicated invariant sets such as Mather invariant sets [START_REF] Mather | Action minimizing invariant measures for positive definite Lagrangian systems[END_REF].

One particular kind of tori that can be destroyed is resonant tori, that is invariant tori foliated by lower dimensional invariant tori. Treschev proved in [START_REF] Treschev | The mechanism of destruction of resonant tori of Hamiltonian systems[END_REF] that if the frequency induced on the lower dimensional torus satisfies some Diophantine condition, then one of the lower dimensional tori is preserved, and becomes hyperbolic (whiskered in Arnold's terminology [START_REF] Arnold | Instability of Dynamical systems with several degrees of freedom[END_REF]). If the unperturbed Hamiltonian is positive definite, it has been proved by Bolotin [3] that this preserved torus admits homoclinic orbits in the perturbed system. Similar results have been obtained by Eliasson [START_REF] Eliasson | Biasymptotic Solutions of Perturbed Integrable Hamiltonian Systems[END_REF] for 1-resonant tori without the convexity assumption. These results are important not only as generalizations to high dimensional systems of old results on twist maps, but also because they are a step in the attempt to generalize Arnold's construction of [START_REF] Arnold | Instability of Dynamical systems with several degrees of freedom[END_REF] and study diffusion in high dimensional Hamiltonian systems. Unfortunately, as is explained in [START_REF] Lochak | Arnold's diffusion : a compendium of remarks and questions[END_REF], there are not enough preserved tori for Arnold's construction to be led directly. That's why the case of resonant tori which do not satisfy Diophantine conditions and which can be completely destroyed is of some interest. Bolotin gives partial answer to that question in [START_REF] Bolotin | Homoclinic orbits to invariant tori of Hamiltonian systems[END_REF], obtaining semi-asymptotic orbits to the Mather invariant set but does not obtain an invariant set with homoclinic orbits.

In this paper, we focus our attention to a general resonant torus no lower dimensional torus of which has to be preserved. We prove that its destruction gives rise to compact invariant sets, the Peierl's sets introduced by Mather in [START_REF] Mather | Variational construction of connecting orbits[END_REF] and containing the usual Mather sets, and that these sets admit nontrivial homoclinic orbits. As we stressed above, this result could have some interest in the study of diffusion, allowing to fill the gaps between preserved tori. Our method, that provides a different and less involved proof of Bolotin's result, is based on periodic orbits: both the invariant sets and the homoclinic orbits are obtained as limits of periodic orbits which are found by the variational methods of [START_REF] Bernstein | Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonian[END_REF]. That the Peierl's set is the natural asymptotic set to be considered has just been noticed by Fathi [START_REF] Fathi | Orbites hétéroclines et ensembles de Peierls[END_REF] by quite different methods.

For convenience, we will consider exact maps instead of Hamiltonian flows, the links between both theories are well known. Let us quote for instance [START_REF] Lochak | Canonical perturbation theory via simultaneous approximation[END_REF] or [START_REF] Bernstein | Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonian[END_REF] for details about convexity assumptions. We also refer to [START_REF] Bernstein | Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonian[END_REF] where it is explained how the local study of any quasi-integrable exact map around a positive definite torus can be reduced to the following setting.

Main Result

We will consider an integrable symplectic diffeomorphism

Φ 0 : T n × R n -→ T n × R n , whose pull back φ 0 : R n × R n -→ R n × R n can be written φ 0 (x, y) = (x + ω(y), y).
In all the following, we will suppose that Φ 0 satisfies the following two hypothesis:

Hypothesis 1 (non-degeneracy) the map ω : R n -→ R n is a diffeomorphism.

Then φ 0 admits a global generating function of type S [START_REF] Duff | Introduction to symplectic topology[END_REF], i.e. there exists a function S :

R n × R n -→ R such that (x 1 , y 1 ) = φ 0 (x 0 , y 0 ) ⇐⇒ y 0 = ∂ 1 S(x 0 , x 1 ) y 1 = -∂ 2 S(x 0 , x 1 ) . (1)
It is easy to see that in our case

S(x 0 , x 1 ) = h(x 1 -x 0 ),
and we will make the following second assumption Hypothesis 2 (convexity) the function h : R n -→ R is strictly convex.

Let Φ be a C 1 exact perturbation of Φ 0 , its pull back φ has a C 2 generating function

S(x 0 , x 1 ) = h(x 1 -x 0 ) + P (x 0 , x 1 ),
where the perturbation P satisfies Hypothesis 3 (exactness) P (x 0 + m, x 1 + m) = P (x 0 , x 1 ) for all m ∈ Z n .

Since we are interested in the dynamical structure of a bounded region of phase space, we can change the mapping at infinity and furthermore suppose (see [START_REF] Bernstein | Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonian[END_REF])

Hypothesis 4 (localization) there is a compact set K ⊂ R n and a number a > 0 such that

S(x, x + t) = h(t) + P (x, x + t) = a|t| 2 + 0 when t / ∈ K.
Let y 0 and ω 0 = ω(y 0 ) be fixed. Assume that ω 0 is resonant, which means that there exists

k ∈ Z n such that < k, ω 0 >∈ Z. Let R = {k ∈ Z n such that < k, ω 0 >∈ Z}
and let r = rank (R), then we say that ω 0 is r-resonant. The free group R has a basis B of cardinal r. It is not hard to see that there is a foliation of the torus y = y 0 in ergodic subtori and a group morphism

F : T n × T n-r -→ T n
such that these tori are

F (x, θ), θ ∈ T n-r .
The unperturbed map restricted to {y = y 0 } is given by

F (x, θ) -→ F (x, θ) + ω 0 = F (x, θ + ω0 ).
The frequency ω0 ∈ R n-r is called the induced frequency. It is useful to define the averaged perturbation

P : T n -→ R
given by

P(x) = T n-r P F (x, θ) , F (x, θ) + ω 0 dθ = lim N →+∞ 1 2N N i=-N P x + iω 0 , x + (i + 1)ω 0 .
The function P is then constant along the ergodic subtori,

P(F (x, θ)) = P(x).
Hypothesis 5 P is minimal on exactly one of the ergodic tori {F (x, θ), θ ∈ T n-r }, which is a non-degenerate critical manifold for P. We will note Γ 0 this minimal torus.

We will use the notation π both for the canonical projection R n -→ T n and for (π, Id) : R n × R n -→ T n × R n and Π for the first factor projection

T n × R n -→ T n or R n × R n -→ R n .
We also set U δ = {q ∈ T n , dist(q, Γ 0 ) < δ}, and we will write U δ for π -1 (U δ ). We are now in a position to state our main result:

Theorem 1 Let Φ be the exact map whose pullback φ is generated by a C 2 function

S (x 0 , x 1 ) = h(x 1 -x 0 ) -P (x 0 , x 1 )
satisfying H 1-5. Let C > 0 be any constant, then there are two nonnegative constants ∆ and A and a function δ( ) with lim →0 δ = 0 such that for any ∆ and any c ∈ R n with c -∇h(ω 0 )

C , there exists a compact invariant set Σc ( ) of Φ satisfying

• Π | Σc( ) is a bilipschitz homeomorphism onto its image. • Σc ( ) ⊂ (x, y) ∈ T n × R n /x ∈ U δ( ) , y -y 0 A √ ,
• Σc ( ) has r + 1 homoclinic orbits lying in the zone y -y 0 A √ of phase space.

• Both the invariant sets and the homoclinic orbits belong to Per(Φ).

The invariant set Σc ( ) is a Peierl's set as defined in Mather [START_REF] Mather | Variational construction of connecting orbits[END_REF], it will be described with some details in section 3. If the induced frequency ω0 satisfies some Diophantine condition, and if the maps involved are sufficiently regular, Treschev [START_REF] Treschev | The mechanism of destruction of resonant tori of Hamiltonian systems[END_REF] obtains the existence of an hyperbolic invariant torus of frequency ω 0 . In this case, theorem 1 becomes:

Theorem 2 (Bolotin, [START_REF] Bolotin | Homoclinic orbits to invariant tori of Hamiltonian systems[END_REF]) There is a constant C > 0 such that if there exists a KAM hyperbolic torus Γ with rotation vector ω 0 , then there is c satisfying the hypothesis of theorem 1 such that Σc ( ) = Γ.

The torus Γ thus admits r + 1 homoclinics.

Remark : The existence of 2r homoclinic orbits may actually be proved in both statements above by noticing that the system is almost reversible in the region of phase space under interest.

In Section 2, we introduce the variational setting for periodic orbits that will be used in all the following. In section 3, we describe the asymptotic set. We first state some general properties of Peierl's sets, proving results of [START_REF] Mather | Variational construction of connecting orbits[END_REF] in order to stress that no minimal measures are needed to introduce the Peierl's set. In the subsections we use averaging methods inspired from [START_REF] Bolotin | Homoclinic orbits to invariant tori of Hamiltonian systems[END_REF] to localize the asymptotic set. In section 4, we explain that certain sequences of periodic orbits converge to homoclinic orbits, and in section 5, we use this fact to get several nontrivial homoclinics.

Periodic orbits

All the constructions below will be based on periodic orbits, that will be obtained as action minimizers. We will first introduce a variational problem for orbits of φ in the universal cover

R n × R n . Let E T w = {(x 0 , . . . , x T ) ∈ (R n ) T +1 such that x T = x 0 + w}, Consider the Lagrangian L T w : E T w -→ R L T w (x 0 , . . . , x T ) = T -1 i=0 S(x i , x i+1 ).
It is not hard to see that there exists a T -periodic orbit (x 0 , y 0 ), (x 1 , y 1 ), . . . , (x T , y T ) of φ if and only if (x 0 , . . . , x T ) ∈ crit L T w . The orbit having a given x-projection is then uniquely defined, it satisfies:

y i = ∂ 1 S(x i , x i+1 ) = -∂ 2 S(x i-1 , x i ).
This variational setting has been introduced and studied in [START_REF] Bernstein | Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonian[END_REF], we will need the following part of their result, in which only estimate (2) is non-trivial.

Theorem 3 (Bernstein Katok) Given C > 0, there exist two non negative constants ∆ and A, depending on h and C but neither on w or T nor on P , such that if dP C 0 C and ∆, the above variational problem L T w admits a minimizing orbit X = (x i ) satisfying

|x i+1 -x i -w/T | A √ . ( 2
)
There are corresponding problems on T n . For any c ∈ R n , let

S c (x 0 , x 1 ) = S(x 0 , x 1 )-< c, x 1 -x 0 > .
The functional c L T w associated to S c has the same critical points and the same minima than that associated to S. We can now define

E T = {(q 0 , . . . , q T ) ∈ (T n ) T +1 such that q T = q 0 } and H T c (q 0 , . . . , q T ) = T -1 i=0 H c (q i , q i+1 ),
where H c (q 0 , q 1 ) = min

π(x 0 )=q 0 ,π(x 1 )=q 1 (S c (x 0 , x 1 ))
In the following, we will usually use q for points and orbits in T n and x for points and orbits in R n . Given T and c, it is easy to see that the function

w ∈ Z n -→ min( c L T w )
is proper. It must have a minimum, and we will call W T c the set of rotation numbers w ∈ Z n realizing this minimum. If X is a minimizing orbit of L T w with w ∈ W T c then it is easy to see that its projection Q ∈ E T is a minimizer of H T c , and any minimizer of H T c can be obtained that way. The minimizing orbits of the functionals H c depend on c, and if w, T and c are fixed, the projection Q ∈ E T of a minimizing orbit X ∈ E T w of L T w need not be a minimizing orbit of H T c . Following Mather, we will call configurations the elements of E T w or E T or in general all sequences of points. Let us end this section with an important remark:

Lemma 1 The function H c : T n × T n -→ R is Lipschitz continuous.
Proof: Let us first notice that there is a constant C depending on c such that for any (q, p) ∈ (T n ) 2 there are covering points (x, y) of (q, p) with (x, y) C and H c (q, p) = S c (x, y). Let us now take two other points q and p and their covering points x and y closest to x and y. The result follows from the inequality

H c (q , p ) -H c (q, p) S c (x , y ) -S c (x, y) K (x , y ) -(x, y)
Kd((q , p ), (q, p)).

The asymptotic set

We will now define and describe the invariant set to which homoclinic orbits will be searched. Let

α(c) = inf T 0 min H T c T .
This function is the α function of Mather. The following lemma will be of great importance:

Lemma 2 lim inf T -→∞ min(H T c -T α(c)) = 0. Proof: Let us set m T = min(H T c -T α(c)).
If for some T we have m T = 0 then calling Q T the minimizing sequence of H T c -T α(c) and n * Q T the nT -periodic configuration consisting of n iterates of Q T we have H nT c (n * Q T ) -nT α(c) = 0 for any n, and so lim inf m T = 0. Thus if we assume that lim inf m T > 0, then there is a real number l satisfying

m T l > 0. Let us fix r > 0 such that |H c (z, x) -H c (z, y)| l/2 when d(x, y) 2r.
There is a real number p with 0 < p < 1 such that, given T points on the torus, there is a ball of radius r containing at least pT of them. Take a minimizing configuration Q T = (q T i ) of H T c , after permutation of the indices, there are K pT points q T 0 , q T i 1 , . . . , q T i K-1 contained in a ball of radius r. The configurations

Q k = (q T i k-1 , q T i k-1 +1 , . . . , q T i k ) of length T k = i k -i k-1
are almost periodic and we have

H T k c (Q k ) -T k α(c) l/2
, adding these inequalities for all k we get

H T c (Q) -T α(c) Kl/2 pT l/2.
This gives α(c) H T c /T -pl/2 which is in contradiction with the above definition of α. The following definition is that of [START_REF] Mather | Variational construction of connecting orbits[END_REF], but we avoid the use of minimal measures.

Definition 1 A Peierl's point is a point q ∈ T n such that there exists a sequence Q n of T n -periodic configurations with q n 0 = q and lim n-→∞

H Tn c (Q n ) -T n α(c) = 0.
The set of all Peierl's points is the Peierl's set Σ c .

That this set is not empty is an easy consequence of lemma 2. Take a sequence

Q n of T n - periodic configurations such that lim n-→∞ H Tn c (Q n ) -T n α(c) = 0.
Any accumulation point of the bounded sequence (q n 0 ) n∈N is a Peierl's point. In order to get further information on the set Σ c , we will need the Lemma 3 Let us fix q 0 ∈ T n and let Q n and P n be two periodic configurations with q n 0 = q 0 = p n 0 and lim

n-→∞ (H c -T α)(Q n ) = 0 = lim n-→∞ (H c -T α)(P n ),
then q n 1 and p n 1 have the same limit q 1 .

Proof: We first take subsequences so that q n 1 , p n 1 , q n T n q -1 and p n T n p -1 have limits q 1 , p 1 , q -1 and p -1 . Consider the configurations

R n = (q n 0 , q n 1 , . . . , q n T n q -1 , q, p n 1 , . . . , p n T n p )
and R n (q) = (q n 0 , q n 1 , . . . , q n T n q -1 , q 0 , p n 1 , . . . , p n T n p ), q ∈ T n , and let

h n (q) = H c (R n (q)) -T n α(c),
where T n = T n p + T n q . We then have

h n (q) = c n + H c (q n T n q -1 , q) + H c (q, p n 1 ).
Since h n (q 0 ) -→ 0 the sequence c n has a limit -H c (q -1 , q 0 ) -H c (q 0 , p 1 ). The function h n (q) thus has a limit h(q) which satisfies h(q 0 ) = 0 and h 0. It follows that the function

q -→ H c (q -1 , q) + H c (q, p 1 )
must have a minimum at q 0 . Taking points x -1 , x 0 and y 1 in R n covering q -1 , q 0 and p 1 and such that H c (q -1 , q 0 ) = S c (x -1 , x 0 ) and H c (q 0 , p 1 ) = S c (x 0 , x 1 ), we obtain that the function

x -→ S c (x -1 , x) + S c (x, y 1 )
must have a minimum at x 0 and the equation

∂ 2 S c (x -1 , x 0 ) = -∂ 1 S c (x 0 , y 1 )
holds true. Let us take a given sequence Q n such that q n -1 has a limit q -1 , then any accumulation point of p n 1 for any admissible sequence P n must have a covering that satisfies the above equation hence the sequence p n 1 must have a limit, which corresponds to the only solution of the equation. We get that p 1 = q 1 by applying this to the case

P n = Q n .
The mapping

Φ c : Σ c -→ Σ c q -→ q 1 .
is Lipschitz continuous. We could prove it using arguments as in lemma 3 but more involved, we prefer to send the reader to [START_REF] Mather | Action minimizing invariant measures for positive definite Lagrangian systems[END_REF]. The set

Σc = {(q, ∂ 1 S c (q, Φ c (q))) , q ∈ Σ c } is Φ-invariant. There is a map φ c : π -1 (Σ c ) -→ π -1 (Σ c ) so that the following diagram commutes. π -1 ( Σ c ) φ / / Π π { { x x x x x x x x x π -1 ( Σ c ) Π π { { x x x x x x x x x Σ c Φ / / Π Σ c Π π -1 (Σ c ) φc / / π { { v v v v v v v v v π -1 (Σ c ) π { { v v v v v v v v v Σ c Φc / / Σ c
The vertical projections Π on this diagram are all bilipschitz homeomorphisms. We will often use the notation Σ c for π -1 (Σ c ) and Σc for π -1 ( Σc ). We have given the general definition of a Peierl's set. In order to obtain asymptotic orbits to it, some information must be obtained on its topology. We now focus our attention to the vicinity of the torus T n × {y 0 }. The following proposition provides us with some Peierl's set in this zone of phase space, and gives us the topological information we need.

Proposition 1 Let us fix C > 0, there is a function δ( ) with lim -→0 δ( ) = 0, and a constant A > 0 such that when is small enough we have

Σ c ⊂ U δ( ) and Σc ∈ { y -y 0 A √ } ∈ T n × R n
for any c such that c -∇h(ω 0 ) C .

In the following subsections, we prove proposition 1.

Averaging

Let us set

S(x 0 , x 1 ) = S(x 0 , x 1 ) + f (x 0 ) -f (x 1 ),
where f ∈ C 1 (T n ). All the functionals defined above on sets of periodic orbits are unchanged when S is replaced by S. We will use this fact to reduce our functionals to more convenient forms.

Lemma 4 For any o > 0, There is a function

f o ∈ C 1 (T n ), such that the modified generating function S o (x 0 , x 1 ) = S(x 0 , x 1 ) + f o (x 0 ) -f o (x 1 ) satisfies S o (x 0 , x 1 ) = h(x 1 -x 0 ) + P(x 0 ) + R(x 0 , x 1 ) with |R(x 0 , x 0 + ω 0 )| o.
Proof: We first formally solve the equation

P (x, x + ω 0 ) + f (x) -f (x + ω 0 ) = P(x 0 ). Let us put P (x, x + ω 0 ) = k∈Z n p k e i<k,x> then P(x) = k∈R p k e i<k,x>
and the function f (x) = k∈Z n f k e i<k,x> , with

f k = p k e i<k,ω 0 > -1 , k / ∈ R ; f k = 0 , k ∈ R
formally solves the equation. Under our hypothesis, the above series can be divergent. Nevertheless, given o > 0, there is

K ∈ R such that |k| K p k e i<k,x> o.
It is then easy to see that

f o (x) = |k| K f k e i<k,x> satisfies P (x, x + ω 0 ) + f o (x) -f o (x + ω 0 ) -P(x) ∞ o.
this proves lemma 4.

It will be useful in the following to use the notations

Sc = S c -α(c), Hc = H c -T α(c),
and Q(x) = P(x) -min(P).

Lemma 5 Let us fix C > 0, there are non negative numbers a and b and a function K(o) independent of such that for any c ∈ R n satisfying ∇h(ω 0 ) -c C ,

a t 2 + Q(x 0 ) -4o -K 2 c So (x 0 , x 1 ) b t 2 + Q(x 0 ) + 4o + K 2 , (3)
where t = x 1 -x 0 -ω 0 .

Proof: For small enough, the function x 1 -→ h(x 1 -x 0 ) + R(x 0 , x 1 ) is uniformly strictly convex thus there exists a > 0 such that

h(x 1 -x 0 ) + R(x 0 , x 1 ) h(ω 0 ) + R(x 0 , x 0 + ω 0 )+ < ∇h(ω 0 ) + ∂ 2 R(x 0 , x 0 + ω 0 ), x 1 -x 0 -ω 0 > +a x 1 -x 0 -ω 0 2 .
This gives

c S o (x 0 , x 1 ) h(ω 0 )-< c, ω 0 > -o+ < ∇h(ω 0 ) + ∂ 2 R(x 0 , x 0 + ω 0 ) -c, t > +a t 2 + P(x 0 ) h(ω 0 )-< c, ω 0 > -o -C t + a t 2 + P(x 0 ) h(ω 0 )-< c, ω 0 > -o -K 2 + a t 2 + P(x 0 )
changing the constant a in the last inequation. We can also obtain the majoration

c S o (x 0 , x 1 ) h(ω 0 )-< c, ω 0 > + o + K 2 + b t 2 + P(x 0 ).
We are now in a position to estimate α(c). The above minoration gives

α(c) h(ω 0 )-< c, ω 0 > + min P -o -K 2 h(ω 0 )-< c, ω 0 > + min P -2 o
for small enough with respect to o. Let x and y be two points of Γ 0 , consider the configuration x k = x + kω 0 , let T be the time between -1/2 and 2 -1/2 for which x T is closest to y and set d = y -x T . Now consider the configuration Y = (y k = x k + kd/T ), it connects x and y and satisfies

c L o (Y ) bT -1 d 2 + T min(P) + T h(ω 0 ) -T < c, ω 0 > + oT + 2 KT because c S o (y k , y k+1 ) b d/T 2 + P(y k ) + h(ω 0 )-< c, ω 0 > +o + K 2 + h(ω 0 ).
Since d goes to zero as goes to zero, we obtain for small enough the estimate

c L o (Y ) T min(P) + T h(ω 0 ) + T < c, ω 0 > +2o √ .
If we apply this remark to the case x = y, Y is a periodic configuration and we get α(c) h(ω 0 )-< c, ω 0 > + min P + 2o .

We have estimated α(c):

|α(c) -h(ω 0 )+ < c, ω 0 > -min P| 2o .
The inequality (3) is obtained by mixing this estimate with the above inequalities.

We have also proved:

Lemma 6 Let x and y be two points of Γ 0 , there is a configuration X = (x = x 0 , . . . , x T = y) with -1/2 T 2 -1/2 and c Lo (X) 10o √ .

Localization of the asymptotic set

In this rather technical section, we prove proposition 1. In the following calculations, we will mainly use inequality (3), thus our result will be true for any c satisfying the hypothesis of lemma 5. We will omit the subscript c.

Lemma 7 Let ν > 0 and x and y be two points of U ν , for o > 0 small enough and small enough with respect to α, there is a configuration X = (x = x 0 , . . . , x T = y) satisfying Lo (X) Cν 2 √ .

Proof : We will construct our connecting configuration via Γ 0 . Let x ∈ U ν , x+d its projection on Γ 0 . Considering the configuration (x 0 , . . . , x T ) = (x, x + ω 0 + d/T, . . . , x

+ T ω 0 + d), we get So (x i , x i+1 ) b d/T 2 + Q(x + i(ω 0 + d/T )) + o + K 2 .
Adding these inequalities gives Lo (X) bν 2 /T + T sup

Uν Q + o + K ,
and since

P(x) min P + C(d(x, Γ 0 )) 2 , L(X) bν 2 /T + T Cν 2 + 4o + K .
Taking T = √ we obtain that Lα (x 0 , . . . , x T ) Cν 2 √ .

In order to prove the lemma, we just have to consider a three part configuration x = x 0 , . . . , x N , . . . , x M , . . . , x T = y, where x N ∈ Γ 0 and x 0 , . . . , x N is the orbit considered above, x M ∈ Γ 0 and x M , . . . , x T is a reversed configuration like above, and x N , . . . , x M is a configuration obtained by lemma 6.

The inequality of the lemma is then satisfied if o is small enough.

Corollary 1 Let x and y lie in U ν , any configuration connecting x and y satisfies Lα -Cν 2 √ .

Proof: Let us consider a configuration connecting x and y, and complete it by the configuration of low action obtained above connecting y and x. We obtain a periodic configuration, writing that its action must be non negative gives the inequality of the corollary. Let 0 < ν < δ, and let W = R n -U ν .

Lemma 8 Let X = x 0 , . . . , x T be a configuration with x 0 ∈ U ν , x i ∈ W for i 1 and x T ∈ R n -U δ . Then for α and small enough

Lα (X) D √ ν(δ -ν).
The conclusion holds true for a configuration with x T ∈ U ν , x i ∈ W for i < T and x 0 ∈ R n -U δ .

Proof: We will give the proof of the first part of the lemma, the proof of the second part goes along the same line and is easier. Since Γ 0 is a nondegenerate minimal manifold for P we have P(x) min P + D(d(x, Γ 0 )) 2 .

It follows that So (x i , x i+1 ) a t i 2 + Cν 2 -4o -K 2
for i 1. We then choose o so that Dν 2 -4o Dν 2 /2 > 0, and obtain

So (x i , x i+1 ) C t i ν √ ,
for small enough and i 1. On the other hand,

So (x 0 , x 1 ) a t 0 2 + Q(x 0 ) -4o -K 2 , So (x 0 , x 1 ) a t 0 2 + Dν 2 -C t 0 -4o -K 2 .
For small enough, we get So (x 0 , x 1 ) D t 0 ν √ , Adding all the minorations we have obtained gives the lemma because

i t i i t i |δ -ν|.
Lemma 9 There is a function δ( ) such that lim →0 δ( ) = 0 and a function a( ) satisfying a( ) > 0 for > 0 such that any periodic configuration (x 0 , x 1 , . . . x 0 + m) of any rotation number leaving U δ( ) , satisfies L(x 0 , . . . , x 0 + m) 2a( ):

L(x 0 , . . . , x 0 + m) 2a( ) if x i / ∈ U δ( ) for some i.
Proof: Let δ be fixed, and let X be a periodic configuration leaving U δ , let ν = δ 2 . If X stays out of U ν , then adding the inequalities

So (x i , x i+1 ) a t i 2 + Dν 2 -4o -K 2 , we obtain L(X) = Lo (X) Dν 2 -4o -K 2 ,
for any small o, so that we have L(X) D δ 4

for small enough. Let us now suppose that X comes into U ν . We can find a configuration Y = x -i , . . . , x 0 , . . . , x j in X with

x 0 / ∈ U δ ,x k ∈ W if -i < k < j and (x -i , x j ) ∈ (U ν ) 2 . The above lemma gives us Lα (Y ) D √ δ 3 .
By corollary 3, we then have L(X)

√ (Dδ 3 -Cδ 4 )
when is small enough with respect to delta. As a consequence of these results, there is a constant D > 0 such that the inequality

L(X) D min √ δ 3 , δ 4
must be satisfied when δ is small enough and when is small enough with respect to δ. The lemma follows.

The horizontal localization of proposition 1 follows. Let us now end the proof of proposition 1:

We are still given a fixed c satisfying the hypothesis of theorem 1, and we omit all subscripts c. Let X k ∈ E T k w k be a sequence of periodic configurations such that L(X k ) -→ 0 and T k -→ ∞.

Lemma 10

The rotation number w k satisfies:

w k /T k -ω 0 2A √
when k large enough and small enough, where A is the constant of inequality (2).

Proof: By inequality (2), we have

x k i+1 -x k i -w k /T k A √ .
Assume that w k /T k -ω 0 2A √ with the same constant A, then setting

t k i = x k i+1 -x k i -ω 0 as usual, we have that t k i A √
. Putting this in inequality 3 gives

L(X k ) T k (aA -o -K 2 )
for any o, which gives a contradiction when o and are small enough because T k -→ ∞ and L(X k ) -→ 0. Using lemma 10 and (2) we obtain

x k i+1 -x k i -ω 0 3A √
for k large enough. Vertical localization of proposition 1 follows, with a different constant A.

Convergence

In this section we will study sequences of periodic orbits, and obtain homoclinic orbits as accumulation points. We fix c ∈ B C (∇(ω 0 )), and we will omit the subscript c in the following Let Q k be a sequence of T k -periodic orbits such that H(Q k ) -→ 0 and

T k -→ ∞. The orbit Q k has a covering X k ∈ E T k w k minimizing L T k w k .
We can take a subsequence to be reduced to the case where the inequality of lemma 10 is satisfied for every k, and where the sequence w k /T k has a limit ω, which satisfies

ω -ω 0 2A √ .
Let X k (m) ∈ E T k w k +m be the minimizing sequence of L T k w k +m , and Q k (m) its associated orbit on T n . We need some informations about these orbits.

Lemma 11 There is a constant C(m), depending on m but not on k, such that

L(X k (m)) C(m). Proof: L(X k (m)) L(x k 0 , . . . , x k T k -1 , x k T k + m) L(X k ) + S(x k T k -1 , x k T k + m) -S(x k T k -1 , x k T k ) L(X k ) -min S + sup x 1, y 1+m+ω 0 +3A √ S(x, y).
Lemma 12 There is a constant C(m) such that for any 0 i < j T k H(q k i (m), . . . , q k j (m)) C(m).

Proof: Let us set P = (p j , . . . , p i ) = (q k j (m), . . . , q k T k (m), q k 1 (m), . . . , q k i (m)). Writing H(P ) + H(q k i (m), q k j (m)) 0 gives H(P ) -max

(T n ) 2 H.
We get

H(q k i (m), . . . , q k j (m)) = H(Q k (m)) -H(P ) C(m) + max H.
Thanks to these lemmas, we are in a position to study the accumulation points of the sequence Q k (m). Let us fix a sequence a k of integers, we can consider the centered orbit

Θ k = θ k i = q k i+a k (m) i∈Z ,
where everything is defined by periodic continuation. There is a subsequence Θ l(k) such that l -→ θ l i has a limit θ i for every i. The biinfinite sequence Θ = (θ i ) i∈Z will be called an accumulation point of the centered orbit Θ k . An accumulation point of a centered orbit of a sequence Y k of periodic orbits will be called an accumulation point of Y k . We have the following proposition, in which as usual everything depends on a vector c ∈ R n satisfying c -∇h(ω 0 ) C .

Proposition 2 Any accumulation point of Q k (m) is the projection of an orbit that is homoclinic to Σ, any orbit obtained that way satisfies the localization y -y 0 A √ of theorem 1.

This proposition provides no existence result up to now since the accumulation point could be an orbit contained in Σ. Proof: Since Θ k are orbit segments, we have

∂ 2 S(θ k i-1 , θ k i ) = -∂ 1 S(θ k i , θ k i+1 ),
taking the limit we obtain

∂ 2 S(θ i-1 , θ i ) = -∂ 1 S(θ i , θ i+1 ),
which means that θ i is the projection of the orbit (θ i , ∂ 1 S(θ i , θ i+1 )). Since x k i+1 (m)-x k i (m)ω 0 4A √ , this orbit must stay in the zone y -y 0 A √ (where A is a new constant). To prove that this orbit is homoclinic, it is enough to prove that the α and ω-limit of the orbit (θ i , ∂ 1 S(θ i , θ i+1 )) lie in Σ. We will prove it for the ω-limit, the other part being similar. It is sufficient to see that for any subsequence θ j(i) having a limit θ, we have θ ∈ Σ c and θ j+1 -→ φ c (θ). Taking a subsequence if necessary, we can suppose that d(θ j , θ) 2 -j . Let us consider the configuration

Z j = (θ, θ j(i)+1 , . . . , θ j(i+1)-1 , θ),
it satisfies H(Z j ) 0 and H(Z j ) H(θ j(i) , θ j(i)+1 , . . . , θ j(i+1) ) + C2 -j .

Adding these inequalities and using lemma 11 yields

i 1 i=i 0 H(Z j(i) ) C2 -j(i 0 ) + H(θ j(i 0 ) , . . . , θ j(i 1 +1) ) C2 -j(i 0 ) + sup k H(θ k j(i 0 ) , . . . , θ k j(i 1 +1) ) C2 -j(i 0 ) + C(m).
We obtain that H(Z i ) -→ 0 so that θ ∈ Σ c and θ j(i)+1 -→ φ c (θ). The last assertion follows from the fact that φ c (θ) is the only possible accumulation point of the sequence θ j(i)+1 .

Nontriviality and multiplicity

As we noticed it in the previous section, proposition 2 does not guarantee the existence of nontrivial homoclinic orbits. Some additional work will be needed. Let us fix δ small, there is 0 such that Σ ∈ U δ when 0 . There also exists η such that d(q, Σ) η ⇒ q ∈ U δ when 0 . Let us define the open set

V η ⊂ R n × R n = {(x 0 , x 1 ) such that d ((x 0 , x 1 ), graph(φ c )) < η} ⊂ U δ × U δ .
We associate to any periodic orbit X ∈ E T w its relative homology

(h b (X) =< w, b > -T < ω 0 , b >) b∈B ∈ Z r ,
recall that B is a basis of the resonant group. We also associate to any point

x ∈ U δ coefficients (h b (x)) = E(< x, b >) b∈B ∈ Z r ,
where E(x) is the integer closest to x, the above giving a good definition if δ has been chosen small enough. These coefficients have of course no topological meaning.

Lemma 13 If δ has been chosen small enough, for any point x ∈ Σ,we have

h b (φ c (x)) = h b (x)+ < ω 0 , b >,
from which easily follows that

(x 0 , x 1 ) ∈ V η ⇒ h b (x 0 ) = h b (x 1 )+ < ω 0 , b > .
Proof: Let us consider a sequence X k as above with L(X k ) -→ 0 and x k 0 -→ x. The sequence x k 1 then has a limit φ c (x) satisfying φ As an easy corollary of that lemma, we obtain that if X k is an orbit such that L(

c (x) -x -ω 0 A √ . We obtain that | < φ c (x), b > -< x, b > -< ω 0 , b > | is small. Since both φ c (x)
X k ) -→ 0 then (x k i , x k i+1 ) ∈ V η for all i when k large enough, thus h b (X k ) =< x k T k , b > -< x k 0 , b > -T k < ω 0 , b > is close to h b (x k T k ) -h b (x k 0 ) -T k < ω 0 , b >= 0, so h b (X k ) = 0, which writes < w k , b >=< ω 0 , b >
for k large enough. We also define the relative homology of a φ-orbit homoclinic to Σ as follows,

h b (Θ) = lim i-→∞ (h b (θ i ) -h b (θ -i ) -2i < ω 0 , b >).
This is well defined because for i large enough, we have θ i ∈ U δ and (θ i , θ i+1 ) ∈ V η . We can now prove the existence of nontrivial orbits. Let us fix some m and consider the sequence X k (m). There is an alternative 1-There exist a k such that x k a k (m), x k a k +1 (m) admits an accumulation point out of V η , and so there is a nontrivial homoclinic, that leaves U η . 2-(x k i (m), x k i+1 (m)) ∈ V η holds when k is large enough. If 2-holds, it follows easily that the periodic orbit X k (m) has trivial relative homology:

h b (X k (m)) = 0.
This last equation writes < m, b >= 0, we can choose m so that this equation is not satisfied, in this case 2-is impossible and so 1is true.

We will now go in some kind of concentration compactness for our sequences X k (m) to obtain multiplicity. Assume m has been chosen so that a nontrivial accumulation orbit leaving U η exists. There are a sequence a k 0 , an homoclinic orbit Θ 0 and a subsequence of X k (m) that will still be noted X k (m) such that x k i+a k 0 (m) -→ θ 0 i for all i. Possibly changing a k 0 , we can suppose that there exists A 0 such that (θ 0 i , θ 0 i+1 ) ∈ V η for i < 0 or i A 0 , and given ν > 0, we can take a subsequence of our subsequence such that d(x k i+a k 0 (m), θ 0 i ) ν for |i| 2A 0 , this subsequence is still noted X k (m). Let us define

a k 1 = min{i a k 0 + A 0 such that (x k i (m), x k i+1 (m)) / ∈ V η } and a k -1 = max{i < a k 0 such that (x k i (m), x k i+1 (m)) / ∈ V η }.
We can extract another subsequence such that there exists a nontrivial homoclinic orbit Θ 1 , and an integer A 1 satisfying (θ 1 i , θ 1 i+1 ) ∈ V η when i A We can continue the process by defining

a k l = min{i a k l-1 + A l-1 such that (x k i (m), x k i+1 (m)) / ∈ V η }
and finding Θ l and A l with (θ l i , θ l i+1 ) ∈ V η when i A l and d(x k i+a k l (m), θ l i ) ν when |i| 2A l . We can do the same for l < 0. Assume now that the process ends on each side, that is we have l + 0 and l - 0 such that a k l + +1

T k and a k l --1 < 0 when k k 0 , then we have found a subsequence X k (m) such that

d(x k i (m), θ l i-a k l
) ν when a k l -2A l i a k l + 2A l and (x k i (m), x k i+1 (m)) ∈ V η when a k l + A l i < a k l+1 or T k > i a k l + + A l + or 0 i < a k l -. Looking precisely at the construction we just performed and applying lemma 13, we see that

< m, b >= h b (X k (m)) = l + l=l - h b (Θ l ). ( 4
)
Lemma 14 If there are only a finite number N of geometrically distinct homoclinics then the process described above ends for each m. It follows that the relative homologies of these homoclinics must additively generate Z r and thus N r + 1.

Proof: If the process always ends, then the above sum formula (4) must be satisfied, and since the function m -→ (< m, b >) b∈B ∈ Z r is easily seen to be onto, we have the second part of the lemma. It remains to prove that the process ends. If there are only N homoclinic orbits, there are at most N leaving U η , one of them must occur at least (l + -l -)/N times in the sequence Θ l defined above, that is there are indices l 1 , . . . , l K with K (l + -l -)/N such that Θ l j = Θ. Let us set -1 , θ 0 .

Y k j = x k
The configuration Ỹ k j is periodic and leaves U η , so there must be o > 0 such that L( Ỹ k j ) 2o. If ν has been chosen small enough, it follows that L(Y k j ) o. This gives C(m) L(X k (m)) Ko which is a majoration for K and thus for l + -l -.

  and x are in U δ , < φ c (x), b > and < x, b > are close to integers, the result follows.

1 and d(x k i+a k 1 (

 11 m), θ 1 i ) ν when |i| 2A 1 .

Acknowledgments : I thank Professor E. Séré for his comments and encouragement. I thank the referee for useful remarks.


DiG: Data centers in the Grid Hardik Soni Inria Sophia Antipolis, France Email: hardik.soni@inria.fr Damien Saucez Inria Sophia Antipolis, France Email: damien.saucez@inria.fr Thierry Turletti Inria Sophia Antipolis, France Email: thierry.turletti@inria.fr Abstract-We are witnessing a considerable amount of research work related to data center and cloud infrastructures but evaluations are often limited to small-scale scenarios as very few researchers have access to a real infrastructure to confront their ideas to reality. In this demo we will reveal our experiment automation tool, DiG (Data centers in the Grid), which explicitly allocates physical resources in grids to emulate data center and cloud networks. DiG allows one to utilize grid infrastructures to evaluate research ideas pertaining to data centers and cloud environments at massive scale and with real traffic workload. We have automated the procedure of building target network topologies while respecting available physical resources in the grid against the demand of links and hosts in the experiment. We will present a showcase where DiG automatically builds a large data center topology composed of hundreds of servers executing various Hadoop intensive workloads.

I. INTRODUCTION

Most SDN experiments having data center and cloud scenarios are performed with traffic traces using emulators (e.g., Mininet [START_REF] Lantz | A network in a laptop: Rapid prototyping for software-defined networks[END_REF], Maxinet [START_REF] Wette | Maxinet: Distributed emulation of software-defined networks[END_REF]) or simulators (e.g. ns-3 [START_REF]ns-3 Project page[END_REF]) due to restricted access to real production environments of companies like Amazon, Google, or Facebook. Therefore, experiment results may be biased or noisy due to modeling techniques of simulators or unaccounted and excessive usage of physical resources in case of emulation.

Many tools exist like Mininet [START_REF] Lantz | A network in a laptop: Rapid prototyping for software-defined networks[END_REF] and Maxinet [START_REF] Wette | Maxinet: Distributed emulation of software-defined networks[END_REF] for running SDN experimentations. Among them, Maxinet is the closest to our work. However, it targets scalable emulation to create SDN enabled data center environments and relies on synthetic traffic generation models. Maxinet is built using Mininet, which has the capability to run real world applications to generate traffic. However, at the scale of hundreds of hosts, running such applications on emulated hosts consume computing resources and hinders emulation's scaling capability of network experiments. Neither Maxinet nor Mininet provides guarantee on allocation of computing power (i.e., CPU cores) for emulated hosts to scale the entire experiment with a minimum amount of physical resources.

In this work, we aim (1) to create data center topologies while respecting computing and network resource constraints and (2) to run real world data center applications on top of it. Since very few researchers have access to production data center environments and the majority of them has access to grid computing environments like Grid5000 [START_REF]Grid'5000 Project page[END_REF], the primary goal of our system is to build test environments for SDN enabled data center and cloud networks in grid physical infrastructures. With DiG (Data centers in the Grid) we can build overlay experimental networks by explicitly allocating available physical resources, like CPU and link capacity, to the requirements of experimental network topologies, allowing to run real world data center applications on top of a grid with performance guarantees.

II. SYSTEM DESCRIPTION

A. Overlay Experimental Network with Resource Guarantee

The DiG system is able to create experimental networks that carry real traffic between nodes running protocol stacks as in real world data center networks. To achieve this, DiG implements a layer 2 overlay network on a grid infrastructure. Having a layer 2 overlay network as an experimental network provides a bare-metal network environment to the SDN controllers and switches.

DiG instantiates data center servers and switches by running virtual machines (VMs) and OpenFlow enabled switches on grid nodes. While creating an overlay network, it is important to take into account the available computing power of the grid nodes and the physical link capacity between each pair of grid nodes. In most of the cases, the physical network connectivity along with the computing power of the grid nodes are known by the experimenters.

B. DiG Technical Description

DiG maps the experimental network on a physical grid network while satisfying the computing power requirements of all the nodes in the experimental network and not exceeding the computing capacity of grid nodes. Similarly, layer 2 overlay links are mapped by satisfying the demand of all the links in experimental topology while not overloading physical links. So, the problem is reduced to the resolution of a Virtual Network Embedding (VNE) [START_REF] Fischer | Virtual network embedding: A survey[END_REF] problem with constraints on nodes computing power and links capacity.

DiG runs in three phases to implement experimental networks on grid infrastructures. Each phase generates an output in the form of text files and these files are used in the next phase as an input. This makes the system more flexible and facilitates modifications and integration of different phases implementations. The names of the three phases are Experimental Network Embedding, Configuration Generation, and Deployment. Note that each phase can be run in an independent way with appropriate input files, without the need of executing other phases. 1) Experimental Network Embedding: DiG solves a VNE problem using the ALEVIN [START_REF] Beck | A simulation framework for virtual network embedding algorithms[END_REF] framework, which is used to generate the mapping between the experimental network and the grid physical infrastructure. ALEVIN is fed with the Fig. 1. DiG Module interaction experimental and grid networks described in DOT language [START_REF]The DOT Language[END_REF] in a text file. The experimental network is annotated with CPU cores requests for nodes, link capacity requests and other application-specific attributes like Hadoop node type for automation usage. The grid network is annotated with CPU core capacity for grid nodes, link bandwidth capacity for physical links and IP addresses for automation purpose. ALEVIN uses CPU cores and link bandwidth attributes for both experimental and grid networks and it generates a node mapping file as shown in Fig. 1. The node mapping file is a text file that identifies the set of experimental network nodes mapped on each physical node.

2) Configuration Generator for Experimental Network:

The Configuration Generator phase takes as input the mappings generated from the Experimental Network Embedding phase along with the descriptions of the experimental and grid networks in DOT format. However, the mapping file can be generated by any means, and not necessarily with the technique presented in Sec. II-B1. This allows (1) running different tools and algorithms for the network embedding step and (2) relaxing the strong dependency on the performance of embedding algorithms.

The Configuration Generator phase prepares the configuration files for each physical host based on the mapping. It contains the meta-data to instantiate the mapped part of the experimental network on physical hosts. The meta-data contains the appropriate commands and the parameters to instantiate the virtual machines and to map the virtual hosts to the physical nodes. It also contains the necessary information (e.g., source-destination UDP port numbers, IPs of grid nodes, tunnel unique IDs etc.) to create layer 2 tunneling protocol (i.e., L2TPv3) endpoints and links capacity information satisfying the experimental network bandwidth demand based on the mapping. Along with the experimental network configuration files, this phase generates files to bring up basic network utility (e.g., assigning IP to experimental network interface, routing etc.) in the hosts.

3) Deployment of Experimental Network: The last phase consists of the deployment of the experimental network using configuration files on the physical machines. DiG instantiates the virtual hosts in the experimental network on grid nodes, creates OpenFlow [START_REF] Mckeown | Openflow: Enabling innovation in campus networks[END_REF] switches interconnected with L2TPv3 tunnels and controls the link bandwidth according to the requirements of the experimental network to emulate. It is also responsible to launch applications on virtual hosts of the experimental network. The Linux Traffic Control utility (tc) is used to control the bandwidth at the tunnel interfaces according to the links capacity requirements of the experimental network.

C. Management Network

As mentioned above, the deployment phase launches applications in virtual hosts. DiG uses a designated node called Manager node in the grid infrastructure to launch the deployment phase in a centralized way. All the communications required for deployment purpose and management of experimental network are carried out on a dedicated management network isolated by experimental networks, as depicted in Fig. 2.

Fig. 2. Experimental Overlay Network with Management Network

Each virtual host in the experimental network includes a management network interface. A management bridge is created on all the grid nodes including the manager node, as shown in Fig. 2. The virtual hosts of the experimental network running on a grid node are connected to the management bridge on the grid node through their management interface. The management bridge on each grid node is connected to the management bridge on the manager node. The Deployment tool is executed on the manager node; it uses the management network to dispatch the commands to launch applications on different VMs in the experimental network. This approach prevents any possible management traffic interfering in the experimental network that could distort experimental results.

III. SHOWCASE

The primary goal of DiG is to create experimental network environments with resource guarantee to imitate real world SDN-based data center and ISP topologies with high level of realism. Such network environments can be used for instance to test and evaluate performance of SDN controllers or routing algorithms with different real time traffic or topologies.

In this demo, we will showcase how to automatically emulate an OpenFlow data center composed of hundreds of servers in Grid5000. The second phase of the demonstration will deploy and run Hadoop benchmark programs. Hadoop is used in many real world data centers and many benchmark suites exist (e.g., HiBench [START_REF]HiBench[END_REF]). Interestingly, Hadoop MapReduce applications generate a substantial amount of traffic during the data shuffling phase and particularly the TestDFSIO and TeraSort are MapReduce benchmark applications. Hence, they can be a primary choice for data-center workload generation to demonstrate the effectiveness of DiG.

IV. EXPERIMENTAL REQUIREMENT 1) Stable high speed Internet connection without blocking SSH traffic. 2) Power supply plug. 3) Large monitor for better visuals (to be provided, possibly by the organizers).

V. CONCLUSION

In this paper, we present the DiG tool to create easily SDN data center and ISP networks on a Grid infrastructure. DiG runs network embedding algorithms to emulate data center infrastructures in a Grid with performance guarantees. DiG automatically creates L2 overlay experimental networks and hosts based on the output of the embedding algorithms and can launch any off-the-shelf application on the experimental hosts to generate workload on the data center to evaluate. We demonstrate DiG by automatically running Hadoop benchmarks in an emulated data center of hundreds of nodes. DiG is available to the community at URL http://team.inria.fr/diana/software/.

In the future we plan to use DiG to study SDN controller performance as carried out in [START_REF] Tootoonchian | On controller performance in software-defined networks[END_REF] but with real applications and traffic. Similarly, DISCO [START_REF] Phemius | Disco: Distributed sdn controllers in a multi-domain environment[END_REF], ElastiCon [START_REF] Dixit | Elasticon: An elastic distributed sdn controller[END_REF], Hedera [START_REF] Al-Fares | Hedera: Dynamic flow scheduling for data center networks[END_REF], DIFANE [START_REF] Yu | Scalable flow-based networking with difane[END_REF], DevoFlow [START_REF] Curtis | Devoflow: Scaling flow management for highperformance networks[END_REF], Kandoo [START_REF] Hassas Yeganeh | Kandoo: A framework for efficient and scalable offloading of control applications[END_REF], Onix [START_REF] Koponen | Onix: A distributed control platform for large-scale production networks[END_REF] or Beehive [START_REF] Yeganeh | Beehive: Towards a simple abstraction for scalable software-defined networking[END_REF] could be tested and evaluated on the DiG system with real network conditions and traffic.

ACKNOWLEDGMENTS

This work was partially supported by the ANR Reflexion Project (ANR-14-CE28-0019). Experiments presented in this paper were carried out using the Grid'5000 testbed, supported by a scientific interest group hosted by Inria and including CNRS, RENATER and several Universities as well as other organizations (see https://www.grid5000.fr).