10

P.LOCHAK, J.-P.MARCO AND D.SAUZIN

too and thus Lagrangian, since they are ^-dimensional. This Lagrangian character

is probably the most important datum in our study, and will prove useful over

and over again; in particular the various notions of homoclinic splitting for these

invariant manifolds appear as particular cases of the preceding definitions (§§1.3-

1.7). Again part of these results are obtained in [Boll], [Bol2] and [BT] in a general

context.

We describe in §1.9 a class of Hamiltonian systems which exhibit the main fea-

tures that we want to analyze in the perturbative setting. That class generalizes

both Hamiltonian (*) and the normal forms one has to consider near (simple or

multiple) resonances. We prove in particular some symplectic straightening theo-

rems for the invariant manifolds, which in turn allow us in Chapter 2 to get simpler

normal forms and to facilitate the application of KAM theory.

In Paragraph 1.10, we address the question of the existence of homoclinic orbits,

in the perturbative setting, from the point of view of Lagrangian intersection theory.

We generalize Eliasson's study [El] of the m = 1 case, and give very simple existence

proofs for all multiplicities, based on the comparison of the cohomologies of the

invariant manifolds. Our approach requires the knowledge of a homoclinic orbit

biasymptotic to the fixed point in the averaged system, the existence of which is

proved by usual variational methods, and we also assume (though it is not really

necessary) the transversality of the intersection along this homoclinic. As far as the

existence properties alone are concerned, variational methods can also be applied

and lead to more general results (see [Boll], [Bol2], [Fa]), whereas we believe that

our method will be more efficient for the sake of locating the homoclinic orbits and

proving the existence of genuine hyperbolic behaviour in the system.

Finally, Paragraph 1.11 is devoted to a geometric description of the splitting of

the invariant manifolds of hyperbolic tori, mainly in the perturbative setting, for

which we examine the connections between our definitions and the usual formalism

(based essentially on the Poincare-Melnikov integrals).

As a conclusion, we note that the notion of homoclinic splitting of manifolds

has in fact several geometric and dynamical meanings. In particular it quantifies

the maximal distance between two nearby hyperbolic tori connected by a hetero-

clinic orbit; passing from a homoclinic splitting to a heteroclinic distance is usually

done by using suitable versions of the Implicit Function Theorem, which, due to

the natural decomposition of the space in horizontal and vertical directions, re-

quires only the purely symplectic interpretation. It also quantifies some dynamical

parameters, such as the hyperbolicity of secondary invariants sets which appear,

under additional assumptions on the torsion, in the neighborhood of transversal

homoclinic tori (see [Crl]), and this point of view necessitates the full geometric

information and thus the Euclidean definition. Recent developments in symplectic

topology make it possible to give more global definitions of the splitting, mainly

based on Hofer's notion of displacement energy, but we will not enter into such con-

siderations here. There are also interesting connections of this ubiquitous quantity

with the notion of Peierls' barrier which is familiar in a variational setting ([Fa]).

1.1. Symplectic geometry: a short reminder

We let (M, ft) denote a 2£-dimensional smooth symplectic manifold and list

some elementary notions of the attached geometry to fix notation.