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.
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