CHAPTER 1
Symplectic Geometry and the
Splitting of Invariant Manifolds
In this first chapter we give an exposition of the geometric ideas involved in
the study of the splitting of invariant manifolds. For the convenience of the reader,
and also for the sake of fixing notation, the first two paragraphs shortly recall
the necessary basic notions concerning symplectic geometry (§1.1) and hyperbolic
sets in a general setting (§1.2). We refer to [AG] and [MDS] for detailed treat-
ments of symplectic geometry, and to [HK] for an excellent general exposition of
dynamical systems theory. We work everywhere with Hamiltonian flows, defined
on a 2^-dimensional symplectic manifold (M, f2) by at least C2 Hamiltonians; most
applications will be to analytic systems. We leave it to the reader to convince him-
self that many properties can easily be adapted to other settings, in particular to
symplectic maps instead of flows.
We introduce in Paragraphs 1.3-1.7 various notions of angular splitting, first at
a linear level (splitting of two linear Lagrangian subspaces in a symplectic vector
space), then for pairs of Lagrangian submanifolds of a symplectic manifold. We will
distinguish between the symplectic notions, which can be defined without additional
structure, and the Euclidean ones, which require in addition fixing an almost com-
plex structure on the ambient manifold. The definitions are given first intrinsically,
and then translated in local coordinates systems; we will stress the influence of the
changes of coordinates on these local interpretations.
Paragraph 1.8 gives the definition and basic properties of hyperbolic tori in the
Hamiltonian framework. We first introduce the dynamical definition of hyperbolic-
ity, which does not require the knowledge of a particular normal form near the tori,
and then take into account the symplectic geometry of the problem. The present
study is thus general enough to be applied, including in a perturbative setting, to
families of non-KAM hyperbolic tori, for instance the ones with Liouvillian rotation
vectors (see [Yl]). Similar definitions are given in [Boll], [Bol2], [BT].
Given a Hamiltonian H on M and an integer m satisfying 1 m £ 1, a
m-hyperbolic invariant torus T is an m)-dimensional invariant torus in M such
that the flow defined by H restricts to an ergodic flow on T (which in applications
is generally conjugate to a transitive rotation) with exactly m strictly positive
Lyapunov exponents, hence also m strictly negative ones (see the precise definition
below). Let us mention that the case m = 1 is indeed special, as far as diffusion
properties are concerned, and there are compelling reasons which lead to restrict
certain aspects of the study to 1-hyperbolic tori. We have tried nevertheless to
give general definitions and proofs for hyperbolic tori, most (but not all) of them
being independent of the multiplicity m. It should be pointed out from the start
that of course, if m j^ £ 1, a m-hyperbolic torus is not normally hyperbolic, for
dimensional reasons, and that the case m = £ 1 corresponds to the well-known
Birkhoff's (normally) hyperbolic periodic orbits. The theory of pseudo-hyperbolic
manifolds (recalled in §1.2) makes it nevertheless possible in all cases to prove the
existence of invariant stable and unstable manifolds for T.
At the symplectic level, the torus T is assumed to be isotropic, and by the
(dynamical) definition of hyperbolic objects, its invariant manifolds will be isotropic
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