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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