In this text we take up the problem of the splitting
of invariant manifolds in multidimensional Hamiltonian systems,
stressing the canonical features of the problem. We first conduct
a geometric study, which for a large part is not restricted to
the perturbative situation of near-integrable systems.
This point of view allows us to clarify some previously obscure points,
in particular the symmetry and variance properties of the splitting matrix
(indeed its very definition(s)) and more generally the connection
with symplectic geometry. Using symplectic normal forms, we then
derive local exponential upper bounds for the splitting
matrix in the perturbative analytic case, under fairly
general circumstances covering in particular resonances of any
multiplicity. The next technical input is the introduction of a
canonically invariant scheme for the computation of the splitting
matrix. It is based on the familiar Hamilton-Jacobi
picture and thus again is symplectically invariant from the outset.
It is applied here to a standard Hamiltonian exhibiting many of the
important features of the problem and allows us to explore in a
unified way the question of finding lower bounds for the
splitting matrix, in particular that of justifying a first order
computation (the so-called PoincarĂ©-Melnikov approximation).
Although we do not specifically address the issue in this paper we
mention that the problem of the splitting of the invariant manifold is
well-known to be connected with the existence of a global instability in
these multidimensional Hamiltonian systems and we hope the present study
will ultimately help shed light on this important connection first noted
and explored by V. I. Arnold.

Readership

Graduate students and research mathematicians interested in
geometry, topology, and analysis.