Amadeu Delshams, Rafael de la Llave, Tere M. Seara
Geometric properties of the scattering map of a normally hyperbolic invariant manifold
(747K, pdf)
ABSTRACT. Given a normally hyperbolic invariant manifold $\Lambda$
for a map $f$, whose stable and unstable invariant manifolds
intersect transversally, we consider its associated scattering map.
That is, the map that, given an asymptotic orbit in the past,
gives the asymptotic orbit in the future.
We show that when $f$ and $\Lambda$ are symplectic (resp. exact
symplectic) then, the scattering map is symplectic (resp. exact
symplectic). Furthermore, we show that, in the exact symplectic
case, there are extremely easy formulas for the primitive
function, which have a variational interpretation as difference of
actions.
We use this geometric information to obtain efficient perturbative
calculations of the scattering map using deformation theory. This
perturbation theory generalizes and extends several results
already obtained using the Melnikov method.
Analogous results are true for Hamiltonian flows.
The proofs are obtained by geometric natural methods
and do not involve the use of particular coordinate
systems, hence the results can be used to obtain
intersection properties of objects of any type.
We also reexamine the calculation of the scattering map
in a geodesic flow perturbed by a quasi-periodic potential.
We show that the geometric theory reproduces the results obtained
by the authors in Mem. Amer. Math. Soc., 179(844):1-141, 2006,
using methods of fast-slow systems. Moreover, the geometric theory
allows to compute perturbatively the dependence on the slow
variables, which does not seem to be accessible to
the previous methods.