Geometric properties of the scattering map of a normally hyperbolic invariant manifold
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hdl:2117/542
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Data publicació2006-10-10
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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
geometrically 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 in
[Delshams et al., Adv. Math., 202(1):64-188,2006] 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.
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