A. Delshams, R. de la Llave, T. M.-Seara
Orbits of unbounded energy in quasi-periodic perturbations of 
geodesic flows
(1313K, pdf)

ABSTRACT.  We show that certain mechanical systems, including a geodesic 
flow in any dimension plus a quasi-periodic perturbation by a 
potential, have orbits of unbounded energy. 
The assumptions we make in the case of geodesic flows are: 
\begin{itemize} 
\item[a)] The 
metric and the external perturbation are smooth enough. 
\item[b)] The 
geodesic flow has a hyperbolic periodic orbit such that its stable 
and unstable manifolds have a tranverse homoclinic intersection. 
\item[c)] 
The frequency of the external perturbation is Diophantine. 
\item[d)] The external potential satisfies a generic condition 
depending on the periodic orbit considered in b). 
\end{itemize} 
The assumptions on the metric are $\C^2$ open and are known to be 
dense on many manifolds. The assumptions on the potential fail 
only in infinite codimension spaces of potentials. 
The proof is based on geometric considerations of invariant 
manifolds and their intersections. The main tools include the 
scattering map of normally hyperbolic invariant manifolds, as well 
as standard perturbation theories (averaging, KAM and Melnikov 
techniques). 
We do not need to assume that the metric is Riemannian and we 
obtain results for Finsler or Lorentz metrics. Indeed, there is a 
formulation for Hamiltonian systems satisfying scaling hypotheses. 
We do not need to make assumptions on the global topology of the 
manifold nor on its dimension.