Perturbations of geodesic flows by recurrent dynamics
We consider a geodesic flow on a compact manifold endowed with a Riemannian
metric satisfying some generic, explicit conditions. It is well known that
the energy is preserved along the trajectories of the geodesic flow.
However, this is not necessarily true if time-dependent perturbations are
applied to the system. We couple the geodesic flow with a time-dependent
potential, driven by an external flow on some other compact manifold. If
the external flow satisfies some very general recurrence condition, and
the potential satisfies some explicit conditions that are also very
general, we show that the coupled system has trajectories whose energy
grows to infinity at a linear rate with respect to time; this growth rate
is optimal. This is joint work with Rafael de la Llave.