Perturbations of geodesic flows by recurrent dynamics

Abstract: We consider a geodesic flow on a compact manifold endowed with a Riemannian (or Finsler, or Lorentz) metric satisfying some generic, explicit conditions. 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 orbits whose energy grows at a linear rate with respect to time. This growth rate is optimal. We also show the existence of symbolic dynamics. The existence of orbits whose energy grows unboundedly in time is related to Arnold's diffusion problem. A particular case of this phenomenon is obtained when the external dynamical system is quasi-periodic, of rationally independent frequency vector, not necessarily Diophantine, thus extending earlier results. We also recover `Mather's acceleration theorem' for time-periodic perturbations. Since the general class of perturbations that we consider in this paper are not characterized by a frequency of motion, we are not able to use KAM or Aubry-Mather or averaging theory. We devise a new mechanism to construct orbits whose energy evolves in a prescribed fashion. This mechanism intertwines the inner dynamics restricted to some normally hyperbolic manifold with the outer dynamics corresponding to two distinct choices of homoclinic excursions to that manifold. As this mechanism uses very coarse information on the dynamics, it is potentially of wider applicability.