Perturbations of geodesic flows producing unbounded growth of energy

Marian Gidea , Institute for Advanced Study
Fine Hall 322

We consider a geodesic flow on a manifold endowed with some generic Riemannian metric. We couple the geodesic flow with a time-dependent potential driven by an external dynamical system, which is assumed to satisfy some recurrence condition. We prove that there exist orbits whose energy grows unboundedly at a linear rate with respect to time; this growth rate is optimal. In particular, we obtain unbounded growth of energy in the case when the external dynamical system is quasi-periodic, of rationally independent frequency vector (not necessarily Diophantine). Our result generalizes Mather's acceleration theorem and is related to Arnold's diffusion problem. It also extends some earlier results by Delshams, de la Llave and Seara.