Perturbations of geodesic flows producing unbounded growth of energy

Marian Gidea (Northeastern Illinois Univ. & IAS)

Abstract: 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.