Geodesics with unbounded speed on fluctuating surfaces

Abstract

We construct C8 time-periodic fluctuating surfaces in R3 such that the corresponding non-autonomous geodesic flow has orbits along which the energy, and thus the speed goes to infinity. We begin with a static surface M in R3 on which the geodesic flow (with respect to the induced metric from R3) has a hyperbolic periodic orbit with a transverse homoclinic orbit. Taking this hyperbolic periodic orbit in an interval of energy levels gives us a normally hyperbolic invariant manifold ¿, the stable and unstable manifolds of which have a transverse homoclinic intersection. The surface M is embedded into R3 via a near-identity time-periodic embedding G : M ¿ R3. Then the pullback under G of the induced metric on G(M) is a time-periodic metric on M, and the corresponding geodesic flow has a normally hyperbolic invariant manifold close to ¿, with stable and unstable manifolds intersecting transversely along a homoclinic channel. Perturbative techniques are used to calculate the scattering map and construct pseudo-orbits that move up along the cylinder. The energy tends to infinity along such pseudo-orbits. Finally, existing shadowing methods are applied to establish the existence of actual orbits of the non-autonomous geodesic flow shadowing these pseudo-orbits. In the same way we prove the existence of oscillatory trajectories, along which the limit inferior of the energy is finite, but the limit superior is infinite.

Peer Reviewed

Postprint (published version)