Geodesics with Unbounded Speed on Fluctuating Surfaces

We construct $C^{\infty}$ time-periodic fluctuating surfaces in $\mathbb R^3$ 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 $\mathbb R^3$ on which the geodesic flow (with respect to the induced metric from $\mathbb R^3$) 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 $\Lambda$, the stable and unstable manifolds of which have a transverse homoclinic intersection. The surface $M$ is embedded into $\mathbb R^3$ via a near-identity time-periodic embedding $G: M \to \mathbb R^3$. 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 $\Lambda$, 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.