Geodesics with Unbounded Speed on Fluctuating Surfaces

Abstract

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.