  We consider an integrable Hamiltonian system weakly coupled with a
pendulum-type system. For each energy level within some range, the uncoupled
system is assumed to possess a normally hyperbolic invariant manifold
diffeomorphic to a three-sphere, which bounds a strictly convex domain, and
whose stable and unstable invariant manifolds coincide. The Hamiltonian flow on
the three-sphere is equivalent to the Reeb flow for the induced contact form.
The strict convexity condition implies that the contact structure on the
three-sphere is tight. When a small, generic coupling is added to the system,
the normally hyperbolic invariant manifold is preserved as a three-sphere, and
the stable and unstable manifolds split, yielding transverse intersections. We
show that there exist trajectories that follow any prescribed collection of
invariant tori and Aubry-Mather sets within some global section of the flow
restricted to the three-sphere. In this sense, we say that the perturbed system
exhibits global diffusion on the tight three-sphere.
