The Connecting Orbit Theory

If we consider the geodesic flow for a small perturbation of the flat Riemannian metric on the torus \(\mathbb T^n,\ n>2\) , the analogous problem of Arnold diffusion is to find an orbit that is nearly dense on the unit tangent bundle. In particular, we have to know how to change the direction of the velocity of an orbit in a prescribed manner. In this chapter, we present the theory of connecting orbits that gives sufficient conditions for such a change of velocity. The key point is that we shall find orbits shadowing those heteroclinic orbits in the Mañé set, which connect Aubry sets with different cohomology classes. Recall that for integrable systems, the cohomology classes coincide with the action variables, thus this gives a way to change the action variables, i.e., a way to approach the problem of Arnold diffusion. When the Aubry sets are hyperbolic invariant sets, this would imply that the stable manifold of one Aubry set intersects the unstable manifold of another. In many cases, the invariant manifolds are Lagrangian submanifolds, thus this gives wild and global behavior of Lagrangian submanifolds in contrast to integrable systems.