Diffusing through spectres: ridge curves, ghost circles and a partition of phase space

Abstract

The study of transport in Hamiltonian and related systems is greatly illuminated if one can construct a framework of “almost invariant” surfaces to organized the dynamics. This can be done in the case of area-preserving twist maps, using pieces of table and unstable manifold of periodic orbits or cantori, as shown by MacKay, Meiss and Percival. The resulting surfaces, however, are not necessarily the most appropriate ones, as they need not be graphs, nor is it clear that they can always be chosen mutually disjoint. Hall proposed a choice based on “ridge curves” for the gradient flow of the associated variational problem, which Golé christened “ghost cicles”. They have the advantage that they are always graphs. In this Letter, we present numerical experiments suggesting that ghost circles are mutually disjoint. Our work has subsequently led to a proof of this by Angenent and Golé. We propose that ghost circles from a convenient, natural skeleton around which to organize studies of transport.