DIFFUSING THROUGH SPECTERS - RIDGE CURVES, GHOST CIRCLES AND A PARTITION OF PHASE-SPACE

The study of transport in Hamiltonian and related systems is greatly illuminated if one can construct a framework of ''almost invariant'' surfaces to organize the dynamics. This can be done in the case of area-preserving twist maps, using pieces of stable 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 Gole christened ''ghost circles''. 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 Gole. We propose that ghost circles form a convenient, natural skeleton around which to organize studies of transport.