Transport in Hamiltonian systems

Abstract

We develop a theory of transport in Hamiltonian systems in the context of iteration of area-preserving maps. Invariant closed curves present complete barriers to transport, but in regions without such curves there are still invariant Cantor sets named cantori, which appear to form partial barriers. The flux through the gaps of the cantori is given by Mather's differences in action. This gives useful bounds on transport between regions, and for one parameter families of maps it provides a universal scaling law when a curve has just broken. The bounds and scaling law both agree well with numerical experiment of Chirikov and help to explain an apparent disagreement with results of Greene. By dividing the irregular components of phase space into regions separated by the strongest partial barriers, and assuming that the motion is mixing within these regions, we present a global picture of transport, and indicate how it can be used, for example, to predict confinement times and to explain longtime tails in the decay of correlations. © 1984.