Turnstiles and Transport: An Introduction
Turnstiles were introduced by MacKay, Meiss and Percival in 1984  as a mechanism to quantify the flux through broken invariant surfaces, such as the homoclinic tangle formed by an unstable periodic orbit or a cantorus that is the remnant of an invariant torus. A turnstile is constructed from the stable and unstable manifolds of a pair of orbits, and allows one to visualize the transport as the flux through a partial barrier, and show that it can be localized to a single “gap” in the orbit pair. However, the flux is essentially independent of the method in which a partial barrier is constructed. Moreover, the variational principle for a Hamiltonian system, or area-preserving map, allows one to compute the flux as the difference between the actions of the bounding orbits. In the case of magnetic field line flow, the equivalent action is the line integral of the vector potential along the orbit.
MacKay’s renormalization theory gives a scaling law for the growth of the flux as a function of perturbation strength as a torus is broken. A related power law appears to hold for the exit-time distribution from a chaotic region containing regular islands, though a complete theoretical justification is still lacking.
 MacKay, R. S., J. D. Meiss and I. C. Percival (1984). “Transport in Hamiltonian Systems.” Physica D 13: 55-81.
 Meiss, J. D. (1992). “Symplectic Maps, Variational Principles, and Transport.” Rev. Mod. Phys. 64(3): 795-848.
 Meiss, J. D. (2015). “Thirty years of turnstiles and transport.” Chaos 25(9): 097602.
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