Just as one can generate an exact, symplectic map by an implicit generating function, exact volume-preserving maps can be obtained by implicit generating forms. Hector Lomeli and I have shown recently how to generalize this idea, originally due to Carroll, to obtain generating forms of different "types" analogous to the four standard canonical generators.

A first step in a theory of transport is computation of the flux though lobes of turnstiles. For the area preserving case, these fluxes are given by differences between the actions of the heteroclinic orbits that bound the lobes. We show how to generalize this to the exact volume-preserving case. In this case the generating forms give rise to an "action" given by an integral along heteroclinic curves.

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