Integrable symplectic maps can be written in angle-action form, with canonically paired variables. By contrast, action and angle variables need not be paired in an integrable volume-preserving map (and of course the phase space need not be even dimensional). The dynamics of nearly-integrable symplectic maps is heavily constrained by the famous KAM and Nekhoroshev theories. In particular, the former implies the existence of a web of tori on which the dynamics is quasiperiodic, and the latter that that transport though around the web is extremely slow—on exponentially long timescales. How does the dynamics differ in the volume preserving case? In certain cases there is a generalization of KAM theory implying the existence of many invariant tori. It seems, however, that transport through the web need not be slow, and ballistic drift and rapid diffusion along rank-one resonances can occur.
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