Resonances and Twist in Volume-Preserving Mappings

The phase space of an integrable volume-preserving map with one action is foliated by a oneparameter family of invariant tori. Perturbations lead to chaotic dynamics with interesting transport properties. We show that near a rank-one resonant torus the mapping can be reduced to a volume-preserving standard map. This map is a twist map only when the frequency map crosses the resonance curve transversely. We show that these maps can be reduced using averaging theory to the usual area-preserving twist or nontwist standard maps. In the volume-preserving setting the twist condition is shown to be distinct from the nondegeneracy condition of used in KAM theory.