Volume preserving mappings model the motion of fluid particles in incompressible fluids and the structure of magnetic field lines. They also provide the simplest conservative generalization of highly studied area-preserving mappings. We have recently been investigating some of the dynamical phenomena that can occur in these systems. The simplest such systems have an invariant which restricts the motion to two-dimensional surfaces. We will show several examples of these including "trace maps" motivated from quantum mechanics, maps that we construct generalizing an idea of Suris, and a fluid stirring protocol based on an idea of Aref.
The destruction of invariant surfaces can often computed by a Melnikov technique. We discovered that the two-dimensional manifolds emanating from fixed points can intersect in surprising ways, and that the Melnikov method can be used to predict the topology of the intersections. This method can also be applied to the stable and unstable manifolds of invariant circles.
Many open questions remain. We hope to leave the audience thirsty for answers.