Invariant Objects in Volume Preserving Maps and Flows
de la Llave, Rafael
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We consider smooth volume preserving maps. Two important phenomena are transport and mixing. We present several geometric obstructions that prevent transport and mixing and numerical methods to compute them. These are quasiperiodic orbits of the maps and their treatment requires KAM (Kolmogorov-Arnold-Moser) techniques for an analytic treatment. The result we present has an a-posteriori format (an approximate solution with good condition numbers implies a true solution) and it also leads to very efficient algorithms (low storage requirements and low operation count). These algorithms have been implemented and run (by J. Meiss and A. Fox) and they formulated to conjectures about breakdown. A novelty of the method is that the topology also plays a role. Depending on the global topology the tori may be obstructions to mixing but not to transport or be obstructions to transport and mixing. This is joint work with T. Blass and with A. Fox.