2014-11-21
,
de la Llave, Rafael
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.