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Research of James Meiss
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Nearly integrable volume-preserving maps with two angles and one action have robust invariant tori
with Diophantine rotation vectors (under smoothness and twist conditions); this
is a consequence of Jeff Xia’s KAM theory. How long do these tori persist? Are they replaced
by “cantori” when they are destroyed?
In Fox and Meiss we study a volume-preserving family
introduced by Dullin and Meiss that represents (just like Chirikov’s
standard map) a generic form near resonance. We show that a generalization of
John Greene’s residue criterion for the standard map holds: tori exist when nearby periodic orbits are
stable---their residues are small, and are destroyed when these residues blow-up.
We study tori with “spiral tails” in their
generalized Farey tree expansions, and look for the most robust torus, in an attempt
to discover a higher-dimensional version of the noble tori of the area-preserving case.
The figure shows a period 17991 orbit approximating an incommensurate orbit with spiral mean rotation vector, just past the destruction of the invariant torus. Is it a Sierpinski carpet?
This research, and much of the research listed below has received support from the National Science Foundation, most recently under grants DMS-0707659 and DMS-1211350.