Renato C. Calleja, Alessandra Celletti, Corrado Falcolini, and Rafael de la Llave
A partial justification for Greene's criterion for conformally symplectic systems
(441K, pdf)
ABSTRACT. Greene's criterion for twist mappings asserts the existence of smooth invariant
circles with preassigned rotation number if and only if the periodic trajectories
with frequency approaching that of the quasi-periodic orbit are linearly stable.
We formulate an extension of this criterion for conformally symplectic systems in
any dimension and prove one direction of the implication, namely that if there is a
smooth invariant attractor, we can predict the eigenvalues of the periodic orbits whose
frequencies approximate that of the tori. The proof of this result is very different
from the proof in the area preserving case, since in the conformally symplectic case the
existence of periodic orbits requires adjusting parameters. Also, as shown in [13], in
the conformally symplectic case there are no Birkhoff invariants giving obstructions to
linearization near an invariant torus.
As a byproduct of the techniques developed here, we obtain quantitative information
on the existence of periodic orbits in the neighborhood of quasi-periodic tori and we
provide upper and lower bounds on the width of the Arnold tongues in n-degrees of
freedom conformally symplectic systems.