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| Invariant tori are prominent features of Hamiltonian and symplectic dynamical systems that are integrable or nearly so. The celebrated KAM theorem implies the structural stability of certain ``very irrational" (Diophantine) tori, and robustness is responsible for the long time correlations and slow transport in chaotic Hamiltonian dynamics.
Each preserved torus has an associated rotation vector. When the rotation vector is a scalar (two-tori for flows, one-tori for maps), the torus can be approximated by a sequence of periodic orbits obtained by the continued fraction expansion. This leads to Greene's residue method for determining the breakup threshold of invariant circles in twist maps. A generalization to higher-dimensional tori is difficult because there is no satisfactory generalization of the continued fraction, and instead of a single ``residue", there are multiple stability multipliers for periodic orbits.
We study three-dimensional, volume-preserving maps with invariant two-tori. We generalize the residue criterion when the map is reversible. More generally, computations of the conjugacy to rigid rotation are used to compute the threshold. The local singular values of these functions gives evidence for the existence of remnants after breakup, analogous to the cantori of symplectic maps. |
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