FLUX-MINIMIZING CURVES FOR REVERSIBLE AREA-PRESERVING MAPS

Approximately invariant circles for area preserving maps are defined through a mean square flux, (phi2-) variational principle. Each trial circle is associated with its image under the area preserving map, and this defines a natural one-dimensional dynamics on the x coordinate. This dynamics is assumed to be semiconjugate to a circle map on an angle variable theta. The consequences of parity (P-) and time (T-) reversal symmetries are explored. Stationary points of phi2 are associated with orbits. Numerical calculations of phi2 indicate a fractal dependence on rotation number, with minima at irrational numbers and maxima at the rationals. For rational frequency there are at least three stationary points of phi2: a pair of minimax points, related by PT-reversal, and a stationary, but not minimal, symmetric solution. A perturbation theory without small denominators can be constructed for the symmetric solution. A nonreversible solution for the (0, 1) resonance shows that this solution is highly singular and its circle map is not invertible.