Obstructions to Topological Relaxation for Generic Magnetic Fields

For any (analytic) axisymmetric toroidal domain \(\Omega \subset \mathbb {R}^3\) we prove that there is a locally generic set of divergence-free vector fields that are not topologically equivalent to any magnetohydrostatic (MHS) state in \(\Omega \). Each vector field in this set is Morse–Smale on the boundary, does not admit a nonconstant first integral, and exhibits fast growth of periodic orbits; in particular this set is residual in the Newhouse domain. The key dynamical idea behind this result is that a vector field with a dense set of nondegenerate periodic orbits cannot be topologically equivalent to a generic MHS state. On the analytic side, this geometric obstruction is implemented by means of a novel rigidity theorem for the relaxation of generic magnetic fields with a suitably complex orbit structure.