Obstructions to Topological Relaxation for Generic Magnetic Fields

For any (analytic) axisymmetric toroidal domain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mn>3</mml:mn></mml:msup> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> . 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.