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1 results

  • Surfaces of locally minimal flux

  • Edited by Albert Fathi, Georgia Institute of Technology, Philip J. Morrison, University of Texas, Austin, Tere M-Seara, Universitat Politècnica de Catalunya, Barcelona, Sergei Tabachnikov, Pennsylvania State University
  • Book:
    Hamiltonian Systems
    Published online:
    10 May 2024
    Print publication:
    09 May 2024, pp 215-228

  • Summary

    For exact area-preserving twist maps, curves were constructed through the gaps of cantori, which were conjectured to have minimal flux subject to passing through the points of the cantorus. It was pointed out by Polterovich (1988) that these curves do not have minimal flux if there coexists a rotational invariant circle of a different rotation number, but if hyperbolic they do have locally minimal flux even without the constraint of passing through the points of the cantorus. Following the criterion of MacKay (1994) for surfaces of locally minimal flux for 3D volume-preserving flows, I revisit this result and show that in general the analogous curves through the points of rotationally ordered periodic orbits or their heteroclinic orbits do not have locally minimal flux. Along the way, various questions are posed. Some results for more degrees of freedom are summarized.