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Surfaces of locally minimal flux

Published online by Cambridge University Press:  10 May 2024

By
Robert S. Mackay
Edited by
Albert Fathi ,
Philip J. Morrison ,
Tere M-Seara  and
Sergei Tabachnikov
Albert Fathi
Affiliation:
Georgia Institute of Technology
Philip J. Morrison
Affiliation:
University of Texas, Austin
Tere M-Seara
Affiliation:
Universitat Politècnica de Catalunya, Barcelona
Sergei Tabachnikov
Affiliation:
Pennsylvania State University
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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.

Keywords

locally minimal fluxHamiltonian systemaction differences
Type
Chapter
Information
Hamiltonian Systems
Dynamics, Analysis, Applications
 , pp. 215 - 228
Publisher: Cambridge University Press
Print publication year: 2024

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