Surfaces of locally minimal flux

TL;DR

This paper investigates the properties of curves with locally minimal flux in area-preserving twist maps and volume-preserving flows, revealing conditions under which such curves do or do not have minimal flux.

Contribution

It revisits and clarifies the conditions for curves to have locally minimal flux in volume-preserving systems, extending previous results and posing new questions.

Findings

Curves passing through points of rotationally-ordered periodic orbits generally do not have locally minimal flux.

Hyperbolic invariant circles can have curves with locally minimal flux without passing through cantori points.

The paper summarizes results for systems with more degrees of freedom.

Abstract

For exact area-preserving twist maps, curves were constructed through the gaps of cantori in \cite{MMP84}, which were conjectured to have minimal flux subject to passing through the points of the cantorus. It was pointed out by \cite{Pol} that these curves do {\em not} have minimal flux if there coexists a rotational invariant circle of a different rotation number, but if hyperbolic they do have {\em locally} minimal flux even without the constraint of passing through the points of the cantorus. Following the criterion of \cite{M94} 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 {\em not} have locally minimal flux. Along the way, various questions are posed. Some results for more degrees of freedom are summarised.