Determine asymptotic laws near infinite-period trajectories

Determine whether the functional mixing, geometric mixing, and curve-growth laws for two-dimensional stationary incompressible Hamiltonian flows retain the same algebraic orders or acquire logarithmic or different algebraic corrections when trajectories approach infinite-period orbits and their limiting equilibria.

Background

The main theorems exclude infinite-period trajectories because their trajectories converge to equilibria in forward and backward time, whereas the action-angle coordinates and uniform estimates used in the paper are constructed in regular regions of finite-period trajectories. Extending the analysis across such trajectories requires quantitative control near the limiting equilibria and a replacement for the periodic coordinate chart.

The unresolved issue is whether the inverse-time and linear-growth laws established away from infinite-period trajectories persist unchanged, or whether the approach to separatrices produces logarithmic or other algebraic corrections.