General transversality for interior Hamiltonian chord evaluation

Determine how to ensure, for a general Hamiltonian H and open subset U, that a Hamiltonian chord intersecting U also intersects U both before and after a prescribed interior time t₀, thereby enabling the interior evaluation map to be treated as a submersion without assuming that U is invariant under the Hamiltonian flow.

Background

The paper studies perturbations of Hamiltonians to achieve non-degeneracy and admissibility of Lagrangian Hamiltonian chords. Corollary 8.13 gives submersivity of the interior evaluation map when a chord intersects the perturbation region U at times both before and after the evaluation time t₀.

The authors explicitly state that it is not known how to guarantee this condition in general. They then identify flow-invariant subsets U as a special case where the condition is straightforward, and the subsequent perturbation arguments rely on that special structure. The unresolved issue concerns extending this transversality mechanism beyond flow-invariant perturbation regions.