Extension of equal action-to-mean-index ratio to non-degenerate flows over arbitrary fields

Ascertain the extent to which property (O2)—that all Q-visible closed orbits x have the same ratio of action to mean index A(x)/μ̂(x)—extends to the non-degenerate case over arbitrary ground fields, beyond the setting treated in Theorem E and Corollary D.

Background

Theorem E (with Q-coefficients) shows that in dynamically convex pseudo-rotations, all Q-visible closed orbits share the same action/mean-index ratio. In Remark 4.5, the authors discuss non-degenerate flows over any field F and note an analogue of part (O1), but explicitly state uncertainty about (O2).

Clarifying the validity of (O2) over arbitrary fields would deepen understanding of how filtered homology and orbit invariants behave with different coefficients.