Symplectomorphism of interiors W and WΔ

Determine whether the interiors of a star-shaped domain W⊂R^{2n} whose boundary supports a non-degenerate dynamically convex Reeb pseudo-rotation with exactly n non-alternating prime closed orbits, and the solid ellipsoid WΔ having the same action and mean-index spectrum, are symplectomorphic.

Background

Under additional non-resonance conditions, the paper shows that the linearized flows along prime closed orbits on ∂W are completely determined by the symplectic homology persistence module and coincide with those for the corresponding ellipsoid WΔ. While dynamics may differ beyond periodic data, it is unknown whether the interiors are symplectomorphic.

The answer may depend on Diophantine properties of the mean indices ∆, hinting at subtle arithmetic influences on symplectic equivalence.