Classification of Periodic Clifford-Torus Parameters

Classify all parameter values for which the dynamics on the corresponding Clifford torus is periodic.

Background

The Clifford-torus family is parametrized by a positive parameter and admits periodic dynamics only for special values related to rational rotation frequencies. The complete classification of those parameter values remains open.

References

More generally, can one classify all parameter values $a$ for which the dynamics is periodic?

— Outer Contact Billiards  (2608.19393 - Caliz et al., 19 Aug 2026) in Section 6, “Open questions,” item 11, “Periodic Clifford tori”