P-CaTherine wheels in the presence of perfect fits

Determine whether, when the corresponding laminations contain perfect fits, the resulting sphere-filling curves form a broader class called P-CaTherine wheels for which most of the theory of CaTherine wheels persists.

Background

The paper discusses CaTherine wheels, a class of sphere-filling curves associated with pairs of circle laminations. Previous work established that Meyer's curves are CaTherine wheels exactly when the corresponding laminations have no perfect fits.

The unresolved conjecture concerns the complementary setting in which perfect fits are present. The proposed P-CaTherine wheels would extend the CaTherine-wheel framework to this broader class while retaining most of its theoretical properties. The examples constructed in the paper, whose laminations contain infinite chains of perfect fits, are presented as lying at the far end of this proposed range.

References

It is shown there that Meyer's curves are CaTherine wheels precisely when the corresponding laminations have no perfect fits, and it is conjectured that in the presence of perfect fits one obtains a broader class, the P-CaTherine wheels, for which most of the theory persists.

— Infinite chains of perfect fits for expanding Thurston maps  (2609.03838 - Loukidou, 3 Sep 2026) in Section 1, subsection “Context: sphere-filling curves”