Exact limiting decomposition elements of critical points

Determine the exact form of the limiting decomposition element $[v] \in \Lambda^{\pm}$ associated with each critical point after taking the closures of the finite-stage decomposition elements $[v]_n$.

Background

For a critical point vv, the paper explains that the finite-stage decomposition elements [v]n[v]_n can be computed inductively from the pair of majors and that their union, denoted [v]∞[v]_{\infty}, is contained in the limiting decomposition element [v]∈Λ±[v] \in \Lambda^{\pm}.

The unresolved issue is whether taking the closure introduces additional points beyond those already contained in the increasing finite-stage approximations. Determining the precise limiting decomposition elements would clarify the full equivalence classes generated by the invariant laminations, beyond the information supplied by any finite number of pullbacks.

References

Note that, even though we know $[v]_n$ for all $n$, we still don't know exactly how $[v] \in \Lambda{\pm}$ looks like.

— Infinite chains of perfect fits for expanding Thurston maps  (2609.03838 - Loukidou, 3 Sep 2026) in Section 3, subsection “Decomposition elements for pullbacks”