Improve guarantees by using attracting neighborhoods beyond attracting blocks

Determine whether the latent-dynamics methodology can be improved to provide rigorous guarantees by lifting attracting neighborhoods, rather than only the attracting blocks currently certified by the approximate semiconjugacy condition.

Background

The main theorem applies when the preimage of a latent attracting block satisfies a residual bound, but the examples show that useful dynamical information can be recovered even when this condition fails. The authors note that a preimage may still be an attracting neighborhood despite not being an attracting block.

Because attracting neighborhoods are generally harder to verify computationally than attracting blocks, the paper restricts its formal method to blocks. The authors explicitly raise as unresolved whether the approach can be extended to obtain more plausible guarantees through the broader class of attracting neighborhoods.

References

This raises the question of whether we can improve our methods to obtain more plausible guarantees.

— Characterizing High-dimensional Dynamics by Combinatorial-Topological Methods on a Latent Space  (2609.01509 - Bailon et al., 1 Sep 2026) in Section 5, Discussion and open questions, paragraph following Example 5.1

The estimates for node $4$ violate eq:loc_semiconjugacy, so Theorem~\ref{thm:main_simple} does not resolve whether $f$ exhibits tristability.

— Characterizing High-dimensional Dynamics by Combinatorial-Topological Methods on a Latent Space  (2609.01509 - Bailon et al., 1 Sep 2026) in Section 3.2, Three-dimensional Leslie model subsection, paragraph following Table 1