Stable dynamical regions outside the Klein combination locus

Determine whether the isolated parameters outside the Klein combination locus at which the modular mating correspondences act discretely are isolated examples or belong to regions of stable dynamical behavior.

Background

The paper constructs or discusses isolated discrete correspondences outside the Klein combination locus, including parameters associated with the conjectured boundary sequence and triangle-group-like examples. It asks whether such behavior occurs only at isolated parameters or persists on open regions in parameter space.

References

What structures are there in parameter space { outside $$? We have seen members of the family $a$ which act discretely' but which havefundamental domains' with more than one component, for example the correspondences ${a_n}$ of Section \ref{structure_K}, and $_{c_q}$ of Section \ref{triangles}. Are these isolated examples or are there regions of stable behavior outside $$?

— Tessellating the discreteness locus for the modular mating family of correspondences  (2608.17243 - Bullett et al., 18 Aug 2026) in Section 3.3 “A possible scenario for $\Psi(^{cut})\subset H$”, third key open question