Equality of the Fatou, projective resolvent, and explicitly identified Fatou subsets

Determine whether the Fatou set of the dynamical map F, the projective resolvent set of the Basilica-group Koopman tuple A_\rho, and the union of all backward iterates of the regions R_0, R_1, and R_2 are pairwise equal, and, if not, determine whether any two of these three sets are equal.

Background

The paper studies the quadratic projective dynamical map F induced by the self-similar structure of the Basilica group and the tuple A_\rho=(\rho(\mathbf e),\rho(\mathbf a),\rho(\mathbf b)). Theorem 1 proves that the union of all backward iterates of the three explicitly defined regions R_0, R_1, and R_2 is contained in both the Fatou set \mathcal F(F) and the projective resolvent set pc(A_\rho).

The authors do not establish whether either inclusion is an equality. They explicitly ask whether the three sets coincide pairwise; numerical simulations are reported to suggest an affirmative answer.

References

This naturally leads to the following questions:

Are the three sets $\mathcal{F}$, $pc(\mathcal{B}_{\rho})$ and $\bigcup{\infty}_{n=0} F{-n}\left(\bigcup_{k=0}2R_k\right)$ pairwise equal? If not, are any two of them equal?

— Spectral dynamics for the Basilica group  (2608.18987 - Liu et al., 19 Aug 2026) in Question 1, Section 5, Conclusions and further remarks

Are the three sets $\mathcal{J}$, $p(A_{\rho})$ and $\overline{\bigcup{\infty}_{n=0} F{-n}(L)}$ pairwise equal? If not, are any two of them equal?

— Spectral dynamics for the Basilica group  (2608.18987 - Liu et al., 19 Aug 2026) in Question 2, Section 5, Conclusions and further remarks