Do orbits from different repelling fixed points fall into the same category for commuting maps

Determine whether, for two commuting holomorphic self-maps q and y of the unit disc D, any two forward orbits under q that start at two different boundary regular repelling fixed points of y necessarily exhibit the same qualitative behavior (i.e., fall into the same category of asymptotic behavior for such orbits, such as converging in finite time to a common fixed point distinct from the Denjoy–Wolff point or forming sequences of repelling fixed points tending to the Denjoy–Wolff point).

Background

The paper studies when commutativity with a single element of a one-parameter semigroup extends to all its fractional iterates and analyzes the behavior on petals. In their parabolic setting, the authors prove structural results (Theorems 1.8 and 1.9) describing how a univalent non-elliptic self-map q interacts with the petals of a semigroup (V_t), including the fate of orbits starting from repelling fixed points of V_1.

They contrast this with the general setting of two commuting holomorphic self-maps q, y ∈ Hol(D), as investigated by Bracci [8], where richer dynamical behaviors may occur (e.g., repelling cycles of y). In this broader context, the authors explicitly note an unresolved issue: whether orbits of q starting at different repelling fixed points of y must share the same qualitative type. This remains unclear outside the specific framework treated in the paper.