Conjectured sqrt(n) lower bound for parabolic zero-step holomorphic self-maps

Determine whether every parabolic holomorphic self-map f: H -> H of zero hyperbolic step with Denjoy--Wolff point at infinity satisfies the universal lower bound liminf_{n->+infty} ( |f^n(z)| / sqrt(n) ) > 0 for all z in H.

Background

The paper studies the rate at which iterates of non-elliptic holomorphic self-maps of the upper half-plane H approach the Denjoy--Wolff point, providing complete characterizations of extremal (slowest possible) rates in the hyperbolic case and for parabolic maps of positive hyperbolic step.

For parabolic maps of zero hyperbolic step, the authors note that no universal lower bound for |fn(z)| is currently known. Motivated by the continuous-semigroup analogue, they put forward a natural conjecture that a sqrt(n) lower bound should hold for all orbits, but existing techniques from the continuous setting do not directly transfer to the discrete iteration case due to potential polar boundary of the Koenigs map image.

References

However, no universal lower bound with respect to $n\inN$ is known for the quantity $\lvert fn(z)\rvert$, $z\inH$, in this case. A natural conjecture would be the satisfaction of $\liminf_{n\to+\infty}(\lvert fn(z)\rvert/\sqrt{n})>0$, for all $z\inH$.

— Extremal rate of convergence in discrete hyperbolic and parabolic dynamics  (2510.12501 - Cruz-Zamorano et al., 14 Oct 2025) in Remark, Section 3 (Extremal Rates)