Optimal escape speed for dynamical non-recurrence

Determine the optimal escaping speed of the singular value for an escaping exponential map f_λ(z)=λe^z to be dynamically non-recurrent.

Background

The paper constructs escaping exponential maps with slowly escaping singular values exhibiting both dynamical non-recurrence and dynamical recurrence. Earlier results cited in the paper show that sufficiently rapid escape, such as escape satisfying a sector or parabola condition, implies dynamical non-recurrence. Hemke’s result establishes that the parabola condition is sufficient, but the authors do not identify the sharp threshold or optimal escape rate separating dynamically non-recurrent behavior from other measurable dynamics.

The question asks for a precise characterization of how slowly the singular orbit may escape while still guaranteeing dynamical non-recurrence for an escaping exponential map.