Dynamical (non-)recurrence of escaping exponential maps

We show that exponential maps could have complicated measurable behaviours by giving two examples in the exponential family both with singular value escaping to infinity ``slowly", one of which is recurrent (in the sense that every positive measure set will return to itself infinitely many times), while the other is non-recurrent. This also complements earlier results of Lyubich, Rees, Urbański-Zdunik, and Hemke. We also give an example of exponential maps with singular value escaping to infinity arbitrarily slow, which is a parametric analog of a result of Rempe.